THE UNIVERSITY OF MICHIGAN COLLEGE OF LITERATURE, SCIENCE, AND THE ARTS Department of Mathematics Technical Report FINITE MOMENT FORMULAE AND PRODUCTS OF GENERALIZED k-STATISTICS WITH A GENERALIZATION OF FISHER'S COMBINATORIAL METHOD D. S. Tracy ORA Project 05266 under contract with: NATIONAL SCIENCE FOUNDATION GRANT NO. GP- 12 WASHINGTON, D. C. administered through: OFFICE OF RESEARCH ADMINISTRATION September 1963 ANN ARBOR

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Science in The University of Michigan, 1963.

TABLE OF CONTENTS Page LIST OF TABLES ABSTRACT INTRODUCTION CHAPTER I. BASIC MATERIAL II. GENERAL RULES FOR COMBINATORIAL METHOD III. COEFFICIENTS FOR ARRAY TYPES IV. PRODUCTS OF GENERALIZED k-STATISTICS V. MOMENT FORMULAE AND THEIR ESTIMATORS SUMMARY REFERENCES iii iv 1 5 50 50 64 75 102 104 ii

LIST OF TABLES Page 1. COEFFICIENTS FOR MULTIPLICATION OF TWO BRACKETS 13 2. ILLUSTRATION FOR k21k2 17 35 k21k2 WITH THE USE OF DISTINCT UNITS 23 4. CONDENSED METHOD FOR THE PRODUCT k k2 26 5. PATTERN COEFFICIENTS 55 6. SEMI-GENERAL PRODUCT FORMULAE 67 7. PRODUCTS OF SEMI-INVARIANT k-STATISTICS 72 8. FINITE MOMENT FORMULAE 78 ii

ABSTRACT When the parent population is not completely known, a general method for finding an approximate form for the sampling distribution of a statistic is to ascertain its lower moments. Frequently these statistics are symmetric functions of the observations such as moments. Many pioneer workers found formulae for moments of moments, but the algebraic complexity of the results led Fisher (1928) to introduce the k-statistics kp as symmetric functions of sample observations whose expected values are the parent cumulants k-. He developed a combinatorial method to express their cumulants and product cumulants. Dressel (1940) introduced generalized k-statistics as sample functions whose expected values are products of cumulants. Tukey and Wishart gave some methods and results for products of generalized k-statistics, which were used to obtain moments of moments when sampling from a finite population. Dwyer and Tracy (1962) transformed Tukey's method to a combinatorial method for products of two generalized k-statistics and gave semi-general formulae for k(i kplp. where ([ stands for any set of subscripts and p1+p2+...< 4. Schaeffer and Dwyer (1963) introduced substitution products for unifying expectation and estimation theory. The aim of this work is to generalize Fisher's combinatorial method to write products of generalized k-statistics as linear functions of the same and to use these to obtain formulae for moments of moments when sampling from a finite population to parallel his formulae for the infinite iv

case. With this aim, the following is done: 1. Additional rules for the combinatorial method for multiple products of generalized k-statistics are stated and proved, although the rules of Dwyer and Tracy (1962) are found to generalize to the case of multiple products. 2. Just as Fisher's combinatorial approach was based on the determination of a coefficient for certain patterns, so the combinatorial approach for the general case is based on the use of coefficients for patterns generalizing those of Fisher. These coefficients are determined and tabulated for the most common patterns. 3. Semi-general formulae for products of k() and kp p kq q.. are provided through weight 4 of the second factor and products of semi-invariant generalized k-statistics are extended to weights 9, 10 and selected ones of weight 12. Checks for these are indicated as well. 4. These results are then applied to obtain moments of multiple products M(P1P2..) = EN(kpl-Kp )(kp2-Kp )..., where EN denotes the average over the sample values when sampling from a finite population of size N. Formulae for M(...) or K(...) are needed, for example, in a study of the distribution of ratio-statistics. These are tabulated for cases not involving the sample mean kl for weights through 10 and for selected ones of weight 12 and also the corresponding K(...) are given where they differ from the moments M(...). A useful check is provided by the fact that these formulae transform to those of Fisher as N -+ oo. Formulae relating moments involving kl to those not involving kl are also given. v

5c Moment formulae can be transformed to estimation formulae rather easily when using generalized k-statistics. This fact is used to obtain estimators of M(...) and K(...), not involving kl, using substitution products, thus extending the results of Schaeffer and Dwyer (1963). The work thus generalizes Fisher's paper for the infinite case to finite populations. vi

IIITRODUCTION After giving a short history of the background material, the aims and objectives of this paper are described. li story When the parent population is not completely specified, a general method for finding an approximate form for the samrrpling dis.ztribution of a statistic is -to ascertain its lower moments. Frequently these statistics are symmetric functions of the observations such as moments and product moments. Thiele, Sheppard, "Student" and Tchouproff were among the pioneers to find formulae for moments of moments, but the algebraic complexity of the results and the amount of work required to reach them led'to a search for simpler methods. In fact, Craig (1928)while extending Thiele's results to write semi-invariants of moments and product semiinvariants drew attention to the need of the use of functions other than crude moments if the algebraic formulation was to be made manageable. Fisher (1928) introduced *the symmetric functions which provide great simplification for infinite populations. Hle defined the k-statistics T1 as unbiased estimators of the parent cumulants )p. On the basis of the simpler forms so obtained, he developed a combinatorial method to express cumulants and product cumulants of sampling distributions from infinite populations. A further development of this method was given by Fisher and Wishart (1931). Georgescu (1932) extended Craig's results and applied Fisher's idea of a combinatory analysis to the sample moment function. Kendall (1940 a,b,c, 1952) systematized Fisher's combinatorial technique by giving rules for the same and their proofso

-2 Dressel (19)i)0) introduced symmetric functions whose expected values are products of cumulants X Xp2... and Tukey (1950, 1956), denoting them by kplp2 and calling them generalized k-statistics or polykays, showed that these are also unbiased estimators of the corresponding finite population parameters KP2 o. Wishart (1952) applied combinatorial methods to actually express products of k-statistics as linear combinations of generalized k-statistics. He also obtained products of generalized k-statistics by algebraic manipulation of the above, rather than by combinatorial methods~ He applied these results to find moments for finite sampling. Tukey (1956) gave some rules and tables for the direct calculation of the products of two generalized k-statistics. Dwyer (1962) studied the properties of polykays of deviates from the mean. Dwyer and Tracy (1962) developed combinatorial methods for products of two generalized k-statistics and gave many semi-general formulae for k i kP2. where {t stands for any set of subscripts and pi + p2 +.,, 4. Schaeffer and Dwyer (1963) gave practical methods for computation, introduced substitution products explained on page75 for unifying expectation and estimation theory and extended the product formulae for generalized k-statistics not including a unit subscript (which are seminvariant in that they are independent of the choice of origin) through weight 8. Objectives The main aim of this paper is to generalize Fisher's combinatorial method to obtain multiple products of generalized k-statistics and thus to generalize Fisher's (1928) moment and cumulant formulae to the case of a finite population. With this aim in view, the following has been done:

-3 1. After a review of the basic material, additional rules for the combinatorial method for multiple products are stated and proved, al- / though the rules of Dwyer and Tracy (1962)are found to generalize to the case of multiple products. 2o Just as Fisher's combinatorial approach was based on the determination of a coefficient for certain patterns, so the combinatorial approach for the general case is based on the use of coefficients for patterns generalizing those of Fisher (1928, pp. 223-226), These coefficients are determined and tabulated for the most common patterns. 3. With these coefficients known, semi-general formulae for products of kti and k p k lq o are provided through weight 4 of the second factor. With the aid of these, products of seminvariant generalized k-statistics are extended to weights 9 and 10 and selected ones of weight 12 (following Fisher, 1928) and are presented in tabular formo Checks for these are also indicated. 4o Formulae adapted for computation of finite moments M(plp2oa) ) EN(kp K )(k - K ).o. are derived, where EN denotes the average I P1 p1 p2 P2 over the sample values when sampling from a finite population of size No One needs formulae for M(o ) or K(...)for example, in a study of the distributions of ratio-statistics such as (whch is used to measure departure from normality), where the denominator is expanded in the -3/2 k2 K 3/2 series K2 (1+ k2 - K2) These are tabulated for formulae not K2 involving the sample mean k1 (pi / 1) for weights through 10 and for selected cases of weight 12 and also the corresponding K( o ) are given where they differ from the moments M(.oo), generalizing Fish.er's (1928)

-4table of formulae to the finite case. Fisher's formulae can be obtained as N --, which helps in checking. Formulae relating moments involving kl to those not involving k1 are also given. 5. Estimators of M(...) and K(...) for pi f 1 using substitution products are tabulated, extending the results of Schaeffer and Dwyer (1963)o

CiHAiPTER I BASIC MATERIAL After defining the terms, some description of the generalized kstatistiicss given. The algebraic method given by Tukey (1956) for writing the product of two generalized k-statistics is briefly described with the help of an example and its modification to a combinatorial method by Dwyer and Tracy (1962), following Fisher (1928), is illustrated with the same example. The steps of the combinatorial method are then outlined. Notation and Definitions Let a random sample xl, x2,... xn of size n be drawn from a finite population of size N (whose moments all exist), the sampling being done without replacement. The finite population itself may be looked upon as a random sample of size N from an infinite population. A sample symmetric function is a function of xJl, x2,.., xn whose value remains unaltered by any permutation of the xi's amongst themselves. The sample augnriented monomial symmetric functions (Kendall and Stuart, 1958, p 276) or power product sums (Dwyer, 1938, pO 12) are denoted by Y1 PI P2 PS pl p2 oo~ pj = 21. x x....o [PIP2 ] 2 - Xj, X X XU where p -Z= is the weiht and the number of parts s is the order of the symmetric function. If the p, (distinct) are repeated 7t times, Tl1 7X2 7s k P P1 P1 P. P P Ps [Pi P2 ~'PS 1J -t X Xi ~ q r t u p =Z T ibeing the weight and 2 7T being the order of the function. 0 -5

-6 The monomial symmetric function is denoted by l7T, 7k A2- 1 A 2 s * ~(Pl P2 "' s..... 2l!1 ~~~ ~st As an example, ~ 0 n n0 2 r ( xi.)2- Zx + xx. -1 I J can be expressed as [ [2 - Il3 J but also, n r ) r I C ) ( xi) xi + 2 X xi Xi so ()2 - (2) + 2 (11). In terms of these, power sums sr are just one-part functions, so that s5r Z = [r] (r). Tables have been provided by David and Kendall (1949) for expressing power sums and augmented monomial symmetric functions in terms of each other through weight 12. Smmetric means or mean power products are defined as the means of products of powers of different xi So Since the sum pP2.. p = Pl P 2 P P Xi xj D o x u is over n(n-l)o. (n-P+ 1) terms, the symmetric mean < P1P2. P)> (termed angle bracket by Tukey, 1950) can be defined as 7The nota1 s The notation (p o P ) is later used for partition coefficients.

-7 <P1P2.. Pe) p] Then, by a basic theorem of finite sampling theory (Dwyer, 1938, Tukey, 1950 ~ "inheritance on the average"), EN <P1P2 " Pp> <p1P2. P>N where EN denotes the average over N) possible unordered sample values when sampling without replacement and (P1P2 oo Pp)N is the corresponding population bracket. In case sampling is from an infinite population (or from a finite population with replacement), (1.1) E <p p p> P2/ 2' t1 P / P2 I'l ~ l-Pr where''s denote moments about the origin. (Kendall and Stuart, 1958, p. 276). A partition coefficient (pl o.. Ps) of p is defined (Dwyer and Tracy, 1962),(Schaeffer and Dwyer, 1963), as the number of ways that the distinct units of p may be collected into distinct parcels described by the specified partition of p. For the p-part partition (P1.0 Ps p ), the partition coefficient 7( 1 s P where.27r - p (weight), 1 n. a P (order). The multinomial theorem can then be expressed as [1] = (p i Pr) PL ~ - Prp]

-8 where the summation applies to every p-part partition of p and g= 1,2,...,p. The p cumulant ^ p of the infinite population can be expressed in terms of the moments of the same by the formula?-1 (1.2) (-p) -l -1) (pi * p).. -/ where the second summation is over all p-part partitions of p. k-statistics The p k-statistic kp is defined (Fisher, 1928) to be the sample symmetric function such that E(kp) = p. In the case of a finite population, E (kp) = Kp, where Kp, the K-parameter is the same function of the finite population as kp is of the sample. Its uniqueness has been shown by Kendall and Stuart (1958) and also by David and Barton (1962) for all distributions. Then, from (1.1) and (1.2), P^ P-1 (1.3) k = 2 2 (-1) (p-1)! (pi... p) p.. p> P p=.\ ^;x<p -l rf ~ as given essentially by Cornish and Fisher (1937, p.5). The value of kp, p>1, is independent of origin (seminvariant) as shown by Kendall and Stuart (1958), and kp, p> n, are not defined. The kp are homogeneous polynomials of degree p and can be written in terms of power sums. Such expressions for kl through k6 are given by Fisher (1928), for k7 and k8 by Dressel (1940) and for k9 and klo by Zia-ud-Din (1954), who has also given an expression for kll (1959).

-9 Generalized k-statistics The generalized. (multiple) k-statistics kp2 are symmetric functions of sample observations which have the basic property of being estimators of products of cumulants (Dressel, 1940), i.e., E(k ) X )\... It is further known that EN(kpp ) K, where is the same function of the finite population as kp is of the sample. Tukey (1956) calls them polykays and defines them by a symbolic multiplication (o) in which products of brackets are replaced by brackets enclosing the product factors. Thus, k _= k ok o... PP2'' Pi P2 (1.4): (-^P'T(-)[ ( ~')( k,'' [k,)'-' h<' t v "''>) where the summation extends over all combinations of partitions p_,.. p of pt. It follows from (1.4) that (1.~5) = P2 PI P2 Thus k is an unbiased estimatorof )Pl... PlP2.'" P~. 2 Again k is uniquely defined for monomial symmetric functions by the implication (1.6) E(kp p ) P X P P2 for all distributions. For suppose there is another monomial syrrmietric function' ^ of weight p whose expected value is also XpX1)p *'for all distr:ibutions, then E(k P2 - k ) - Oo Bt k -k' is a monomial symmetric func-..P.P.. Pl2...2"

-10 L.:ionl being the difference of two such functions and can therefore be expressed as the sum of terms 2xPi, Z xi Px etc. Thus its expectation is a sum of terms each of which is a moment-product. The vanishing of this series implies a polynomial identity relationship between the moments of x which is impossible except perhaps for a particular subclass of populations. Hence kplP2 - k' vanishes identically and so k p _ k' lP2...... PlP2..... Some work of Halmos (1946) on the uniqueness of estimates is interesting in this connection. It can also be noted that a generalized k-statistic kp P2. is seminvariant when no pi = 1. For, by Taylor's theorem, if we write z for Xl) **)XnA k 2..~ (z+h) - k... (z): hf(z). PlP2 —- PlP2092 Taking expectations and remembering that when pi 1, Xpi is independent of origin, we have (1.7) 0 - h E f(z) In view of the remarks above, since a polynomial identity relationship among the moments for all distributions is impossible, f(z) = 0 and k is seminvariant when Pij 1, all i. If one or more pi is 1 the expectation of the corresponding generalized k-statistic involves W1 as a factor which certainly depends upon the origin. An advantage of the generized k-statistics derives from the fact that we can express them in terms of augmented symmetric functions once and for all (Wishart 1952) and hence derive non-linear functions of them as linear functions to which the Irwin-Kendall principle (Irwin and

Kendall, 1944, p, 138),(Kendall and Stuart, 1958, po 301) will apply. This principle says that if for a symmetric function f, E(f) = fajXj, then EN(f) - Za.K., for otherwise, EN(f) could be expressed as some other function of K's whose expectation E would be the same as that of T-ajKjo This would imply a polynomial relationship among the K's. The polykays enable us to write down unbiased estimators of products of cumulants. In fact, the operations of taking expectation and estimation become trivial after the functions of the observations xi are reduced to linear functions of generalized k-statistics. Combinatorial Method Fisher (1928) tackled the problem of writing the sampling moments and cumulants of k-statistics in terms of parent cumulants by algebraic as well as combinatorial methodso The problem is essentially that of finding mean values of powers and products of these k-statisticso To any number p with partition p,' o.o p,, there is a moment ^(P E(o k ) and a cumulant (p' o0 pa, ) related to these moments by the usual identity in t's, (Kendall and Stuart, 1958, p.282) z;...>7)TT= i (91 *m)T where Pi, i 1,2,..., s and qj, j 1,2,..., m are column and row totals repeated ti,. times respectively ( =.;= ) for the two-way array

I 1.:.. --. 5 i - - - - - _i - - i a ) P1 ~ ~ P1' P I. _ p where a row corresponds to every ) in X X and a column to VrI VA every part in ~(pl oo p a ) and we consider all the ways in which the body of the array can be completed by the insertion of numbers whose column and row totals are the respective pi, qj. To take an example from Kendall and Stuart (1958), when seeking the coefficient of KMXC in (42 2), we consider such arrays as 2 2 2 6 2 31 6 3 3 0 6 11 0 2 11 0 2 1 0 1 2 1 0 2 1 01 2 0 1 2 42 10 4 4 2 1i0 1 4 L o Fisher's (1928) empirical rules were stated more formally by Kendall (Kendall and Stuart, 1958) for writing the sampling cumulants of k-statistics using a combinatorial methodo A proof of these rules is also provided by Kendall (Kendall and Stuart, 1958, Chapter 13) by employing an operatoro The algebraic coefficients (po25) of many useful patterns of arrays have been provided by Fisher (1928)o Wishart (1952) modified the FisherKendall rules in order to obtain products of k-statistics and also used

-13 combinatorial methodso Then he proceeded to find products of generalized k-statistics algebraically and has listed all such products through weight 60 He has also given products of single subscript k7s through weight 30 Products of seminvariant generalized k-statistics(independent of the origin., ioe not having unit parts) up to weight 8 were given by Schaeffer and Dwyer (1963)o Tukey (1956) gave a method for finding expressions for products of two generalized k-statistics using a table for multiplication of brackets (from which we have adapted Table 1) and using a rule involving the number of unit partso Table 1 COEFFICIENTS FOR MULTIPLICATION OF TWO BRACKETS Number of Number of parts in bracket whose coefficient parts in multi- sought plying brackets 1 2 3_ 2_______ 1xl1 1 ln n 1x2. - i 1x2. n nD 1 x 3 1 I n n 1 _ 1 4 2 2x2 _l-+( 2) n-n) T n 2 i 2 i 6 2 x 3 ------------------- ------- J —------------------------------------------

As an example of bracket multiplication, a~><b> 1 xaX 2 x b n2'i I 3 j 1 a b x, a+- b jb - x. -+x 2 x n2 1 J k 1 i ab] - [ab] j n n The first row of Table 1 presses ths result In general, one has to The first row of Table 1 expresses this result. In general,, one has to obtain "all products which can be obtained by matching some (including none) of the letters in one bracket with letters in the other and then replacing matched letters by their sum." (Tukey, 1956, po46)o Tukey (1956) also observed that when a bracket with g unit parts is written in terms of polykays, only polykays with at least g unit parts appear and vice versao He used 0(1g) for any set of terms each of which, when expanded linearly in brackets or polykays, contains at least g unit partso Also, he used the term unit weight of an expression for the maximum number of unit parts appearing in any term of that expression. In terms of these, the rule is that while expressing a polynomial in polykays as a linear combination of the same, the unit weight on the linear side can not exceed the unit weight on the other side Thins implies that the coefficient of every kooo having more unit subscripts than the set of original subscripts is zero.

The multiplication of polykays is carried out in three steps: (a) expressing each koo, as a linear function of brackets, (b) multiplying out the brackets, and (c) reconverting the resulting brackets to polykayso For example, Tukey (1956) considers k21 k2 k2 o ) k2 [<21> - l111>][<2> - 11>, using (1.4) (1.8) < 21><2> - <111> <2> - <21>< +111> <11> (1 -) <221> + 1 <32> + <41> 2(1 1 )<221> n, n n n n2) j 2 ) < 32> + 6) <221> + 0(12), using Table 1 and the rule of unit parts "4 (1 +8 i 12I 1 2 ( 4 " - )) k221 0( ) + k( ) k+ 3k k2121 + 0(12) + 1 k + 3k221 + O(l2)J + 0(12) (1 2 )k 1 a(1 2+ 2 )2 + 0(12) ( - )) 32 + k l + (1 + + ) 22 0 (1o9) 3 n+l n(nl-) - k32 + k4 1 + n1 221 Tukey, however, believed in "sparing the use of combinatorial techniques as much as we are able" (Tukey, 1956, po37)~ Dwyer and Tracy (1962) have modified Tukeyss algebraic method to a direct combinatorial method using arrays which consists of the same three stepso

The first step, as in Tukey's method, gives (18) which appears at the top of Table 2, The multiplication of brackets is achieved more directlyo To express 421> 2> for example, since (1.10) ( 2 xx)( Z x,) - x'x+ x'x3 + 2xx2 I3 jj a' ii k or, f2lJ L 2 L4i] L32 + 22l or, n2(n-1) (21><2> n(n-l) <41> + n(n-l) < 32> +n(nl)((-2) <221 > we have (1.11) <21><2> 1 &41>+ 1 K32 + n-2 <221> o n n The coefficients in (lo11) are termed n-coefficients and one need not actually go through all the steps from (Lo10) to (lo11) in order to obtain them in a given caseo For a bracket having P parts, the ncoefficient in the expansion of kp P k ~ ~ ~ s simply n(F) where r, s are the total number of parts of the parti0 n( n tions of the p. and the qj respectively In the example, then, the (2) n-coefficient for <41> and <32> is n,. 1 and that for 4221> (3) n (') n is n n-2 n(2)n( ) n All these results are indicated in Table 2o Both partitions of 21, ioeo 21 and 111, are matched with both partitions of 2, iLe, 2 and 11, in all possible ways (except for permutations by rows), filling in O's where neededo The process leads to the 12 arrays which appear in Table 2 and in each of which the first column represents a partition

TABLE 2 ILLUSTRATION FOR k21k2 k2jk2 = [< 21 > - < 111 >] [< 2 > - < 11>] = < 21 > < 2 > - < 21 > < 11 > - < 111 > < 2 > + < 111 > < 11 > Array No. 1 2 3 4 5 6 7 8 9 10 11 12 22 4 20 2 21 3 21 3 12 3 20 2 20 2 11 2 20 2 10 1 11 2 10 1 10 1 12 3 11 2 10 1 10 1 10 1 11 2 11 2 10 1 10 1 10 1 10 1 Array |01 1 10 1 02 2 01 1 10 1 01 1 10 1 10 1 10 1 01 1 02 2 01 1 1 1 01 1 Formula Coeff. 1 1 -1 -1 -1 1 -1 1 -1 -1 1 1 Comb. Coeff. 1 1 2 2 3 1 2 6 1 1 6 1 n1 -Coeff 1 1 1 n-2 1 n-2 n-2 1 (n-2)(n-3) n-3 n-3 (n-3) (n-4) n n n(n-1) n(n-1) n n n(n-1) n(n-1) n(n-1) n n(n-l) n(n-l) 1 1 -2 -2(n-2) -3 n-2 -2(n-2) 6 -(n-2)(n-3) -n-3 6(n-3) (n-3) (n-4) -Coeff. | n n n(n-1) n(n-1) n n 2(n-1) n(n-) (n-n-) n n(n-1) n(n-1) 1 2 2 1 2 1 1 4 1 2 1 1 1 2 1 1 2 1 3 1 1 1 2 1 3 1 4 1 12 1 Transformation 5 1 2 1 Coefficient 6 1 1 11 7 1 1 1 1 8 1 2 1 9 1 1 10 1 1 I I 1 1 12 1 1 1 -2 2 0 0 1 0 0 0 0 0 n n n(n-1) n-l1 I 1--

of 21 and the second a partition of 2 while the marginal column represents the resulting bracketo In general the arrays appear in the order of expansion of bracket products, but for convenience in subsequent steps, they are grouped according to the number of rowso The arrays for the product <21>) 2> are thus numbered 1, 2 and 6 in Table 2, The coefficient obtained from formula (1l8) at the top of the table is termed formula coefficient and shown in the row directly below the arrays, To avoid extensive repetition~ equivalent arrays resulting from the permutations of the second column entries have been grouped together and a compensatory combinatorial coefficient supplied in the row so labeledo The n-coefficient is obtained in the manner described above or from Table lo The product of these three coefficients is the.coefficient or bracket coefficient for the moment product or the bracket indicated by the marginal colum=nn More than one array may lead to the same moment product or bracketo The result after the first two steps is il~l~) kelk2: ~C41)+ 1 2 i2n-2) (1012) k21k2 4 41>4! 432> 2 42 - <311>-n + n n(n-1) nR) * These results are equivalent to those obtained by Tukey (1,956) by the direct algebraic method though, since he does not use arrays, the coefficient of a specified array can not be identified in his result (lo9)o The third step in the derivation requires the expansion of various brackets in terms of generalized k-statisticso Tables are available (Tukey, 1956, po44), (Abdel-Aty, 1954), (David and Kendall, 1949) for assisting in thiso However, the device of introducing the parent cumulants, recommended by Kendall (1952, pol5), and obtaining the final

formula by estimation is used. In this example, for instance, from (1.12) or from the h/Acoefficient row in Table 2, (1o13) E(k21k2) n, i + g - n) - n Now, to transform from /L s to )'s, the two components of the y's corresponding to the two columns of the arrays need to be distinguishedo This is achieved by using a multipartite notationo Thus the A of the first row of array number 1 is treated as the bipartite /2o Then the expansion in terms of bipartite X's is (1.14) f -2 Al^+2 XZKOll —2 K,2+ K Z0 -t- 2, o,,, 11+ 0KXOIOl 2, 0 + 4 \11 o oli + Ko^ oiaAOI X01 and the transform of i2s/ n10 is the right side of (lol4) multiplied by X,,o The coefficients appear in the first row of "Transformation Coefficient" of Table 2o Similar transformation coefficients for the / s indicated by array numbers 2,3, 0o, 12 appear in row numbers 2,3, 0o, 12 of the "Transformation Coefficient"0 Coefficients from the unipartite expansions are available for checkingo The calculation of the ) - or k-coefficient is then straightforwardo We observe, for example, that array number 4 has non-zero transformation coefficients in rows 1, 3 and 4 (which means that array numbers 1, 3 and 4 yield array number 4 as a separate(page 2)o To obtain the k-coefficient for array number 4, then, we multiply the transformation coeffith l cient in 1the i row of this column by the p -coefficient of array number i and foimthe sumo For array number 4, the k-coefficient is thus

-20 1 ( 2) + 1 ( -2 ))+ 2. 0o v n(n-17 n n-1)17 / n A simple way to obtain the k-coefficient of an array is then to multiply th each transformation coefficient appearing in that column in the i row by the C'-coefficient of array number i and form the sum over i. In practice, the arrays and rows need not be numbered and this step can be achieved by multiplying the transformation coefficients in a column by the fI-coefficients of the columns indicated by the diagonal terms of "Transformation Coefficient" and forming the sum. We find in this example that (115) E(k21k) k n. 1 r )2 + 2 E(k21k2) ) = X ~ z~ f i~i+ AK~~ + n — ~~KI so that, taking estimates, we have n1 _f 312 n(~1 32 221 + 221 (1o16) k21k2 k1 + n k32 - J k32 +k221 n- k221~ It can be seen that (116) is in good form for approximation with large n and does show the contribution of each arrayo Establishment of general rules applicable to the contribution of a given array makes possible further condensation and the development of a true combinatorial method, We observe that the k-coefficient of every array having a marginal partition with two or more unit parts is zeroo This agrees with Tukey's (1956) rule that the coefficient of every koo having more unit subscripts than the set of original subscripts is zero, This eliminates array numbers 4, 5, 9, 10, 11 and 12o The zero k-coefficient of array number 7 is very noticeable nowo This case is not covered by Tukey's rule, but follows from the rule of

proper parts,* established later for general products that the coefficient is zero for any array having at least one row with a single nonzero element which is a proper part of some integer subscript. The 1 in the third row of array number 7 is a proper part of the 2. Tukey s rule follows as a corollary since any additional unit subscript must come from a row with a single proper unit part. For array number 8, although the combinatorial coefficient is 6, the two arrays resulting from the matching of the unit parts of 2 in 2 with unit parts of 21 are the ones contributing 1p each to the coefficient, while the other four arrays feature proper parts and hence do not contribute anything. The first two then belong to one array type and the other four to anothero The distinction becomes more obvious if we think of the elements of the arrays as composed of distinct units. For example, let el, e2 be the two units comprising the 2 of k21 and let e3 constitute the lo Again, let e4, e5 be the units in the 2 of k2o Then, array number 8 with a combinatorial coefficient of 6, is actually representing the following six arraysel e e e e e4 e e e e e4 e2 e5 e2 e5 e2 e4 e3 e5 e3 e4 e3 e5 e3 e4 03 0 e3 0, e2 0 e2 0 e1 0, el 0 The first two belong to one array type, the contribution of each being 1 n -, whereas the last four have coefficients zero since el and e2 are proper parts appearing alone in a rowo * A proper part of a partition of an integer is any positive integral value less than the integer.

The need of combinatorial coefficients is eliminated when we consider distinct units and the transformations coefficient may be looked upon as a separations coefficient indicating the number of ways an "amalgamation" can be separated into a particular "separate" (as used by Fisher, 1928)o In Table 2, array number 4 is a separate of array number 1 and array number 1 is an amalgamation of array number 4- With the use of distinct units, of course, each separations coefficient is either 1 or 0, depending upon whether that separate can be obtained from the given amalgamation, and Table 2 transforms to Table 3o Array numbers 3a, 3b originate from array number 3 of Table 2 and similarly array numbers 8a, 8b from array number 8, The vertical lines separate array types from each other. The k-coefficients of arrays of the same array type are the same, We can observe that array nurmber 2 can be obtained from array number 1 by permuting the entries in the second column, and similarly array number 3b from 3a and 8b from 8a (leaving the row e3 0 fixed), The contribution to k2 k2 from the first two arrays which is 1 (k(el+e2)+(e4+e5), e3 (el+e2), e3+(e4+ej) can be expressed then, as I k(e+e2) e3+ (e4+ e5)' 0 n 21+ 2,0 n 1 2 3 4 5 n where kab + cd indicates the sum of the k's with sums of subscripts a,b c cd permuted, ioeo k -k -k i Thus, ab + c,d as -c b+ d a+ d b+c k21+ 20 k41 + k23 (the commas may be dropped when unnecessary') Further, for contribution from array numbers 8a and 8b, where the row e3 0 is fixed and the entries in the other two rows are permuted, we

TABLE 3 k21k2 WITH USE OF DISTINCT UNITS ARRAY NO. 1I 2 | 3a b 3. 6 | 8a 8J> el+e2 e4+e5 e1+e2 0 el+e2 e4 el+e2 e5 el+e2 0 el e4 el e Array e3 0 e3 e4+e5 e3 e5 e3 e4 e3 0 e2 e5 e2 e 0 e4+e5 e3 0 e3 0 Formula Coeff. 1 1 -1 1 1 1 1 1 1 n-2 1 1 n-Coeff, 1 n-2 1- n n n (n-|) n(n-1) n n(n-l) n(n-1) /' f 1 1 - 1 - 1 n-2 1 1 n n n(n-1) n(n-1) n n(n-1) n(n-l) 1 1 1 1 1 1 1 Separations 1 Coefficient... 1 1 1 -Coefficient 1 1 - 1 - 1 1 1 (distinct units) n n n(n-1) n(n-1) n- n- n I ro

-24 use the notation ke,e2;e + e,e;O to indicate the sum of the k's with sums of permuted subscripts prior to the semi-colons~ Then, (1L17) k1k k 1 (lol?) kk2 - n 21+20 n-1) k21ll + k221 1 + T kll; 1+11;0 where k21+20 = k41 + k23, k21+l1 2 32, kll; 1+11;0 2k2210 We can now treat kH Pp k2 on similar lines. We use Pil, Poi2' oo to indicate parts of po Then, k p k2 [<P P2> - (P11 P12) 1l 12 P2> 2(P21 P22)<P1 P21 P22> +.~ ~ ~l<2 > -< 11 <p1 p2><2 - P1 P2>")< (PP11 Pl12)Pl P. 12 P2> (1.3) <p1 p2><2> <p1 p2><11>+2T<p11 p 2 p2><11> +2 T <1 P21 P>22K11>+ ~, where the summations are over all 2-part partitions of p1, p2 respectively and T (Pl1 p12 P2>' called a bracket type, symbolizes (Pl P12) similar brackets. When we consider E(k p zk2), only the first four terms appearing explicitly in (1.18) contribute non-vanishing coefficients, other bracket products having zero coefficients by the rule of proper partso Hence, in order to obtain a formula for kp p k2, we need only consider the bracket products corresponding to the four terms explicitly indicated in (1.18)o We now need the concept of a conditional amalgamationo It is an

-25 amalgamation in keeping with the conditions of addition of the rows, eo.g Pi2 can not be added to Pjl. The only conditional amalgamation of the array type P1 qcrl -i2 qr2 Pi 3 Sc ^Pi3 qsl Pj qs2 is Pil- Pi2 qr (r = rl+r2), since only parts of qsl Pi3 ^ Pj qs2 the same pi, ooo can be added together. Hence, for two rows to be additive, the non-zero entries in a column should be parts of the same subscript Thus, 2 2 is a conditional amalgamation of 1 1 while consi1 0 1 1 1 0 dering the products k3k2 or k21k2. but not while dealing with ki k2 For a two-column array type having prows, with r r.i r non-zero entries ky in one column and s.j s non-zero entries qc in the other, the formula coefficient is (-1) (r-l) iT () (sl)! and the n(f) a t a J n-coefficient is n( A) Let us absorb these two in the n'-coefficient n ns l) (ri - 1) f { 1 (s- - 1)! n() n(r) n(S) Then the algebraic coefficient of the array type is obtained by adding the n1coefficients of the array type and all its conditional amalgamations

-26 (Rule 1, Chapter II) Thus the algebraic coefficient of pll 1 in 1 0 considering kplp2k2, with p1 2 as the only conditional amalgamation, is 0 1. + ii~n —l 1 1 n n-l With this setup, a condensed method for obtaining k lPk2 is presented in Table 4o Table 4 CONDENSED METHOD FOR THE PRODUCT k Pk2 ~Plp 2 Array No. 1-2 6 3 8a 8'b P1 2 p pi 0 P1 1 P 1' 0 pI O 2 0 Array type p2 0 p2 0 1 p 1 ^ 21 2 P2 0 p22 1 Algo Coeff, 1 1 1 1 n n(n-l) n-1 nl T Comb.Coeff 1 (PilP12) (p21P22) k-coeff., 1. 1 -1 (Pl P22l n; n(n-l1) nl nl-1 A general formula for k k2 is then PlP2 k k k + 1 21 p1Pp)k;0) kplP2k2 -k= H I+O k kl 2 nk Pl^p22 n H P1P24+az~o P ppnr T p lP21n+P +ll + lPi2 plP12; P2 + 1 i (P 2122KP21P22;Pi + 11;0 (1o19) n

2Y)7 If one subscript, say P2, is 1, the last term in (1.19) vanishes since 1 has no 2-part partitions~ The formula is also applicable to the case with p2 = 0 if we drop the pp from all terms containing p2 as a subscript, and drop all terms containing other functions of P2 as subscriptso Thus, (1.20) kp 1 - k (p1 1 kpl2l+l; P14 p1 2 p2 P n-l 1p Semi-general Formulae To obtain more general formulae, I is used to represent the set p1, P2' ~.. and 1qi\ indicates the group of array types in which q appears in a row with any element of o Similarly, ql.q2i indicates the groups of array types in which ql., q2 appear in different rows with elements of }o3 The notation j tqlq2 is used for the array type in which qL, q2 appear alone in additional rows with the initial array,ypoe Such an array type is termed an extended array type and is defi ned as one which consists of an initial array type plus additional rows in which elements are p's (but not proper parts of p's) matched with zeros, or q's (but not proper parts of q's) matched with zerosO The array types (1 Pll q1 are some examples of P12 q12 P12 q12 P2 0 P2 0 0 q2 extended array types when the initial array type is Pll q]1 1 P12 q12

The notation qlj q2 symbolizes the array type with ql in a row with some element of I and an additional row containing q2o Double subscripts indicate partitions~ Thus qlq 125 q2 indicates an array type in which the 2-part partitions of ql are added concurrently to two elements of j and the row containing q2 is addedo Also, tPllP12+ q1 12 is used to indicate the array type in which the 2-part partitions of p1 appear with the 2-part partitions of q1, the other p's appearing aloneo Similarly, IPllP12+ +qllq12 q2j indicates an array type like the previous one with, in addition, q2 appearing in a row with some pi, i l1 And Pllp12+ qllq912. q2j q3..o is used to indicate the extended array types with additional rows of q'So In this notation, we can write the semi-general folrmula (1,21) k k kk }2-4 k - 1 (P P) Dwyer and Tracy (1962) have provided formulae for k, k, for 2iq. 4 From (1o21), to write k k2 for instance, we have T1P2 2 \P2k2 1P2 + n l 20 D plp2 T ) Pl211+ 2)PllP 224110 +n-l PP221 P22 P21P22;P1+ 1;0 + ^ (P21 }5p2pl ll o which is the same as (119)o To get k)2k2 now, we can use either (119) or (121) and obtain n k 445 k2 + 2) kk2 n(n-l) n - 422 3 1 1 2 6!o = k62 k44 n(n l) k53 + n-+ k332 +

Steps of the Combinatorial Method The steps of the combinatorial method for products of generalized k-statistics can now be stated. lo Write each generalized k-statistic of the product desired in terms of bracket typeso 2o List all possible arrangements of the products of bracket types in which the bracket type components of the first factor are placed in the first column, those of the second factor in the second column, etco, to form the array typeso In so doing ignore any array type which has a proper part as a single nonzero element of a row, 3o Compute the combinatorial coefficient for the array type by forming the product of all partition coefficients associated with every partition appearing in the columns of the array type. 4, Compute the algebraic coefficient for each array type as indil cated above and using the rules of Chapter II and the results of Chapter IIIo 5o Multiply the algebraic coefficient by the combinatorial coefficient to obtain the k-coefficient for each array type. The listing of the k-coefficient in the column for the k-term gives the result in combinatorial formo More explicitly, 60 Write the formula for the sums of the products of the k-coefficients and the k-termso 7~ Expand each of the k-terms to feature explicit k's if more explicit form is desired.

CHAPTER II GENERAL RULES FOR COMBINATORIAL METHOD We now consider some rules which are helpful when it is desired to express products of generalized k-statistics as a linear combination of such statistics, (eogo for purposes of taking expectation and estimation), using a combinatorial methodo The first four rules are generalizations of the rules of Dwyer and Tracy (1962) which they used for double products, Rule lo Algebraic Coefficient Rule The algebraic coefficient of an array type is obtained by adding the n -coefficients of the array type and all its conditional amalgamations, In order to establish this rule, we need to minimize the effect of the combinatorial coefficient. We need not be concerned with the formula coefficient since it is fixed for each array type, We use distinct units to eliminate the combinatorial coefficients since each of them then becomes unityo Using bracket types (1.18) to indicate all the brackets with distinct units^ we have lp2. l/(2.1)I2 f)-)> with -f()= (-1)'T((-I)!,a formula coefficient. Then (2.2) klp2`o~' =f(fs)J1)' Tk<.1.o2 k,' - > T<1,, 9,L 12T.i>...) T< where..~).(Ai)ooo is the formula coefficient and the combinatorial coefficient of every array is unity. With distinct units, as in Table 3, the separations coefficients for every conditional amalgamation is unity -30

and the algebraic coefficient is simply the sum of the n'~coefficients for the array type and its conditional amalgamationsO Releasing the condition of distinct units so many brackets in (2o2) may be indentical, this quantity is multiplied by the combinatorial coefficient for the collectiono Rule 20 Pattern Rule Array types with the same pattern have the same algebraic coefficiento Array types are said to have the same pattern when the various groups of the partition parts correspond in locationo Thus, the array types pll 3'l - 2 and pl 4 3 P12 q 2 1q2 r 2 P12 1 9l q P2 p l i P2 2 2 2 1 2 in the expansion of kl5k42 and the array type plla 5 qll 4 P12 4 q12 ~ 3 P2 8 q2 =2 in the expansions of k98k72 have the same patterno For this reason, array types and patterns are used in a synonymous senseo This rule follows from Rule 1 when we notice that array types with the same pattern have the same n-coefficient and similar conditional amalgamations ("similar" means having the same pattern)0 In the example considered, the algebraic coefficient is ( in each case since the only n(n-2) conditional amalgamations result from adding the first two rows0 Rule 30 General Rule of Proper Parts The algebraic coefficient is zero for every array type in which there

- 32 is at least one row in which a proper part appears aloneo A proof can be given like that for the case of products of two generalized k-statistics as shown by Dwyer and Tracy (1962)o Let the proper part appearing along in a row be 1 and let all other non-zero entries be greater than lo Then each of the k oterms arising from the array type has a unit subscripto Since the product expansion does not have any k oc -terms with unit subscript, the k-coefficient and hence the algebraic coefficient must be zero (since the combinatorial coefficient can not be zero)0 A more formal proof is now presented. Consider an array type with a proper part p.+, of p. appearing alone in a rowo Let the other entries in this column consist of the remaining r * A I p Kparts p. o0o p~ of p,, s zeros and t. "" ~other non-zero entries (indicated by crosses in the figure). If we let A absorb the product of (-1) (Pl)i for all n(1) f |) ~ columns except the one considered (P 0- -- 0- 0A- number of non-zero entries in a column) and the product of all partition coefficients of p ~ P., then the contribution to the algebraic coefficient from this array is lt) r n(t (&i, 000 pa) A = (lf r- (Pl.. p,,,,) (n-r-t-l A. n(~+trl) If we consider only additions of the row with the proper part) the resulting conditional amalgamations belong to two array types"

-33 1) s amalgamations of the type when one of the s zero entries of this column is replaced by the proper part p. +, kf.~ | resulting by adding that row to one of the s: I I — i~-J -rows having a zero entry in this colurnnu Since the contribution to the coefficient of'-" each such amalgamation is (1)r ro (p op. (n-r-t-l) A, the contribution of s such analgamations is (g1) r s (p, oop].)(n-r-t-l) A. 2) r amalgamations resulting from the addition of the proper part row - I ~0to a row having a part of pi in this columno!C — | For example, if we added the proper part row to the row having Pil in this column, th.e >.,-' 1 amalgamation will yield a separations coefficient (pl sP., ) to be multiplied by __ I __.._ _ r ) -1) ar(l) (P, + Pi+,P >o~,p r ) n- r-t) A. The contribution of such an amalgamation to the coefficient is then (1) rl (r-1)' (PA,,p + )(n-r-t) A, since (p,,P, )(p,, p!,p, ~ o s (p;, ooo P,,+)~ Thus the contribution of r such amalgamations is (-1) rl (pt, ~ o +, )(n-r-tJ A. Hence the total contribution is (-1) r(p, oo)p(l)( n-r-t-l) A [(n-r-ts) s t (n-r-t)j which is zero~ This total contribution is similarly zero for each conditional amalgamation involving additions of sets of the other rf s + t rows among themselveso Thus the coefficient for this array type is Z 0,

-34 Corollary: Tukey's Ruleo When expressing a polynomial in polykays as a linear combination of the same, the unit weight on the linear side can not exceed the unit weight on the other sideo This is so since in the linear expansion of any term of the polynomial, any additional unit subscript must come from a row with a single proper unit parto Rule 4o Rule for Extended Arry Types The algebraic coefficient of an extended array type is the same as that of the initial array type. Let us consider an initial array type extended by a row containing a single Pi (not a proper part). Let the column containing this entry have s zero and r other non-zero entrieso If C indicates the product of signs and factorials for all columns and 1 for all columns but this particular one, the contribution of the array type and its s conditional amalgamations resulting from adding the new row to the initial ones, but not involving any amalgamations of the initial rows, is C. n(++' A) + sC ) n+A -C. -1, n-r-s) + s CO I ) (n-r) CO n'A which is the contribution to the algebraic cotioneffcient for the nitial array typeo Since this equality holds for every conditional amalgamation of the initial array type and the corresponding contribution of the extended array type, the two algebraic coefficients are the same. The argument applies when more rows of this type are addedo

-35 Rule 50 Rule for Augmented Array Types If the non-zero entries of some columns of an array type are such that they are carried all the way through in all conditional amalgamations and do not impose any further restrictions in the addition of rows, the al.gebraic coefficient of the array type is the product of the coefficients (-l) (P-l)' of these columns (r being the number of nonn(o) zero entries) and the algebraic coefficient for the rest of the array type~ (An array type augmented by these columns is termed an augmented array type when the rest of the array type is looked upon as the initial array type), The reason for this rule is obvious as each column of this type contributes a factor equal to its own coefficient to the contribution of each conditional amalgamation, and all of these conditional amalgamations are just conditional amalgamations of the initial, array type augmented by these columnso Let us illustrate this rule by considering an augmented array type having one such columno The algebraic coefficient of the array type,, q,, r,, is times the coefficient of p q,, i eo, p o r2 n(n-1) pI q,_ 0 P21 q12 P,3 4q3 0 PI q13 _,-1 1__ o (The coefficient of n 1n) -lT ~ R(n2t) (n-l)(n-2u p,, q,, is the same as that of p, q, by Rule 6). Each amalgamaP 0 P,. q, P,1 1p3 q13 tion of the two-column array type is just augmented by the extra column

-36 having r,, r,, and its effect is only to multiply all contributions by -1 Since no addition of rows in p,, q,, is restricted by the nn-l) P2 0 121P, 42 p3 q3 presence of this column, each and every amalgamation of the two-column array type appears in the augmented array type with this column appendedo Hence the overall effect is a multiplication of the coefficient for the initial array type by -1 n(n-l) ~ In contrast, the coefficient of p, q,, r is 1 0 rz n(n)(n-2)' Pi ~ r2 -. 112 0 ~ 0 Po 3 which is not 1 times the coefficient of p, q, (which is zero n(n —) p 0 Py,, q, PZ 12by the rule of proper parts)~ The reason is that although the last column is carreid in all amalgamations, it imposes restrictions in the addition of the first two rows. These can be added in the two-column array type, but not when the last column is augmentedo In p, q, r, however, the first two rows of the two colurn p1, O r, q 0 P22 ql ~ p3 ~ array type p, q,, can not be added anyway, so the last column does P, ~0 p93 ql3 not impose any additional restrictions in the addition of rows,, Also it is carried through in all amalgamations as such, hence the coefficient for the array type is 1 times the coefficient of 1.

-37 p, r ql i.ef, zero (by the rule of proper parts), All the zero coefficients 0 P, 3q,3 in Table 5 are attributable to this phenomenon (pages 55-63). This rule takes a specially simple form when these columns are composed of single non-zero entries. It can -then be stated as. Rule 5a. The algebraic coefficient of an array type having r columns with single non-zero entries is 1 times that for the array type with these nr columns deletedo For, suppose C is the coefficient of the array type with these r columns deleted, Since no additions for conditional amalgamations depend on these r columns, their only effect is to contribute a factor 1 each, n ioeo t, towards the contribution of each amalgamation and hence the n algebraic coefficient of the augmented array type is 1 C Rule 60 Blocks Rule The algebraic coefficient of an array type which falls into separate blocks is the product of the coefficients for these blockso An array type is said to fall into separate blocks if the columns can be divided into two or more classes, each confined to different sets of rowso Fisher (1928) was able to ignore array types consisting of blocks (Kendall and Stuartt, 1958, po283, Rule 3) in the case of single subscript k's as he proceeded straight to cumulants0 Wishart (1952) has given this rule for products of single subscript k,'s For a proof, we first consider the case of two blocks A and. B, having a and b rows and c and d columns respectively. Let the c columns

-38 of A have al, o., a non-zero entries and the d columns of B have b,0., bd non-zero entrieso At this stage, we do not consider any addition within the blockso Let the symbols A and B absorb the signs and factorials of the blocks A and B respectivelyo Let us suppose a > b (no loss of generality in doing this due to symmetry)o In a typical amalgamation, r rows of B are amalgamated with r rows of A, there being a(r)b(r) such amalgamationso The contribution AB (a+b-r) of each of these to the coefficient is n (_ a n Thus the con-' a tribution to the algebraic coefficient from this set of amalgamations, allowing no addition within the blocks, is' ~ n( _ )a n-a) _ AB3 J- a() b() nffr) - AB na) b (r))(t-r) TTTn c )1 n() t=O na AB n(a)n(b) by Vandermonde s Tnr ) T[ (6b) Theorem n 11 n (Riordan,,1958,p 9) An(a) Bn(b) ioceo the product of coefficients for the two blocks If we now allow addition within blocks, let the sets of conditional amalgamations be denoted by Apt, B }o Then, the total algebraic coefficient for the array type A 0 is 0 B Contribution to algebraic coefficient for amalgamation A- 0 0 B Ft O B, (Contribution of Block Ap)(Contribution of Block Bq) ( ZContribution of Block Ap)( Z Contribution of Block Bq) (Algebraic coefficient of A)(Algebraic coefficient of B)o

39 If now there are three blocks A, B, C, we can treat A 0 as one 0 B block and C as the second. Repeating this, we find that the rule holds for any number of blockso Rule 70 Rule for Column-nbordered Blocks The algebraic coefficient of an array type consisting of two blocks __ _ a.. A and B connected by a common colurun C, whose A CA portions going with the two blocks are denoted by CA and CB, is given by the following rule*0 CB B (a) When no parts of any pi occuring in CA appear in CB and vice versa, the algebraic coefficient of the array type is the product of the coefficients for the blocks A CA and rC B (b) Wlhen there is at least one Pi which has parts in both CA and CB, the algebraic coefficient of the array type is zeroo It makes a difference in the proof whether the coemon column C is solid or not. Solid means -that there are no zero entries in theLi coluimnC The case of a solid column is simpler, so we consider it firstO Case lo Common column C is solid~ (a) If we just consider the array type A CA 0 0 CB |.. B...,.. without considering any conditional amalgamations, it is fairly easy to see that the rule holds. For if no pi is common to CA and C, the co* So far a general proof is not available, but the rule has been enstablished for all cases needed in compiling the products of generalized k-statistics through weiLht 12,

-40 efficients of the two blocks A CA and CB B are independent of each other. Also, if the contribution of A to the algebraic coefficient of A CA is denoted by A and the signs and factorials for CA are absorbed in CA and similarly for CB B, the algeO C braic coefficient of the array type A CA is AB AB (a+b) B ---- n _(a+b) o C, B n" ABCACBo Also the algebraic coefficient of | A C is A A n(a) ACA and similarly that for B B is BCB. Evidently n(a) then, the rule holds when we do not consider amalgamationso The argument is extendable when amalgamations are allowed, For then, if we denote sets of amalgamations by tA~, {B 3, the total algebraic coefficient for the array type A CA 0 i A 0 CB B I Contribution for amalgamation Ap CA 0 C Bq q (Contribution of A l~" )(Contribution of %C B ( Contribution of Ap I )( Contribution of C~ Bq, (Algebraic coefficient of ( A ]C )(Algebraic coefficient of j B't (b) Let now a particular p have a parts in CA and b parts in CBo Let us again suppose a >b, without loss of generality, Not allowing for any other addition of rows except those involving parts of this particular p, a typical amalgamation consists of;r rows of B added to r rows of A, (r b), there being ar brr) such amalgamations.~ If the c columns of A have a1, ooo,a non-zero entries respectively and the d columns of B have blooo bd non-zero entries, the contribution

- 1 of this amalgamation to the algebraic coefficient is (-ab-r-1 ABCACB (a+b-r-1) TEn xP) mbnY where A and B absorb the signs and. factorialsfor blocks A and B respectively and CA, CB for entries in CA, CB which are not parts of this particular po Then the algebraic coefficient of the array type is Z (-1)a+br-1 ~r-57 a(r)b(r) ABCACB (a+b-r-l) r.ff n)4 nbi) aUb4 ABCAC a(r) (r) Lb-,rJ (-l)a+b1 ABCACB (a-l ) 2 (-1)- r a r) ab7l~pTn('O, n (-)atb-1 ABCACB (al) (-i)b Z ()a ()( [b-r] b-rb (Jb-r) since a = (-1) (-a) 0, since Z ( r)a (-a) (a-a) by Vandermonde s Theorem 3-5rm - 0. As before, if we now allow for additions which were earlier restricted, the total algebraic coefficient of the array type A CA is 2 Contribution for amalgamation AAp C 0 2 0 ~ CB B O C Bq Case 20 Common column C is not solido (a) Let the number of rows in A, B be a, B and let CA, CB have e, f non-zero and g, h zero entries respectivelyo It may be noted that e +g = a f+h bo Absorbing signs and factorials in A. B, CAY CB, the

-4 i-2term contributed by each amalgamation is ABCACB multiplied by the /fn )| F1 a) appropriate factor in the following scheme~ We are not allowing for additions within the blocks at this stage, Number of rows of B added to Number of Amalgamations Multiplying those of A Factor 0 1. 2 3 4 1 eh - fg + gi. g( 2)fh t h(2)eeg f(2)g(2) g(2)h(2) + 2 2- 2' e(2)h(2) + - + efgh e (3)(3) f(3)g(3) g(3)i(3) (3) f(2)h _3 + 3 _ + - (n-ef) (gh) (n=e-f) (g+h- ) (n-e-f) g(3)f(2) Jh(3)e(2) 2 2o + h3)eg2) 2 + efg(2)h(2) ef(2)(2)h 2' e(2)h(2)fg 2 f ) g() e( 4 4)h4) (3) )(LIeg(3)ih( 4 3),(3) 4)i, f (4)h(3) 31 3~ 3~ e(2)g(2)h(4) f(2)g(4)h(2) - - - + - I —.... 4 ~ 6 efg(3)h(3) ef(2)g(3)ih(2) e(2)fg(2)h(3) _' 4 --.. _... 2,) 2,e(2)f(2)g(2)h(2) 6

Th en, the cI efCfjici.ent ABC C. I1 n ~ )In ( n-c- r) ( z-i~I ) -t- (ch - + 1 + - 1) (n-e-f) (A -l) +. ]) A bC^ C.;. r~t, iD if we denote the quantity in the square brackets Now consider the product (n-e)() (n-f)(h) by An ef We have (n-e)( )( n-)^. ) f) n(+) n_1l ( "n + r +fe- l) + ( e+g)(f (+h) n11 ( - - - - ) ^e"5717)f) (e +g) (f)28 + ) 21 n ( cf+h2) + C o o'by Vanderi11monde s li eo eil (e47) f n nL (2) (e+g). ( 2o I(2) (nef)( + +( ge+:g) (+h)(n e.:f)r-g+h1 )+ f +.h)(2)r ( n-e-fr)(+h2) + ~ ] (nn-e-f) - l.. (e- -+ + +-:' +-ei) (n- -fr) " + -f.*) (e+f) -T 7T) ) (2) (2 g(2)fh + h(2) eg+ 2) g(2) 1:k; ( 2 ) I. Q) I., 0 4 0g(2)(2)e, w " e:2)(2 4: 2 2 e~ ( e2)f(2) p f +(2) +f(2)eg Ieng) ( (.gh.-2)) -2) 2ef ( 2 + e)fh+ 2) +e j n-e- * +..)!An, e, f-+ef (n-e )(+-l)+ (e,, I1) -. I., N (2) [o^ ):.. -... (n-e-f) - i oo.4'1"'3 (n-e f) kjtu. — I {re,,-I, ~.:: e'~'""~ fL''"/ 4 Ij, ~ cf (e)+ f) eAf 1 n. ef 777f) fAnef+n e —f+ - ('c-f.e-..+l )( -+h} e-1)~+ (flg g-+ gh) (n-e-.-fl~~. (. i (2 T(2) -t)) II + c: Oc(( 0- c f 4

n(etf) I[ ef n,e:)n-( L) Ane,f + n.e-f.l An-l1e- f-l + (Je) (f) n^e~f n-e-f+l e(2)f(2) 1 2,(n-e-f+2)(2) An-2,e-2,f2 + We find that Ane,f An.l,el-,f-1 An-2,e-2,f-2 = o for all values of e, f, g, h e 3 that we require in order to derive products of seminvariant k-statistics through weight 12. Hence, (n-e)^(g)(n-f)(h) (n2f) A 1 + ef e(f(2) ] n(e) ( An,ef 1 n-e-f+1 2(n-e-f 2)(2) Anef I(e+f)+ e(2)f(2) (ef-2),oo n,e,ff +[n(ef) +efn(etf-l) + e___ (ef2 n n nef) f n(e) n(f) (2~ 3) An, ef Thus, the required coefficient is ABCACB (n-e) () (n-f)(h) Xn' Tln(ij)4 ACA n(a) BCB n(b) Tfn n In T n = (Coefficient of A CA | )(Coefficient of C -.| B ) Hence, as in Case 1, the total algebraic coefficient of the array type A C 0 is the product of the algebraic coefficients

-45 of A CA and C B (b) Let a particular p have e non-zero parts in CA and' in CBo Let the number of zero entries in these portions be g and h respectively and let there be k other non-zero entries in C, so that e f g +h +k a +bo Let A, B absorb signs and factorials for blocks A and B and let CA) CB do the same for entries in CA, CB which are not parts of this particular po Then the term contributed by each amalgamation is ABCACB multiplied by the corresponding factor in the following schemeo Again, additions within blocks or involving parts of other common p in C are not considered at first, Number of rows of B added to Number of Multiplying Factor those of A Amalgamations F4a..-..I -..;.., I g|0... 0 - 3 1 ef eh+ gf+ gh e() f(2) 2 e')fh + ef( )g + efgh fg ( + egh-) f (^ g g() jh 2o 2o e 3 f (c3) e( h)+efgh e(O f() 31 -1 1l ) k >S1 1( (-1)' (e+f-l)'(n-e-f) ( lSt (e+f-2) (-C e-f+l (^-l) e+f -1 n- e- f) (-l) (e+f-3) (n-e-fi+2-2 e+~6+r- Z (+t-l) (-1) (e+f-2).(n-e-ftl) efS- %+h-2) (-1) (e+f-l) I(n-e-f (-1 e (e-4) (n- e- f+3 )

Number of rows of B added to Number of Multiplying Factor those of A __ Amalgamations.}...... 3 (con't.) e (3) f h h e ( f(3)g e() f(2) gh 22+ 2I e (3fh) ef ) g (+ e f gh +ef g h+e h - e fg 21 e() h(3) f( g) g(3 h( f ^ - -- + + f(3) h(2) (2) gh(3) fg hi e ghi +~T + T ef2 )( e()f g 2t 2o eg ()h( + 2 of + + efg(2) h() e'f33)gh 3 II I (-1) (ef- 3)' (n-e-f+2) e4fC-2 2.Vi-2) (-1) (e+f-2)'(n-e-f+l) ( r'(e+f-l).(n-e-f)((-1) (e+f- 5) (n-e-f4i) (-1) - (e+f4) (n- e-f+3) 4 e(4) f(L) TT —4 _3 +3 eg 3) o 3 o e4 -)f_(h_) f(4)e(2)g e())gf)h() 4-+ 2 —' — + (e =%)g)-' egf)K, e f )gh 2' + 2 ((-1. ( e+f-3)' (n-e-f+2-) eg )f(4) _- + 1 (3)?f geh 2 2 ()fh(5) (3)h( e fl e hPgf + 37 2 2 + efg% + +2 3; 2 + e g e(2S(zhf) 2 2 i I I I j t t (-1 ~(e+f-2) (n-e-f+l +e() T ) gTf * TT T 3;'"3 (( e(.4) g(.4,.) e(gh.9 e,? I (- (e+f-1)o(n-e f fJh^ nfp^ eW^^^4W hI I 3 + -I~ + — + - +'

-47 Number of rows of B added to Number of Multiplying Factor those of A Amalgamations_____..... 4 (con't.) e4(3gf f) geh efg — 2 — + 2 6 e2f ( e. 2 gAh 3, e O O O O O O O G Then, the coefficient (-1 (e+f-l) [ (n-ef) + (eh fg4gh) (n-e-f)I) +...] +(-+1) (e+f-2)'ef [(n-e-f+l) + {(e-l)h (f-l)g+ghj (n-e-f+l) +ooo + (-1) (e+f-3) -i - [( n-e-f+2$i + 2(e-2)h+(f-2)g+ghj (n-e-f+2) +2. -(~~~0 (e+ 3)(-+f () a4 () (k). (-l -') ( e+f-)(n-e)n-f)(- ef-2) ef(n- e)(' (n-,,f (.f.-1 (e+f-3) (n-e) (n-f) +i-., by (2 3) _ (-i' (n-e') (n-f) (e+f-1l): -ef(elf-2):' +~ e +f-3) -... a (1o) o (n-e) (n-f) (0), by Vandermonde's Theorem 0. Allowing now the additions which were earlier restricted, it can be shown as in Case 1, that the algebraic coefficient of the array type is zero.

Rule 8o Rule for Row-bordered Blocks The algebraic coefficient of an array type which falls into t blocks with a common row connecting each two consecutive blocks is 1 times the product of the coefficients for the blocks, nt-l We prove this rule by induction, starting with the first nontrivial case of t s 2, (trivially true for t = 1), in which case the algebraic coefficient of the array type falling into two blocks with a common connecting row is 1 times the product of the coefficients for the blockso Let there be a, b rows and c, d columns respectively in the blocks A and Bo Again,. at first, let us not allow additions within blockso In a typical amalgamation, r of the (b-l) rows of B can be added to r of the (a-l) rows of A (can assume a b)o There are'al)_ bl) ro such amalgamationso The contribution of each of these to the coefficient is AB (nt~_-i* where A, B absorb signs and factorials for _ff ("aO j)n lin!!n the two blockso Then the algebraic coefficient of the array type is AB z (a- b-l)n Tn(~) n( Y rABn() E D I ABn) ( al)) b (na) ln bn ABn (.j) (n by Vandermonde' s Theorem 1 AnC) Bn() n nC(a) Tn(ba)

149 Then, total algebraic coefficient for the array type is 2 Contribution for amalgamation 0A 0 B 1 2 (Contribution of Ap)(Contribution of Bq) n ft p q 1 (CQontribution of Ap)( 2 Contribution of Bq) n 1 (Algebraic coefficient of A)(Algebraic coefficient of B)o Now let the array type consist of t —l row-bordered blocks A,, <Al 1?~i i 1,2, ooo, t+lo Let A denote the --, 0 portion of the array type consisting of the blocks A1,,oo, At and let B con-.\ _i | sist of the blocks At, At+l~ Suppose A0 the rule is true for to Then the co( — efficient of A is t —-1 T Coefficient t+,, n 4J of A o If' instead of At, we now put B whose coefficient is l(Coefficient of At) (Coefficient of At+l), we find that the coefficient of the array type is 1 ( T Coefficient of A9)(Coefficient of B) tl ~=~ n n1 ( (T Coefficient of A.) 1 (Coefficient of At)(Coefficient of At+l) ^'T"T <( Coefficient i/~n LU^n ~+1 lt 7 Coefficient of Aio n A= Thus the rule is true for t+l if it is true for to But we have seen thatit is true for t - 2, hence it is true for all to Most of these rules are the result of a study of coefficients of particular array types which are listed in the next Chapter, On the other hand, once the rules were known, they were very useful in determni nig coeffi.cents of additional particular array typeso

CHAPTER III COEFFICIENTS OF ARRAY TYPES Fisher (1928) gave the algebraic coefficients of some commonly occuring patterns when considering the products of single subscript k'so They need to be generalized when we deal with products of generalized k-statistics and each of his patterns gives rise to several array typeso These are systematically tabulated for his first twenty-four patterns and coefficients of several general array types are studied, Generalization of Fisher's Patterns Fisher (1928) while listing some commonly occuring patterns and indicating their algebraic coefficients, had a simpler situation in that all rows can be added when we consider products of single subscript k-statistics (or their cumulants as Fisher did)o When we consider products of generalized k-statistics, addition of rows is restricted as a part p,. of some p. can only be added to another p./ or to a zeroo So the patterns can no longer be indicated by filling in xss for the entries as Fisher did and in fact each of his patterns leads to several patterns distinguishing the location of parts of one p. from those of another~ Fortunately, by the pattern rule, we do not have to distinguish for numerical values of p, so long as the location is not involved, Also rows and columns of an array type can be permuted at will, We follow the practice of indicating the subscripts going in the first colurmn by p's, those in the second by qjs and so ono Where the number of columns is generalized, we change to p l p2 p3,00

Fisher (1923) has recorded that in the case of two rows and r "y -./ 1 cm -Y.-l columns, the algebraic coefficient of the array type xo0 is _-._ IX where -r1 We further observe that the array type n-1 I Z, /P- p p o opo p p o 2.! 12 12 2. (encountered, for example, in the generalized product k: k o ookpA k k.2.6 o o kr ) has an algebraic coefficient h h h h h. r. 1 r-l (I) 1 X s <r n (3ol) } (1 gi) s r (Fisher s case)o n The distinction emerges from the fact that when s 4 r no addition of the t.wo rows is possible, whereas s = r makes this possible, In the case of three rows, when there are r columns, each consisting of three parts of a single p,, the algebraic coefficient is found to be 1 (2 * 34' + l1) where x n - -1 n- 2 n n — We get the special cases listed by Fisher by putting r - 2, 3 4, the algebraic coefficients being n for x 2 Cn + 0 for xxx and 3.3 2 x3xx lX n - 9n3 + 33n - 60n + 43 for XXXx n(n-l)3 (n-2)3 XXX

But now when we consider three rows and two columns, we have to list all the variations such as Pi, q,1 PlI q,, P, q11 P,l q, P,, qlI P,, q P, q P,. P, 9 Pz q PI, qq Pp c qp P,, q2 P,,,q P2 q PI ql, 13 3 ql PI q I Pi 13 q3, p2. 12., 12 3 3 3 the algebraic coefficients being n 1 1 2 n(n-i) (n-2)'-"'n-l)(n-2)' n(n-2)' n(n-l)(n-2) 9 1 _ 1 respectively n(n-1)(n-2)' n (n-l)(n-2)' n(n-l)(n-2) (Table 5~4)~ No additions are possible for the last four array types. Generalizations are, of course, possible in both directions, For example, the algebraic coefficient of the array type, q, obtained R, Ci PR q3 P:, (a^ xx by adding a row to Fisher's pattern xx, is n-a 1 (Dwyer and Tracy, 1962, p.35), which sheds addin(a+2) (a ia-l) tional light on the coefficient of p,, listed earlier as 1 as'i 9q n(n-2) P q. n-l actually being -r- o The formula holds formally when no rows are added, n(3) ie,, a = 0, and gives the coefficient of Fisher's xx as ) - - Similarly, the coefficient of

-5,3 P. 412.. P13 q13 Pv q2. (n-a)2 Ta+3 n.' / (Dwyer and Tracy, 1962, po35) h which, for a z 0 P,+l q -Ia+t P 4s n gives n-i( n,-2 For p,, q1, p1. q8 pz q, r3 p~ q3 r3 3. 3, we find the algebraic coefficient to be p a4 ra n-a)2 - 2(n-a) [n (cJ which, when a 0 gives n2 - 2n [n(n-l)J2 n-2 n(n-l)2 as the coefficient for Fisher s XXX xxx ~ Similarly, for p q rnz s2 P,2 q r s3 P3 q r3 S3 -^ar~ aI 0c+l ~k^+ the algebraic coefficient n23 a 0 O gives n -3n-3 as n2(n-l)3 [is Ln -(a a22 ) 2 3) [n'^ which, for the coefficient for Fishers xxx.AXXXX ~ For the general case of r columns and a + rows with two rows additive, we get 1 -a I [ (1 ~o}g ) as the coefficient where =: -:]- 1 This n for a 0, gives Fisher's result (3,1).

.51 Now generalizing in the other direction, we consider the array type p,, q,, r,,.... The algebraic coefficient is found to be P,2 q1 r2 Pl3 q2. rz~ ~ ~ n(n-)1 - 1 n-2 -1, when there are r columnso Use of Table 5 The more common of Fisher's patterns (first twenty-four, Fisher, 1928, ppo 223-24) are now taken and elaborated into the several cases they yield for products of generalized k-statisticso The results are presented in Tables 5.1 - 5024~ The top left entry in each case denotes Fisher's patterno Taking Table 505 as an example, Pl, q,, rl is P3 qt r13 successively transformed to all its variationsc The top left entry n (n26n+ 10) multiplied by the of the table indicates that the multiplier 1 at the t( coefficient of p,, q, r, is P2 q2 rr p 13 r3 op n (n 6n+10) n26n+ 10 The second entry in the first row n2(n-1)2(n-2)2 (n-l)2(n-2)2 is for a pattern where the column q of q's is changed to q,, and so on q12 q)lu q1,3 q2 till the last entry in the last row represents the coefficient for p, q, r, (multiplied by 1 ) The changes in the P2 qz r, n2(n-l)2(n-2)2 P3 q3 column of r's are indicated at the left of the rows and r,, etco are r,2 1.2

-55 written as r,, rj2 r just for the convenience of presentation. The choice of indicating changes in p and q columns by columns and those in r columns by rows is purely arbitrary, except for some -thouigt to the shape of the table. Although a pattern like p q, r, does not p9 cj1 r. 93 q2 t 3 appear specifically, its coefficient is the same as for p, q, r, P,2 q r. P, q3 r3 for reasons of symmetryo In bigger tables like Table 5.13, the saving is considerable due to this as whole rows and columns for certain variations can be ignoredO TABLE 5 PATTERN COEFFICIENTS Tabl e 501 1 n(nl P,, q,, q, Piz..ia _._ pil 1. Pi P2 -1 1 _- i

-56 Table 5.2 1 n2(n-1)2 Table 5.3 1 n3(n-1)3 Table 5.4 1 n(n-l) (n-2) Pll qll rll s l 1 rl 8 P12 ql2 rl2 S12 2 r2 s2 n(n2-3n.3) -1 1 ql q2 -1 1 -1, q 2 1 -1 1 Pll Qll qll q P12 %12 12 q2 P13 q13 q2 q3 n -n 2 PllPl2P2 -n n-1 -1 PlP2Pl2 -n 1 -1 P1 P2 P3 2 -1 1 Table 5.5 1 n2(n-1)2(n-2)2 Pl q.11. r qll qll q% P l 2 11 Pl2 PIIql P1 ql P12 %q12r12 q2 2 2 P12 q12 Pll q12 P12q2 P2 q2 P13 ql13 rl3 q2 q12 q3 P2 2 P12 2 P2 q3 P3 q3 n2(n2-6n4^10) n(n-4) n(n-4) 4 -(n2-4n+2) 2 -2 2 rl r2 r2 n(n-4) -(n2-4n+2) 2 -2 (n-3)(n-1) - 1 -1 rll r2 r12 n(n-4) 2 -(n2-4n+2) -2 - -1 -1 -1 r r2 r3 4 -2 -2 2 1 1 -1 1 Table 5.6 1 n2(n-1)2(n-2) Pll qll rll qll qll q Pll ql1 PlIll P1 1 P L Pll ql Pi ql P12 qq2 rl2 ql2 q2 q2 P12 q12 P12 2 P2 2 P12 2 P2 q2 P13 13 q2 q12 q3 P2 22 P2 q12 P12 q2 P2 q3 P3 q3 n2(n-3) -n(n-3) n -2 n2-3n*l -1 -(n-1) 1 -1 rl r 0 2n -2 -n 2 1 1 n-l -1 1 12

-57 Table 5.7 1 2 (n-2 n (n-l) Pll qll rii ql 1 1 P P P120 r12 P12 P2 P2 12 P2 P13 q12 0 q2 P2 P12 p3 P P 2 q 3 q2 12 n2 -n -n -n 1 n -1 r r2 0 -n 1 1 n -1 -1 1 Table 5.3 1 2 2 n (n-i) 0 qll rll q 0 0 o ll 11 ql ~ ql *PI P 12 ~ ^0 Pl P1 0 P12 q12 0 q2 P2 2 q2 2 n -n -n 1 r r 0 -n 1 1 n-2 1 2 Table 5.9 n3(n-1)3(n-2)3 Pll qll rll 1 Slrl Sl1 rl l S 1 rll 2 r Sll 11 rl ll rS r s1 P12 ql2 r12 S12 S12 s2 r12 S12 r12 2 2 s r2 S2 r2 2 s2 r2 s2 p13 q13 r13 S13 s2 s3 r2 s2 r2 s12 r12 12 r12 s12 r3 s12 2 r s n2(n4-9n3+33n2 -n(n2-60n + 43) 6n +12) 8 n3-6n2+12n 4 4 n3-6n2+12n - -4 -4 I I —4 qll q12 q2 -n(n2-6n +12) n3-6n2+ 12n-4 -4 -(n3-6n -2 -2 2 2 -2 12n-6) q. q^2 43 -4 4 2 2 2 2 -2 -2 P P12 P2 n3-6n2+ 12n - (n3-6n2 (n-1)(n2- 1 1 1 -1 -1 1 11l q12 q2 -4 +12n-6) 2 5n 7) PII P12 P2 4 -2 2 1 1 1 1 -1 -1 1 Pll q2 1l2 q21 q2 q12 P1 P 3 PF 2-4 2 -2 -1 -1 -1 -1 1 1 -1 41 2 p3 P1 p2 p3 -2 2 1 1 1 1 -1 1 1 q2 q3

Table 5.10 1 n'(n-lD(n-2 )2 Pll qll rll 1 Sl r11 r1 r'2 r rll S1 rll S1 r2 S1 rl S1 P12 q12 r12 S12 s2 r12 r2 rll r2 r1 2 s2 2 s rll 2 2 s2 P13 q r13 r3 r2 r12 r2 r3 r2 2 r12 rl2 0 0 n2(n-3)(n2- -2n(n-4) n(n2-5n+8) -n(n-4) -n(n-4) -4 -4 n(n-4) n(n-4) 4 4n + 6) 1ql q12 q2 n(n2-5n+) -4 (n-3)(n2- -2 -2 2 2 2 2 -2 2n+2) qll q2 q12 -n(n-4) n(n-4) -2 (n-3)(n2- -2 2 2 -(n-3)(n2- 2 -2 2n+2) 2n+ 2) q q!2 q3 -4 4 2 2 2 -2 -2 -2 -2 2 Pll P12 2 -(n3-5n2+Sn 2 n3-5n2+3n 1 1 -1 -1 -1 -1 1 qll q12 q2 -2) -3 Pll P2 P12 n2_-4n+2 -(n2-4n 1 -(n-3) 1 -1 -1 n-3 -1 1 Pll P2 q12 +2) Table 5.11 1 n3(n-1)3(n-2) Pll qll r1l Sll qll qll ql 11 1 Pll qP11 Pll2 ill P11 ql P1 ql P12 q12 rl2 S12 1q2 q2 q2 P12 q2 P12 q P2 2 Pll q2 P12 q2 P2 q2 P13 q13 0 0 q2 q12 q3 P2 T2 P2 q12 P12 q12 12 12 P2 q3 P3 q3 n2(n2-4n+ 5) -n(n2-4n -n 2 n3-4n2+ 5n 1 n- 1 -1 1 +5) -1 r1 r2 0 -2n 2 n -2 -1 -1 -(n-l) -1 1 -1 rl r2 0 2n -2 -n 2 1 1 n-l 1 -1 1 s1 s2 0 Table 5.12 Table 5.13 1 n3(n-1)3(n-2) Pll qll rll Sll qll qll ql Pll qll Pll qll Pll q Plql P12 ql2 0 S1 2 2 q2 q P12 q12 P12 q2 P12 2 P2q2 P 13 q13 r 2 2 q12 q3 P2 q2 P2 q12 P2 3 P3q3 n2(n2-5n n(n-3) n(n-3) 2 -(n2-3n+l) 1 -1 1 +7) s1 S2 0 n(n-4) 2 -n(n-3) -2 -1 -1 1 -1 rl 0 r2 n+2 -2 -2 2 1 1 -1 1 s1 s2 0 1 2 1 n3 (n-1)3 Pl ~ rll Sla 0 P1l P1 Pll Pl0 P12 110 Sal ql P12 P2 P121l P2ql P13 q12rl2 0 T2 P2 P3 12T2 3q2 n2(n- 2n n -1 -n 1 3) r1 0 r 2n -(n -n 1 n -1 +1) s1 s2 0 2n -(n -1 1 1 -1 -1) r1 0 rI -(n+l) 2 1 -1 -1 1 1 s2 02

-59 Table 5.1l 1 3 3 n (n-l)3 Pll ql rl11 S1l 11 P P P2 P1 Pllql P 11ql P 2qi Plql P120 r2 S12 0 P12 2Pl P 2 P120 P2 0 P10 p20 P13 ~3 0 2 P2 P12 P12 p3 P2 q2 P12q2 P12q2 P3 n2(n-2) -n(n-2) -n(n-2) n2-2n4 0 -1 n(n-2) -1 0 1 r r2 n -1 -1 -(n2-2n+2) 0 1 1 1 0 -1 r1 r2 0 -n 1 1 n2-2n+2 0 -1 -1 -1 0 1 s1 s2 0 Table 5.15 Table 5.16 1 1 n3(n-1)3 n3(n-1)3 0 qll rl Sl q 0. al Pll q01 rll ll l P1 ql Pl10 r12 s12 0 P1 P 0 0 0 r 12 s12 0 00 P12i2 0~ ~ 2 P2 P2 q2 P12 q12 0 0 q2 P2 q2 n2(n-2) -n(n-2) -n(n-2) n2n-1 n2(n-1) -n(n-l) n(n-l) rl r2 0 n -1 -1 -(n-2) rl r2 0 -n(n-1) n-l -(n-l) r1 r2 0 -n 1 1 n-2 r1 r2 0 n(n-1) -(n-) n-l Table 5.17 1 n(n-l) (n-2) (n-3) Pll qll.q.. q2 qll qq1q qll qll q'l q2 ql P12 q12 q12 q11 q12 q21 T12 T2 T2 q3 T2 Pl13 ql3 ql3 q12 q21 q12 q2 q3 q12 qi q3 P1 T1ql Tq2 qi3 22 q22 3 12 T3 q12 4 n2(n4-1) -n(n+l) -n(n+l) -n(n-1) -n(n-1) 2n 2n 2n 2n 6 PjlP12P13P2 -n(n+l) (n-1)2 n+l n-l n-l -(n-l) -2 -(n-l) -2 2 PllP12p21P22 -n(n-1) n-l n-l n 2-3n+l 1 -(n-2) -1 -1 -(n-2) 1 IP11p12P2P3 2n -(n-l) -2 -(n-2) -1 n-2 1 1 1 -1 PlP2P3P4 -6 2 2 1 1 -1 -1 -1 -1 1

-6o Table 5.18 n2(n-1)2(n-2)(n-3)2 rll r rll rll r 11 II 11 11 11 r12 1r2 r21 r12 r2 r13 r r r r r13 r21 r12 2 3 r2 r22 r22 r3 r12 n2(n4-12n3+51n2- n(n3-10n2+25n n(n3-lOn2 n(n3-10n2 4n(n-6) 4n(n-6 74n-13) +12) +29n-12) +29n-12) n(n3-0n2+25n+12) -(n -10n3+31n 2(n2-6n+3) -2(n2-6n -2(n -6n 12 -24n+42) +3) +3) n(n3-10n229n-12) 2(n2-6n+3) n -lOn3. -6 -2(n2-6n 6 41n2-84n +6) 66 4n(n-6) -2(n2-6n3) -2(n2-6n+6) 6 2(n2-6n+6) -6 36 -12 -6 -6 6 6 -(n4-n+31n2- n4-lOn3+34n2- -(n-l)(n- -(n2-6n-7) n2-6n-5 -4 24n+42) 42n+17 5) 2(n2-6n-3) -(n-l)(n-5) -(n2-6n+7) 2 n2-6n+7 -2 -2(n2-6n+3) n2-6n5 n2-6n+7 -2 -(n2-6n+7) 2 -12 4 2 2 -2 -2 n41On341n2- -(n2-6n+7) n4-1On3- 1 n2-6ni8 -1 84n+66 35n2-48n +19 -6 2 1 1 -1 -1 -2(n2-6n*6) n2-6n+7 n2-6n+8 -1 -(n2-6n+8) 1 6 -2 -1 -1 1 1 -6 2 1 1 -1 -1 2(n2-6n+6) -(n2-6n+7) -(n2-6r+8) 1 n2-6n+8 -1 -6 2 1 1 -1 -1 ) 4n(n-6) -2(n2-6n +3) 4n(n-6) 12 36 -12 6 -2(n2-6n+6) -6 -6 6 2 n -6n+7 -2 -6 6 -4 -2 6 -6 -2 -2 -2 -1 n -6n+8 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 -1 -6 6 -6 2 -2 2 1 -1 1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 -1 1

-61 Table 5.19 n2(n-1)2(n-2)2(n-3) r Pl1 q11 r11 rl P12 q12 rl2 r12 r2 P13 q13 r13 r2 r3 P14 q14 o O O............................ qllql2ql3q2 q112q2 q13 "llql2921q22 qllq-21ql2q22 "llql2q2 q3 Qll 2 q3 q12 l q2 3 q4 P P P P p11p1213p2 q11 12q2 q13 PllP12P13P2 PllP12P13P2 P11P12P13P 2 q q2 q3 1q4 llPP 2p21P22 qllqi2q2122 PllP12Pl21P22 qllq21ql2q22 PP12P 21 22 qrllq2 3 q2 PllP12P21P22 ql q2 q3 q4 PllP12P2 P3 qllql2q2 q3 PllP122 P3 q2 q3 qllql2 p11 p12P2 P3 qllq.2 q3 q12 PllP12P2 P3 q1 q2 q3 q4 P1 P2 P3 P4 q1 q2 q3 q4 LA^ I n2(n3-8n2+ 17n+2) -n(n4-8n3+22n2-23n.-30) n(n2- 5n-2) n(n2-5n+2) n(n2- n+2) -2n(n-5) 4n -12 n4-8n3*20n2-13n.2 -(n2-5n-2) -(n2-5n+2) 2(n3+1ln2-128n*240) -2n(n-5) -n(n2-5n+2) 2 -n(n2- 5n+4) 2n 2(n2- 5n43) -2n 6 n2- 5n+2 n2-5n 2 36 -12 -4n -2n -2n 2n -6 -12 -12 n2-5n+4 n -5n+2 -(n2-5n+4) -2 -2 -(n3-6n2+9n-2) n3-6n2 lOn-4 n-2 2 -1 2n n -5n+4 -(n2-5n+5) -2 -1 -1 1 -1 - (n2- 5n4) n -5n+5 -1 1 -2 1 -1 2 -1 1 L

-62 fable 5.20 1 n2(n-l)2(n-2) (n-3) P11 11 ll 1 Pl 1 P2 P11 P1 rl Pl rl P2 r1 P1 rl P1 P 2 q12 r12 P12 Pl P12 P2 r 2 Pllr2 P12 r2 P2 r2 P3q1 q 3 P P13 2 P21 P3 0 P130 P120 P21 0 P3 0 P14 q14 0 P2 P13 P22 P4 0 P2 0 P130 P22 0 P4 0 n2(n2_4n-1) -n(n2-4n-1) n(nl ) -(n-4)(n2-2n 6 4n2 -4n -n(n+l) -2n -6 +3) qlqlq2q3q2 -n(n2-4n-1) n2-4n 1 -(n 1) n2-4n+l -2 -4n n+l n-l 2 2 q1qq12q2 ql -n(n2-4n-1) n2-4n-1 -(n+-l) -2(n2-4n+2) -2 -4n 4 ntl 2 2 qllq2 q12q3 -2n n-l 2 1 1 2n -(n-l ) -2 -1 -1 q2 qllql2ql 3 n(n+l) -(nli) -(n-)2 -(n-l) -2 -n(n+l) n+l (n-l)2 n-l 2 qllql2q1l22 -(n-4)(n2-2n+3) n2-4n+ -(n-l) n3-5n26n-1 -1 -2n 2 n-l n-2 1 qllq21q1222 n(n-l) -(n-l) -(n-1) -1 -1 -n(n-1) n- n-1 1 1 qllql2q2 q3 2n(n-4) -(n2-4n+l) 2 -(n2- 4n2) 1 6 - -2 -1 -1 q2 q3 qlql2 -2n 2 n-l n-2 1 2n -2 -(n-) -(n-2) -1 ql q2 q3 q4 6 -2 -2 -1 -1 -6 2 2 1 1 Table 5.21 1 n 2(n-1)2(n-2)2 Pllqllrll rl1 rr1 qllrll qllrll qllrll qllrl qllrl qlrl P12 2rl2 r12 r2 q12r12 q2 r12 q2 r2 q12r2 q2 r2 q2r2 P13q130 0 0 q2 0 ql12 ql12 q2 0 ql12 q3O P140 r13 r2 r3 r2 0 r2 r12 0 r3 0 r3 0 r3 n3(n-4) n3-6n2+16n -8 n(n-4) -2n -n2 4 2n -4 -2 PlPP2p13P2 -n2(n-4) n3-4n2+2n 2n -(n2-4n+2) n n -2 -n 2 +8 PllP2 P12P n2 -n n 1 n n -1 n-1 1 pllP2P21P22 0 0 0 0 0 0 0 0 0 PllP21 P12P n2 -n n 1 n-l n-l -1 -(n-l) 1 P1 P2 P3 P4 4 -2 2 1 1 1 -1 -1 1 Table 5.22 1 n (n-1) (n-2)2 p1, q rll ll rl11 r11 r2 rl1 P12 q12 0 0 0 0 P13 0 r12 r12 r2 r11 r2 0 13 r13 r2 r12 r12 r3 n3(n-3) -(n3-2n2-4n+4) -n2(n-3) n2 n(n-4) qllq12O q2 -(n3-2n2-4n.4) n(n2- 3n+l) n(n-3) -n -n(n-3) qllq2 0 q12 -n2(n-3) n(n-3) n(n-3) -n -n(n-4) q2 qllo ql2 n2 -n -n -n(n-1) n q1 ql 0 q3 n(n-4) -n(n-3) -(n-4) n n-4 PqlpPp 0 p PllPl2P2 0 qllql20 q2 n(n2- 3n+l) -(n2- 3n+1) -(n2- 3n+l) -(n-l)(n-3) n2- 3n+l PllPl2P2 0 q2 qll~o q12 -n 1 1 1 -1 P2 PllPl2p qllql2~ q2 -n 1 1 n-l P2 PllP120 -n 1 1 n-1 -1 P2 P1lP2o~ q2 q10 q12 -n n-1 n-1 (n-1)2 -(n-l) PI P2 P3 0 q(1, q p3 n-4 -(n-3) -(n-3) n-3 1

-63 Table 5.23 1(n 2 n2(n-1)2(n-2) pl qll rll Pl P1l P2 P11 P1l Pll Pll P2 P2 P2 p1 P12 0 r12 P12 P2 Pll P12 P21 P12 P2 Pll Pll P3 P2 P13 112 0 P13 P12 P12 P21 P12 P2 P12 P12 P3 Pl P3 P14 q13 P P2 p13 P 13 P22 P22 P3 P3 3 P12 12 4 n3 -n2 -n2 0 -n2 0 2n n 0 0 n -2 qll0 q12q2 -n2 n(n-2) n 0 n 0 -n -(n-l) 0 0 -1 1 1q10 q12 q -n2 n n 0 n 0 -n -1 0 0 -1 1 02 ~ qllq2 q -n2 n n 0 n(n-1) 0 -n -1 0 0 -(n-l) 1 ql 0 2 q 3 2n -n -2 0 -n 0 n 1 0 0 1 -1 rl r 0 0 -n2 n n2 0 n 0 -2 -n 0 0 -n 2 q110 q12q2 r r2 0 L n -(n-l) -n 0 -1 0 1 n-l 0 0 1 -1 1 r2 q 12 rl r2 0 0 n -1 -n 0 -1 0 1 1 0 0 1 -1 02 O q-l_ l, rl r20 012 n -1 -n 0 -(n-l) 0 1 1 0 0 n-l -1 q, ~ q2 q r r20 0 -2 1 2 0 1 0 -1 -1 0 0 -1 1 Table 5.24 1 n2(n-i)2(n-2)

CHAPTER IV PRODUCTS OF GENERALIZED k-STATISTICS With the use of machinery developed so far, two types of formulae for the products of generalized k-statistics are worked outo First, formulae for multiplying k<, where 4} is any set of subscripts, by products of kooo's up to weight 4 are obtainedo Next, these and at times direct combinatorial method are used to write specific formulae for weights 9, 10 and 12 not including unit subscripts o Checks are indicated for both types of formulae, Semi-general Product Formulae A generalization of Fishergs combinatorial technique as developed by Dwyer and Tracy (1962) into a combinatorial method for products of two generalized k-statistics (outlined in Chapter I) is used to obtain semigeneral formulae for products of more than two generalized k-statistics, The rules obtained in Chapter II are very helpful in determining algebraic coefficients of most array types and, by virtue of the rule of proper parts, quite a few array types need not be considered, Also, for a number of patterns, algebraic coefficients as listed in Table 5 (Chapter III) are already at hando Products involving k1 have been further checked by the use of the rule of multiplication by k1 (Wishart, 1952), expressed by Dwyer and Tracy (1962) as (4.1) k ki k + 1 Formulae for multiplication of' k< by products of k, Ios u-p to weight 4 are presented in Table 60 The case of weight 1 simply entails one for -64

-6.5 mula, ioe., (-cl), For weight 2, Table 6ol gives formulae for k 3 k2, 2 kt kll and k) k 2 The first two have already appeared in Dwyer and Tracy (1962) whereas the last one has been used as a checko It is a special case of the formula (4.2) k, k: oi, n where the second summation is over all partitions (rl' ooor) of rs r-I, not involving unit parts and I is an integero Formula (4C2) is a generalization of (4.3) k k ) k) ~ 0, /., n in Dwyer (1962), Also, in Tables 6o2, 6o3 where the multiplier of k has weight 3, 4 respectively, the formulae for double products like k,)k3, k) k21, kl k22 have appeared in Dwyer and Tracy (1962) and are included here for completenessTo read ery formula from Table 6, the coecffici enCI s appearing in the appropriate row are multiplied by the k-teima at the head of the columns and the sum formed, An illustration is presented to clarify some further abbreviation used at the head of final columns, The formula for kok, in Table 6o3 reads 6 12 k 3 + (4) p.,,, + ) (3) k 1) 131 (-ln n (p( n-12) ) n(n1 2=L] ++ n-1 z p 2 (P +- 2_ (P: Pz) kf:.*'nl

4(n+l) 13 (; p 2p ) 6n Checks It may be observed that checks are availableo For example, for weight 3, it is known (Wishart, 1952) that k,.Ik + 31 kl + and also n k k 1 k_ k k 1,,k o Hence, it can be checked that k) k 13 k 1 k k + 3 k andl n k k k k +k k k kk n 3 2 Formulae for polykays of deviates (Dwyer, 1962) are also helpful in checkingo To take the same example, we should have kk k3 + 3n k d2, * n k+n d! a 0O n where d,,,ooo represents the corresponding kooo of deviateso This is so since d, = O and rn

where the summation is over all f-part partitions (r,,ooo r) of r, and pr 1,2, O,ro As the weight of the multiplier of k ) increases, more and more cross-checks become available, in view of the increasing number of lower weight formulaeo Also the zero coefficients for certain array types for particular products can be obviously expected by some rules of Chapter IIo TABLE 6 Semi-general Product Formulae Table 6ol Weight 2 (1..a,( 2) |11 fl2^ tl 1 ill2 1' i kl k 1 -1 1 1 f 2 1 ninf-l) n n-l k k 1 2 1,1 | l 11n )|1n nnnn-) n i k, k 1 1 1 2 n n — 2 nnn n2

Table 6.2 Weight 3 I^W I _ 2 iPi2) iZ(PilPi2Pi3)' 11f2l 2 53 t 11 3 2 filil 1 f21)'3 + 1.1 +I1 +' +1:J k, ki 1 2 -3 1 -3 3 n U" ~ 3 ~n~n-() |1n(n-1) n (n-) (n-2) n-l (n-l)(n-2) k k21 1 - 1 -1 - 1 1 n+l -1 -1 n n(n-l) n n(n-l)( n(n-l) n-1 n(n-1)(n-2) n(nl(n-n-)(n-2) k k 1 3 3 1 -3 -3 2 I 111| n -n n(n )(n-2) n(n-))(n-2) 3 k k 3 1 3 1 3 3 3 3 1 3 1 n n2 n n2n2 n2 n3 n3 n3 k2 k1 1 1 1 -1 1 -1 n-3 1 1 1 2 n n n(n-1) n n2(n-1) n2(n-1) n2 n- n(n- n(n-l) k kk k 1 2 3 2 3n-2 2 1 2 -1 -1 -2 11 1| n n n2 n2(n-1) n2 n2(n-1) n2(n1) nn-) n2(n-l) n2(n-l) I

I 0\ \,0 Il - 7 1 9- - Ir Tu 1 (- t-' (F4X t (. (2 ) ( r)(l) (tt ) - ( - T/S - 9 (i)p ^^) rn)'^1^,1-u C~,- u 0- {;-,.)Q "0-), F(t )(- -")" ~ i-" ((A- (-uA ( (-) - ) W)-I ~t)( _ ( ^U (O1) I(4,U (t) (T- (u-t) - 4 F y - -T ( _ — =- - — F- -— _T ~- + —' i —=- -X-,., t,, - " - <"'-.&u.et~ (")""(',~-") ":(: 1-")kC- =" S — C(;- - r I-"y p-")^ c~~l^")^ l 0- o-")* 4.C-") 14 " ).( r- 2 v-ce >-1- e"?* * -?G r - - T rs (~- )i (I-" -U iU (I-")t~ (f-1)cU (i-_y^U ('-X)t t' -"; (I-'Ct (i-C (A (t)" (A)*:-r - -7- -^ -Y)QU (T^S-~ ~ ()-1-T -?^^'"^ I-u (i-:' ~-:4 ) (: c -: u. (. U (t) FU" (t) — ( (-"U" z —U (i.u)u ~~) (.e E " ^^-'- 4 ^ - Tu - O")' (-),,t)2 o,~.__~._, 14 (f_~w4>( _ _- OSu)T t(1-4)(A (- 1 - TI 2.1 47. r * ('47U U(f —Ct I r-;- T w (F"\> (I")" o u I-u T r Tu u T~ u T I:e'lb "' (7 (C} Cou (1{ T(Ie l'1 } III) (C (IF;, ('FC41 ""y! (rr31 ^w[' r"F[' ^*^<^ I 9?9'9 -u (w" (F-U)" IT T-~~I C.t- 9 q 41 9 If!::.+ (:IC+ (:t=. I-,:-t f:fI+-:+ I{il:+: 1: i(,:1 - tf:11+ 11{:11+ l — - { (TZ [1 l [IIr [:,{f l,[ I It {I I[,, I [' I l11<,<.t'.l;,,].{fl IrT. 4.,1,1, r tr i ( 1111 ( } f7 LN- W4(V\ - C- - 1 0 Y L

-70 Particular Product Formulae The semi-general formulae are next applied to write particular ones by substituting for({ o Products of seminvariant k-statistics of weights 9 and 10 and powers of these for weight 12 (following Fisher, 1928) are presented in Tables 7ol, 7o2 and 7.3 respectivelyo The products computed are listed in the first columns of Tables 7ol and 7~2 and to read a formula for some product desired, the coefficientsappearing in the corresponding row are multiplied by the k-terms at the heads of the columns and the sums formedo In Table 7o3, rows and columns have been interchanged for convenience of presentationo Here each product is allotted a column. and one reads a formula down the column, multiplying the coefficients by the kterms in the first column, and forming the sumo Formulae for weight ~ 6 and for products of single subscript k's for weights 7 and 8 are given by Wishart (1952) while Schaeffer and Dwyer (1963) give formulae for products of seminvariant generalized k-statistics for weights 7 and 8s Checks Again several checks are availableo For example, since. n+l 1 (4.4) k = n k + n kZl2 we should have, multiplying both sides of (4i4) by (n-l) klk2 and transposing, (4~5) (n+l) k4kz2k2 - (n-l)(kkz - kl, Formula (4.5) checks expansions of k3k2k k k. 3 ki2 k2 simultaneouslyo k4k22k2, k42 k 2

-71 There are further checks possible for products of single subscript seminvariant k's as illustrated for k, k4 o In view of (114), /'(3 2^) -^(32242)2X(322)X(2)2X32)x(3)+H(3')X(22)+2x2(32) +(3 ) X(2) +x=(3)X(22 ) + 4 (32)X(3)A(2)+)X( 3) )<(2) 1 4, 2n + +35 X 2n2 + 98n-160 + 0 -7 —r 2- + 7N-57 —-VT — x^^ n ~ + (n-l) n (nl) 7 with the help of formulae given by Fisher (1928)* for cumulants and product cumulantso Hence, taking estimates, k k2:1 2n+35 k 2n+ 98n-16o + ~ o nF lo n) 2 n l 73k as in Table 7,o2 * Some errors have been corrected by Kendall and Stuart (1958)o

(,r —),("-.f) (b- U)(:-+-) (c'- (r-A(- u ( ) L /T 8~-' Cf + 4rIA Q -LU (t-IU)(- U) I- ) bt_ (' I -' A,) (-u,)+"-u) (-+(U ( oZ - UL'+}zr(S - )S ('- -)( I —~ u (g -9)S~_-W1(P) I1-( (g__-)u -FE(Iu) C. a,(r )T^FI (r- (, r- )' (L'z: -"(' O-" (z-u)( -L) ('t r — — ~) (r- (A)(I-) (I-QU) ( — (1-a) 1 4 U1 oo C-A -t't),.'~-t /8L-'LI +? (C-U)(-I') T u ( -4 (r-)O-")'ob +L(1- UcrA (90 + -_5 U) A 7. Or(.- u)u -01(- U))U.Ie-utb_ I-u..-.)L-A OL ( 1-'A (t-'"5 — u Tt (4 (4 ct "7 tuc -F \J pI (t 7 o- U) (A ('u I-" s-'T 6S'+- I -L4 -o " I-LCt 7F 1('+AH -' C- 0 Ul-r- " + (Co-^trf-,A) (oL -+"Lr +A) ((-^4) — )(fob I; -r 7 z TV ~^ *7g} 7t L V I IFfT^ 9 I I~F? IC CTar -C -z - v^ i19 7I L (I-A) ab+t -(b+ (I- U) ((-k) U I-L 8 W.:( I - u U U uc ~ r^ T c s 2 r |T | s7 ^b^, at 6 li-M - I L 31S9vlJ SfLF(LVL-H- ~NVlYVANt(W3g JO L 3 ]V.L sL'tnfojd

TA5LE 7-2 -WEI~HT i 0 48. 473 4$32 43743 44 1 Q6 i1 3 A. cA y1 4352'k.1 2i35kl,ke *333. 43 *4 4,. 4203i323 4S 43t 435 A2 03 431{52 ^-c 24A4 ta;2 ^z2 3 412 z Sl- I Zl n-1,-I 24 n_, _L5,_~L 2(n,+13) 213-I33 (n —0) 3~-) 5_ -l' F-483. n - 9' 3- I,lo0 -!P-L n,-l i2(7-o —9 C3r)rZ n (35-1) 6(13-17) 2 (-0-55n-74; n ("-I) 112,-I 16% r r+.,)-I 200 5(37 - 5 ) 3I3-I)z 3/.,. — 70,,Jos ",(, -7) n(19 -,)3 I/ 3 _, X,._.Y-...- QI6n (. -)(n - (-06(n-2) (n-l )' 1 M(0 -33) (53-4-11') 753-700 I 630,1200'n (n —0(n-2) (n-0%)1 + 57 4-117 r,-2 6 40 -?'(,l'? + ~;6 n —5 4)..- 03(n-.).4,4S t)nq ^ 6c^,1 111 i#4 { ____33 (r-l)(.-2) j, - )(n — 350 n 1500 n ( + —)(-j) (n-l)(n-2) b-^-0~ (n-I) _3o( -+ 12-03 10 (23,'1:9 —305) 3- (857n,-2 8 | );n- ) (n-l,)'-(3 ) (-O —— )q(.7_5~sa/ _2. x tic..4. _. ~ 13, Sa:L~.n _.6LI 44z^. z 433.2 ZLn I 5 I.(3 -+4-1 ) I+1803- j 3 5n. ((-2 | -g+5I3-ISI+I;2. I 1'IA3'~!..-(..O~JZ ~ T(la' + 35f+Z' / n —/ M.4-112/ A1 I ~:- ~~. II 3(-I)~+l 5ILI) ",32(3-) 56+7 W20 n37) r'("'-O' ~3 (3-I)z 0I -4) 3 (30/) 4f3-1 _ ll7n+80 n30(34)5 3(3-I),3,3-I ) 3(3-I) -- A.t n(n-l9 3)0-) n(3-4) 2 n(.-4) -15 n(n-". ) n(-4) ~nt -l) 33.. 4-+7 -s 30(^3) n~n-l ) 3.(03+l3 30.6-4)3 3)(3t I)2 45 _G 30(34)^ 3X (n () -I *(n-.1)3 II (9n-,L+- 1 7) 73* (-1)4 -=( - 13)'nin-3 -I -n (.-JyX,,-:z) 3 ( 3+_n-)( I +(3-I) n-I n (n-,?)n-,) n) (I -)3 (3 —) S(nt7)o(339;13n'-63 4S) 40( r3 +.0o n-7+- 7o),.(3-1)4 3 (n_)0I n-I 230,-I n-2 n 2n -) (n-)(n-.2) M-i l -z _2_ 1 - - -n-(3 3(n-I) -4 nitn-l ) nmn) -I:) 30(34) -12-=i2-1 y33(-4) -4 33(3- 4) 3 33(3,-4)3 3'go-nto)' -33( 1- )......1- -I ---, a.l. 3n-4-4n-(0-3 (' -Qt~n-l) (n-I.(n-2) (3-I)(-, ) n(n-0) f:2-4I(n-70 (n -l)(n~- ~-) 24(n3-7s (nX-)(n-1) _-4 3(.-IX:_) n(n-l) 6e6(n - L, n-1(n- 7. n(n, —) 3)C0)0-0) 4(3+-4223). 4(n3-3) 3(3-I)A -2) 72(n -5) 3'3I nt(n-l) I 4Go3-+o7,,-72.n o-6 I (Cn-1)>1) tGn-0t) 9(51)49 1 -3g_ 30 -_l 3(3,-.4) (30-+(n- Z0) n(n-1)0-2) ^(n-0)(7)-1 -72 (3 -I5n2) -13 3(-(-I) ( -)( n - )_ ) _(.,-,)-2) — 4(~n-) 3(3-I) (n-I)Z-),'(n-l):-iCnt^l3) n (n-fC-z ) 6(,-0X(.-2) ~4(n2+03) 3(1-63-+80 ) n(n-)-C(n-2).(.+53'-033-2a) -3C-30- +0) | r3(3-4)3(3 1) 437 3+t -6 3 g +) -60 |(n-1)ln-2) (+-0(,-22) (.,-6.,-273 (.-1)(n-2) | 7(n-*)I (3.n(-27) (n -i X,-z) 13-IX"-2) 2(53+06 33-4_(l_-2) z(.n +5~ 7nL} I (n-^l )(C - 1 7(n-4)(n-) 3(3i-91-23) -__]{.:~2_ll3,(.- X-l)~-) h(n -1)(n-~_) -IZ(n-5) n(0 33-331) 0n2[n -I31)~ ) (n0)I-2) +46- 3-4)3 n3 + &7nz_ ~_.q~ + 51 D n (,-0)(3-). 2)7n - 195n3+46(3'(3-0)'( -,-2) - ( n3+4'-I6 )n -+2) 3l(3-_I)3(oo-2 4(3n - 5434+'1) n (n-1)3 (t-1 ) n-1)-s 3o^3(+ —-3o -4) (n-l) O~,-l) (r-) H)(, - 1)(_)..._.3~4.201, -470.) 7X3+2l3x+2S5I+(87 (n3-4-)3 0(,-30)(n —+5)(-n+7) n(3_,)3 3 -I 4 I03 0n I(n-4)(n-2) (n-)( —z) 3)3r>- 00) (3-0()-3-2) 124)3-4))3n-0) (.-/() - x){-3) I-4;5(n+0) (n-DICT- 2)(>-3-) IC( -J)(X —) (n-33,-3) __<+<~ n -?1_t=.+-2___.,)(3-2)[X-50) | (n-I)( 3-)(-3) ^(._1)2 (.-I^D.) l(3-.373n - 1)2n'- (o - -2)(n 3) I n (n-l -2( -) O 3 yi53-_3 | ( (+53-4)3lqn 08oo0 (,-+ _) | (-,,)0-,)(.-3) (3-,X3-2X —-)..,, (2n'+- I o 6 —3) I (n-_)-[._)-2 I (-5-1On- ) | n(n+ n,-,-I Ge (n-I) -1 |.7-!(n-0z A-1 3)-3 1 6 1 ( n-l) -^ (n-)X(n.-) (:-0)05_-2 1+ 3+-SO I (-0X3r,-2) |72 2. -_93-0) 003n-03+ >>( 30 I,-(X-X-)(n-3) | 18(lS~n^^l -1hn+3 2;) -(3^-(3^3-0) | (3-003n2)(n-3) 3(3-03'- X,(-3) 14 n-In l;+64 (n-j3.(-2)(n-3) n+ _n5-61t(n 1 n (n-I)(n- z)(,~-s) -,(3.,-3. +") n(n-lJ(n-l)(~- ) 9o()(> IIr+ *79 (-+)(n-+ (n- 3) (n-1)^-0(n - (.+0(3+:(3,n(n-i)'3+2.5n3 +1643-3814 (n4-7 -; l-^0 n- 2t) I') (3-)^(-2) 3). -4)0(3_%Y-3)03-3)+7z 2( 0 4-+18n - 5~9) 3(,,,) - 1 123.3(n+5) |(r>-iX(-Ta-X3)*^-) | 2 4 r3( + 0)( +7) | (n+0(n-+?___] (3-o)0:.-zo(.-0) (.43 +)"(._+. _n(_+'5)(n+7) 63- f3)(rt22+S) 3+7) 1+ 7 n.-I) (3-03(3-0) |%h-n(m + I) 243)3+0) (6n-)Xn-z-X,-~) -7Y-2(-J(n-3-03) _-72 (n-x)(n — ) 13-23(-0.- (. - z)(n-3) 1 (n-l)(n-)(n-3) (3-2)(3-43)(3-4) - |C)~+__ (..3 (,n 7) 4,,).,~ 6+_7. 1),(r.-4^nt1) _(C-l)(n-a(-3) (^t73')(3-s3-2In+)(-. - 4)3 I C(7-l)'-Cn- [)(7-3) |(n tf3)(~+5)(n t7) (n -3 ) 5(3+Z-0o) (I-(X"-2) 3"( -0 I3+n20t-50 n(3-43-1) 7 3+43 | (3.() I I<

TABLE 7-. - WE)CHT (2 _[3,4 A' _____ ___ ^ ___________k ^ ___ ____________x________________^ z_____ &12,lo,2. 10,2. 752 7az'633 k5, A545k622.2. {44+1 44332 43333 4 31212 3322.z?ZIZ^^Z n-l 180 n-I 450n (r4)(n-3 360a0_ 2.16-0-z) (^-l"n-3 ) 45o0 ^ (n- t)n-2z) 2i600 n (r-IXn-G-).06 2(.n+l') -! O0t n(y>n-l) n-tlX,-2z)("-~) 15'30 n (n +__ (-l )(n-.2X)(n-3) )QOO n (n+) (n-1X)n-a)(n-3) (n-O(,n-2Xn-3Xn-4 2. 1.6(! O-na,+- 5)n-) (n-( n-1X(n^-;L)n-X -n-) n(n -0) - "('- 17) 12. ( 7n -70 ) -C,,-I -7)Y (n-0 0 +-2) ) ("4)~(n(-a) 1j40 ( +n —I ) (,~?- (,n-;.) 44~(3+ 17/.n_-4. n+16o ) 54 n n-I) n tS n-gO) A7(17n -49qn 35) n^7-41, 1o C7-;r2- n +/6) 3(3+15-0 2- 42o0 +350) x7C37,,-70) +101' +53_ 9,,-6 7) r(.n-,l- (n-2 ) ~r~~-+ mr,- 732,+'7 ) n"5 6(n-1)' (,,-t15- r+ In- 77. + 63) ( n- ) 5(rt7) (,+: ) <3 (n-1)a CoG(:-9)6)(3-H7nZ- 15n3-35) 80(r-2)(n2t 6o,-n-105) r)3 (n-,) 6(yn-2)(ny>X6 y^-l-nt 7) - 4 (n+ 564 +109:3-7774' ^+1-360,ot20o6) _.6.Y -n (-It )'(n -2)L10ot5 2'-.L2Q0,+t24) n (, -1) 3(n-L) a2j6 (2,4'+,_3- 676, n+.(,8 n -410o) - ~(,,o 8r, _ ) (n-.)^(n-:)" -4. +%n+ 5175 ~ — e-47+- n 52)4 C -13,,3140 ^.+200 +4) _-2^+^h200w^1 - I t77 +10 n 0+ 9 Ti _v \ -.,. _.,. _ I7(J17_3,- 15033n a, 42.^-73vn,, — 42.0 0) = (n,+ r - 7n-l36(3+-42oj9n-784 +700 4(2n4 4-10S -57-OOn +5- 0) 243 (n34-I2h)- 57, +7Q0) (,n-) (-.a2)' 5^'6+ I_-n — +1300nY3- 10 60 n'4- L.7472-n-2a2o00O. + A6, h 49 2,(n3). 73( -, 332 8'+ II - l~+ 63 5 3- 7^ _6..^ 2 03.3.28n-17:200.f _ 93. & &L. _..._ 4goGr —), 4+7)(,+*) ~,-(.-r)'1 (n+7)(-n+9 )c/ 3,3 4 3 n- 63: + — ) n- (.n_l) at (,,+q)(32-On -7 7+70)) (n -4)s " ('n-"-0 | 0.r-2;7- 27r)-70.) j(0 _- i) (n+5) (r+7) (n + ) "' (n-1U5 (,n l )(n +3)(n + 5)G(+7) (n,+ ~) LI ( —. (i -r.L?(n-_ --- 7.v,.( +.l. n -.717> -2.6) (n-l)^(n-2)Cr_ - 3) 1728 n(29n3- 196n + 317n +6L ) 17,3 n (n+ )(nl-5n + 2) - ( -q.%,-zn- 3)' _ -.... (n-1) (n -Z).-24.. (, 4-,,z- O1r..,+~) (, -1) (&n-z)3 log n,(n,-+27,- 70) (_ )3 (,-2)

CHAPTER V MOMENET FORMULAE AND THEIR ESTIMATORS The advantage of formulae expressing products of generalized kstatistics as linear functions of the same shows up in problems involving expectations and estimation. We use M(rsoo) E EN(kr-Kr)(ks-Ks)oo to denote the moment M(krkso o o) and K(rsoe ) for the corresponding Kparameter* of a finite population of size No We first study these finite moment formulae where none of r s, o c is lo Then we find expressions for finite moment formulae involving ls (ioe. the sample mean kl)o We also give expressions for the estimators A A M, K of these moment functions~ The fact that these forrmulae approach -the infinite formulae of Fisher (1928) as N->o is used in checking; One device used is that of substitution products introduced by Schaeffer and Dwyer (1963)o lKrKs ooJn denotes the expansion of KrKsooo with N replaced by n and similarly [krksoo O-N denotes the expansion of krksooo with n replaced by N, the expansions being feasible by Table 7o Obviously, [KrKso o oN KrKs~ ~ and LkrksooJn kk...n estimation, complex substitution products like LkI] nkgN appear Finite Moment Formulae Although we need the linear expansions of products of k,.:s for writing moment formulae, the best computational form need not necessarily be linear in Ko.o Schaeffer and Dwyer (1963) in fact point out that "the completed expansion in terms of Ko,, does not yield a formula which is most desirable for computation since it demands the use of the expand ded form for some products when the actual values of the factors of the * K(o oO) ) M( o - ) up to triple productsc For more factors, K(.. O) are given in terms of M( O o ) on page 77for selected values through weight 12o -75

products are knownc" For a simple illustration, we consider the variance M(22) of k2. M(22) EN(k2- K2)2 2 2 EN(k2) K2 EN (I k4 + 22 ( K4 N- 1 K22 N) K4 + (n-1 N 1 ) K22' using EN(kooo) Ko b 2 2 (5.1) 1 K4 + ( - ) K22 n N n-1 N- ~ The expression (5ol) was given by Tukey (1950) and is in good form for estimation, being linear in Koo., but the variance of k2 can be better computed as (5.2) M(22) [K2]n K4+ 1 K22 - 2 since K2 is better calculated as K20 K2 than as 1 K4+ -l K22 2 2 2 N N-1 K ~ We give similar formulae for M(rso0 ) for weights through 10 and for special cases of weight 12 in Table 8, generalizing Fisher s (1928) table of formulae for the infinite caseo The second column headed "Est" for Estimator and the row starting with 0 can be ignored for the presento Use is made of these while dealing with estimators latero* The terms are grouped by the number of factors in the K-productso These formulae can be written directly, like (502)(Table 81),2 when the expansions for * To obtain the expansion for a particular M(o o ), we multiply each coefficient in that column by the corresponding entry in the "Expo" (expectation) column and formthe sumo

-77 products of k-statistics (Table 7) are known. For a strict generalization of Fisher's (1928) table, we should write the finite K(...) formulae rather than the M(o..) formulae. These are, however, equivalent up to triple products. For products having more than three factors, we use K(24) M(24) - 3M2(22), K(323) = M(323) - 3 M(22)M(32), K(3222) M(3222) M(32)M(2 2M2(32), K(423) M(423)- 3M(22)M(42), K(34) = M(34) 3M 2( 32) K(25) M(25) - lOM(23)M(22), and K(26 M(2) 15M(24)M(2) - 10M2(23) + 30M3(22). Columns at the end of Tables 8.5 to 8.8 give expansions for K(.1.). It is useful to note some general results for the finite moment formulae. Schaeffer and Dwyer (1963) give the variance-covariance formulae as 2 _ Jn 2 (5~3) M(r ) = nr- ]nK (5.4) M(rs) KrKs]n - KrKs and also give expressions for M(r3), M(r4), M(r s), M(r s2) in terms of substitution products. In direct generalization of (5~3), we can write M(ra) =EN(kr Kr)a This result can be used to write M( M(23) M, M), M M(43) and M(62)o

-78 TABLE 8 Finite Moment Formulae TABLE 8.1 --- WEIGHT 4 TABLE 8.3 --- WEIGHT 6 Exp. Est. M(42)1 M(33) M(23) K6 0 1 1 1 n n n2 K42 0 n17 9 3 (n4-3) n-1 n-1 n(n-1) K33 6 n+8 4n-2) n-1 n-1 n(n-1) K222 0 6n (n-l)(n+) (n-l)(n-2) (n-l)i 0 1 k4k2 k ~ K4K2 [k4k2] -1 -3 n K32 [k32] -1 K2K2 [ik22k2 -3(n-l) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _n - i K3k23 2 TABLE 8.2 --- WEIGHT 5 Exp. Est. M(32) K5 1 n K 0 n+5 32 n-1 0 1 k.k K3K2 [k3k2] -1 TABLE 8.4 --- WEIGHT 7 Exp. Est. M(52) M(43) M(322) K7 ~ 1 1 1 n n n K2 n+9 12 2(n+7) K52n-1 n-1 nn(n-l) 2 K43 0 20 n+29 n +22n-35 K43 n-1 n-1 n(n-l)2 K 0 36n (n+5) (n7) 322 (n-i)(n-2) (n-i). 0 1 kk k4k, k- k K5K2 [k5k2] -1 2 n K4K3 1k4k3] -1 1 K32K2 [k32k2 -2(n+5) n-1 K22K3 [k22k3- n]l..32.]. n-1 3K2 3k2 2

TABLE 85-WEIGHT 8 ~Ex. | Est. M(6z) M(53) M(4(33z) MCz | K(z) 1K 0 __L I I I I I'-i'^ - "'iii-I K62 0 2L 15 +_ _ _ (n+) (-f20 +4-(n+5) s c _,n-_l \, -1,-. ( ) n(n n -: ) 2 n- ) K53 0 30 YT n-0A 8|C| -7) d(2 2n 3.2,-3) 32(n-): 3,-_(n-_ ) y)-1 -- n-1n(n'-._ - (n) —-) ( -1)3 - (n_) 2 K... 0 20 30 f4~33 ao 26-3 9'I0-51 3223)-03^45 3 3!~ -_5r' ~~y"K 3K=) -n4-O- -i4 K4z 0r. _- 2,+~ 71 ) _...q nn:)( + n 3X(n2t) &b)3)()+s+) 0 1 Xn4)(nl)(n-)) ( (n-O —- - | (n- ) |-(,n-l)- K332.n f - t- C 1 - i-l4y -4I-0 | Y 2-Xn-t5(4 1}-k^ _.. (r,-|Xn- fc (~J-l; 1(? n n -2) l^-1 | -r( zKa O [<;5 | 22i _ 4lr+7J (- 5 _ _ ) _ (+0Z3(n+3) 0 -45 A 3 4<3 ______ ^ K K; I |_ I A- I I K]5 K I A3 - i I I- n s"nnt1Ze K Z/^K^ ) Li) 9 1 _ 4(n+1)2(nt3)32 _ 2 )2(y)~+3) 2 K4 K < 7 9 |(n+1 K33 KZ IJ2 12 -2)_ 0 ^K31 K [K3 33^]- 2 — ~ n( —J )n-3! *h-I __K2.2.___ K [cia < ) ___ _____ -___ K K2 jkRn] 2. K1.2 KL 13 2. K K^^ ^ ^ji ~lJ.......... I —, \O I

-80 TABLE',6 - WEIJHT 9 Eit. 1 M(72) M(63) M(s+) M(52M) M (45z) M) (39) K(332) KI 0^5- _4r r ^Ir n(-I^ n3 -) 7n -) I 0 3Ln >Sn62 0 7) 20(3n -4) 1 +-70^- 5- 3(n+2Sn-7 -n 15 60nW I7 - 1560n- 7) K| 0c0,n-l n- r 3n-l ) n4(-2)3 n( —4 n4(n-l-) (j-L GSS O 7-2_ loS ~ L 9 — )+.11 Sn+qF -10 S7(4,,L. 3(n3- + 3 _ 3)! 5n-l r-l n-l [n(nJ)*(n-l)k 4n-.) n- (n-l)' - 3. Ks= - 0_ San 1 0^n +C l 2) IZ_-XnF 12 n-g5) 54{4n-7) 3(n -7)(? 3(n+-7_+- 3 K L -I 2366 600n _ (l n +40 n-+O^ n-5 7(^.n 2(7n+9 61.Q,7n)e - G17n+ l57) |(J'(.^:) (n-')Cn —) (n-)^ 7 )n( — I)- n (n-I) ~(>q)-) T J(n- ~) ([n-l) (.n-) (Ln- l) n(- )n-3)2.(n —43 n(n-.) 7)-I K, 0 (n1.-0.) | -L 36.n- ) n(-t70) n_ 1-t 55n Et&? -I I n*t —5n+3Oq' K30 K,6 2^3e; 12~~ [ ^Ji;2 ~ ~ ~((n-n) ) (n-4)1 )- ( -t) | 3233 | LaiJ - 36) 3 3| _ 6y(nIF.-4 27) -._0 70(n.7) 0L r.1| o-, 4.(n-ln() -I (n-li). — K, [,. _ K_ ___ _ K,2., K] 1'- 1 3 (n-) n(n+^ KK3 kA43_ n n|1 31 4 [q12R3t] g n+S _ 3(w+57 I I re 1- I I I - (-'n) nr 1 1 1| r — m-l n - (.- - (.-n') L^31 3 ^ [^.3K Il z1! I _ 36n 3(n.+95)(n+7_)_ -3 7(n- +7) _12|_..._________ (_-)-_)_)- (n )-l n-) K| 3,, | (n-K3 I _'n-] K, 1 ^ [3 ] 2, - ^^ ("[m^J1I ) (, -I) K'32. K* tKK [41s n-l i 3(r)l K^ K3 2f K.(^1)t C23~l2. 14311 2

TAbLE 8'7 -WEIHT 10 E,4. Est. M(gz) M(73) 1(64) (l A(s5) M (i) M(53) M(4) M(M ) M (4m) M(2) K(,') 3;(4^) K q ) K)o 0 __ _ _ _ _ I _ _ -,,3,I4 2-iL O Zl~r ~ k 25:2_c~+l'r 1~ — __- _ 5i +7_ K73 O 5,, %6 JO ia,-9),94*^-1oi2 0(330'~2(ns,-4+4t)n- 3 09423-%O) J-373) 8Q0(n-J) 2('-i-49"-80)'4('(",-7) _80_rl 7-K — LO *Jn- lo - 5.- If 4 (n 1l 24,4- C-K16^ -l("".(-'2)' - (-5'), - s.7., ) - (( ) 1 Co-if (T-|' 5.n.- y)3( 3'( n(n-l) 3 (n-l):" a+-< l yK 4, 0 A26n. I SO.20 I OSO^ 4c +-I5 35>*^'<,,'-i-2' i qt12'-. ) 7 -3, -i-3 ) 2J ( o 4-.+r37) t — 1 7.3:(3O' -' 15C-"111- - IZ(J'- I7l) J(3."-'2z3'-1 — 3) 8tdo+ soKC^i(., 3 33 -.'o.G 3(3I+K'a- 4s L(IC)L. os ^-I-o! 3('(0.,fs J )-4i- 3 0.oIo )+,4700, - -I74 4 7- I 57) <^5)(n-2 (.rlW)-2$ (,,_ ~(_3 ~(.-&^A ""'^-0~-"35),7 26. 7);(-,) 24ooL,,+3X 2+7)L- C~Cn~L nr+~54~CI.231o' J,3~52,, 2L ~)t~5o-7~~+1(~.1)1~-3.1-n7 pl (,i-L)(.o7) J2.-i-3X,'3)14-I_) 4+7) K442., i4 AS lygr &01OM5 &a 43 A( -X-.) - ( -,?3-O) -if. ( — -.,l-_)' " 2 - --.433 1500 I, I - - s(,,++o) - +&6 1 t) - 014+5) Sf. +s ) - a( +9 - 0( 0- %+ 7) k^ [0, Aso,,-~ (,,+I) 400,1 + _Z,5.- I _ 4311 3C,3tb- ELL fig 10(i) -Rtn,-+~L+?nt C,7 o )(-)-)()n+5)C,+7) XK533 1I -nto -(i+ )-D 4 & —,, t) 3-7+,- i3l(zI) 0 - Ar, [0 7) _ 414+ o ) 14441, [&r&A j.40 00 01+33 40(2"-2) -3(n'+z(- 2 (n' -i - (2-0 - T K- i) K<^u K. ~ 2(,,+7)-,.- JL.. - af(,-.-o -o-,)-.)+ ~,~.KM^~~~~~~~~~~~~~ l"~~~y2*-ij- J^.- J^.i-fcJ^-_ ^(n-^-7)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 2 XK,, -K -a _,~__.2. -&o,+ 00(03.30 23o[i),I(o4' Ks-j K., fL4,(n-r(7)r 7nM-ni(n-.- )(-) - ^K iK4-yl [A~&.- ('-.9 - i)(n-' K K,, [ & 0.1 1. 3C) A(({o, _*-._ 3)_ 1( 41 tl- l r ^ ^ i-ffn-H) - 3(r.-t -)I )7' ( I)'(n —I)3 K+, K3 [zSi^^J- "y9- 3l^'n.-^^-H~r^+^^-5 3J>-3Cf~+'?') -8^-2)(+, - S) 5)-'Kn(,-HOC —'tO —,+^ -,0 (n-X,F~_ K3. fI, 1)(^ (k( - t 4("-p K,,. K, [A0 A,],.-,- -i7. ( IIJl l1 ['*m i.'1!]-S.'i-^ -W -— <3.n+nj-S)-5^n-*-lJ3n->-3)Cn-29 3 - n-'1~.^ )+0L () -1-5)( + K,.az Kx [L,,, f, _-. -"')" - - +-. + K Z K4 1,33d+17 - I4104 -3 I, -, x _Q)~Tn~' (n-I~L,.,-4?- (,.-4),'-9",-2' I 0o t-' O

TABLE 8-7 (CNCL]D.) ~E e 4. Est. M(8(2) M(73) ( 54) ) M(3 M() F1(4) M32) M(35)) Mt_3'2) _(__) ((25).K4^ ^]. 2] 2 _..3 I~^ 4 L~[-] 9~ 3+7) 30orf3) 13 i sOL n+_ | t 34 |[it3@t, | l l ll l — l ||| l n (" K33K [^^Jr.^^ 8-&-50 K,K- K, 3~-l [ - 0:-5), 3. ^,Cn-l- Cn.-I) n-)-I - (n-) K K, K K ], K ] Kzt~~~~~K*[I^J________ ________ __________ ___________ __________ _ _ -_._.-____+,o,_3__t 3(rr 2o,,4+ k3 ^ -3 - an l K;1~[ J -- ) 2. ^^ K5 Z III III rD I

TABLE 8. -- WEI4HT 12 4. EAt. M(61) M( ____ M (.34 m (__) K(^)K(2) K 0 -- 4 -8 K,, 4-5' 1,? 2 2 I74 -3_ _ _-4_(_-i_)_ -- KC4 0(W+.50. 0 1J2(2,70) 3(,3+ 50,'-40 —350) 2~5+^ol,-Qn0,(. n1- 1575.-525) 3 +1510-4'L,0 350) (5Ha,,,'- oo,?-gs s K22. 0 4. 2 7(07~,,"' 27(37-.s_ 701) i 5'(- 7)( n-+q) K^ 0,^ 5 n^ )- --,,*( —0' K,7s 0?S0 sO0 A 1o4 0'-77.. ( (,11.Z0,+g6 0C,-sX7-/ } _ losC7.-. -0,~ I Q qd(.-.zX7.___ (n —"-4' "-' " n-) 7~o0l ~ ~.,i(on-22I' %(,h-9' -*() —' C,'(.-s) ___________I(_7_-_7o _ 54('nh +50. Iq, o,'7 +-3 ) S o(,+9+1 ol,,%cf,?+I17.5) +. (5,-5 -Z 61-/32+,07?5- 5- _ K52 0 450n l72. a7 (11) A7/7- 0 7(n' 7 0)C+ ]. 7 ^5-3JL2 0 a^OQ^^+Qir~~~~~~~~~7- I+ aC0n' -O S(r-X+) C -sX~-A &"O'L"-3) (.-I A<"-0& x)''j —iY' C' ("-3Oe K43.:~32. 0 *', 000 I ('-( (:_"~'+~'t;'z+1~qq1-77?+is90~"-4 ~' $0oC5*()(4+, 0n-4053 o —,- -00) (o+ 0 (C I, p0,? q ls +7b) 7747/ -,,.22*0 974 2- 0 o1L7,~+70) K3 0 S)0(0 1,~-38 4.o 12-8z, ).3 4-,(-q"- C (-"73 I &,'t 1-i Co -. 2)-0 - -, C,,.'-7)",4o'J5Z,+.,,3 (-73'T -Z,,".,3,-J15" I -0)&?4 -70 5405,"6x+5)(, L2,+3I),-24 (.-,- ) a ^ Z 0 SSS^&^S19A 5i7,a 1,i- +2 C- g -yz X4q-4n-1,-, K,11C1 ^ ^___ ^_____ ^ ___^ - _ _ -- -K"-0'("'~)) -'~("-~~("-z~ C".- )$ _ —?_ kg5 K.,- 4ooC~-s) 4886. -" K72 K3 [-, ",- C' - log'"^ C^J- CC-n-g7)-17.-z)35-,K*C2 r —- -" - - 6,43) kJ3 K r N,. ^Z^ L^2~~~~~lt~?n+' 1 7.,(q,-F/i," 517,,+(-),,+,5+~f~,~Bf —/f)SCz(+),)~~).%j I~5+f.q-,'8'g —Iu ^ i/ ri 4, ")'n~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~cw-_o(_pG-)n~X~ \K: L^zzJz-:^ ~ ^ ^ [^ 5^ 3]~ ~. 1 ~ +' ~,~-~-.-.3/? Z -(-' X ".T+.), 05J'n+.7- 6.^6H 3 ( +5 ~ 7 6 ~lq o r('n x+.7.^ -S ).+ X ~3 ( + g.~ 7 (~ ~ ____ _____________________ [,-X,-X, ~X,~X.~J C'O"C-)~5~ - - (-)'(-p-.) c._~p.___._s__(._0 I 03! ODJ I-l

TABLE g.? (CONT:) t | ~ | (Et. I ) 1 M(4m) 1_I MC2') K(-) I KC) r I _ -- r I I i K6 K3 K55 K, Ks, K, K53 K+. K44 K4 K,, K Ks-, K, K,'3 K,. K,'2 K3 K442 K,. K,,4 K.1 K4. K422 K+ K453 K1 K,42 K5s K,,* K4 K,3, K3 K,, K42,,,. K, K4,, K. Kz K.,., K.2.,2- K,, k, K.,, KL. K1, K2, J 33I(2.,.2 /caa....2 Kj^ [4,, *. 55 k,J [C] [kn L ~] [4t, 2,] [b kJ [h^3^j [4324Ft [AL ^j3 ItMLZ2 [^U,,t Lil l] I -14L 3n-3 3(n +332 n-l _ o8(4n-7) (n-),(n-2) _ loS(n,+'-7n-70) (o s t-J- (n-OL). _)'.n- (-.l) _ 60(n+5)(n+72 _9 0 (n-2.X+7) n'(n- )3 - &^^-7 _ 3 anh +7)(3nS+ 3a/-n 6 -5) _.tao(n+t. o ns-7, 70, ) nZ~n-J)4 - Oty+3)C(n + 5)(n+7) n (n-1)3 _ A+O(,-ln)(n+7) n(n-O)$ _( (ni)(n+3)(ne t7) (n- 0* 108((4n-7) 36 _ I08 (? -7) Cn-) ( — ) 5 —(n-+8) _ lo8( n- + - 7, -70) a (n- ) _ 324 (n-m)'(.-r) 36n(n+8) 72n (n^+X7,-J7) (n-r)L(n-;.)L -M2n-2) 9_(n-2)(6nl-12n+7) _ 480(n-z) - 15-(3o 23L-33+45) _ 60(n+-)(-+5) oc(-, —) - - - (n- ("t7) n (n-1)3 _ 40(.n-2n{l) n_(n-l) _ _30oCnt7-t-)(3n + 637.t+5) - (n-l) _ or ) (,, n + + 7) ) nL(n-/)J - 90(n, 3)(n-5) -t (_On — - 7 —+70 _ a3o(0-7X+^7) r^(n -1)3 _ a4 0Cn -z.))(n+5) ( n -1)4 _ 160o (n-++) _ 60(ntl)+OX) 40(n -3:(>5-0(n(+7) _ a*o(n-r)(n+lXn+5) n(n-l) _ 80OC- )()h(H j _ ( n+l) (rne3) (n+5)(n-t7) (r-l)+ I (0 4-I 72n. (-n+ ) (~-1) (.-z)(.- 3) KK, [K. ^Z j,, K6 K q [A,-&4 ] C J 5) 4' [1 2 30 ~t,,-I) —., ( ) I ---

TABLE 8a (CONCLP.) rK,,]( -lik e) [ S4 kll, K, ia,5XA. F{ (n+loRt3) K, KK. ^ KZ [ffEst. t i 1 M(5(3) M('>0-K_ (3(43)3~ _ _ L, - L/- [i i 8 K1 U KL [4o | g _ 1(irr~nX 2, I K K. [_ _,,_ __on+ __,a~n —-- K [.3 K ^ g- ", O-Dr+,44k; ~l,, ] ____ _____ i- -[tn L-a~ --- -JA -3120.j -Ic~~~~~~~~~~~~~~~~~ ~~~~~~~~-, (w — 4 ( 1~1-5(, — 2 ^KJ^^ o,-o,-) (("-:,,-') e 3 - [- i-j- -- ^K [; - -KJ K(. -12 Q0nl( 12 K,, ("- K(n-0 __ ___ L _ __] - -' -_o I L I I0 I

-36 Also, since (5.6) kpkq - 2?(n) 2 (p,...r )( )...p%) k p Cl(p,... r ) q,.. c,..P~clop where we assume p>q and f(n) is Fisher's (1928) function () )i n(i) associated with a two-column pattern having Irows ( (oP) being the leading ith advancing difference of the series Of, i, 2, ) and the second surmation is over all pairs of r-part partitions of p and q, we have (5.7) M(~pq) = Z (n) r — (P, pf ) (q, -'e o ) Kp p+ %qr' %Kq b A special case is (5.8) M(p2) = p2 + Kp 2 + n- 2 (PP2) Kp+ 1, p+ - Kp2 In generalization of (5o7), Lk M(p... ) = EN IT ( ^) = EN 1 (-1)) T IC? - -z (5.9) = 2 (-)i I7T Ka. ICr tC-1 I where I is a subset of P - l p,... We can also write A t k[ Kp. I and I the cardinality of I Pu and I is the cardinality of I.o tU an- (5.10) M(P1i. Pu) = (-1)h Kp. Kp LKp *)Kp (.o. )KpK(.. =Kp} 1' Al AA Y where the sum is over all subsets I i (, =,' Df P = I -,^jand Pi i,.... ^ 7' for the product in the square brackets.

-87 As an example, (5.11) M(p PP,) = [Kp Kp Kp 2?3JI - K [K K p p, P3 Pn - K LK K 3 2, P, P3 n Kp LKPK + 2K K K p PL 3 At times, however, semi-general formulae of Table 6 can be used to write a more general moment formula. For example, from (5.11), M(p22) a [LKpy - K K2 - 2K2 [ KpK2 + 2KpK2 = K22 + K +2,2 + n p4 nT1) p+l,3 + (P2) n^^D ^nn-) p+3 n- 2 1 12 (p+2), (p~2) K n- Kp22 +n Kp+2,2 + n2 K+4 tnT-i)2 (p2 ) (p K(,+,(2)+ +- 1 n n(n-1 1, + 2 2 ( 2 ((5.12P2) 2 Kp ~n-+ p 2 + n (n-7i) (P KPd 3 P-+ 4 4 n(n-1)2 p2+ 2(n-)4 (n-_)2 ~ (PlP2) ((+~-),, (P2+l)K) (P+ +), ((Pl))l 1 n+l _ 2 KpK4'n- K22- 2Kp2 Kp 2 - 7 —(PlP2) p+ l,p +1 K2 (5.12) 2 +2KpK2 where ((p+ l), (p1+ 1) denotes the partition coefficient of (p+l), etc. This formula can be used to write M(322),...,M(62 ).

-88 Formulae involving the Mean Some general results are now presented for determining expressions for product moments involving the sample mean klo For Fisher's (1928) infinite case, a very simple rule was found, which gave for example X(4IA)u o.The relationship is not so simple for the finite caseo 7) Formulae of the type M(plr), M(pqlr), M(p2 1 ) are sufficient for writing all the finite formulae through weight 8. These are now obtained. Dwyer (1962) has shown, using deviates from the mean, that M(lr) EN(kl), assuming K1 L 0 = EN L. _r k q,, where the summation is over all p-part partitions rl,..r, r (j= 1, o...r) of r (5o13) - (r, v'., rt )t,..." t_-, Kq,.).,<, where the summation is over all non-unitary partitions rl,.oo,r. of r (i.e. partitions not involving unit parts) and = (-1) ( r ) ( r ) ( ( ) ) j ( (3 n N N Barton and David (1961) have also shown that (5.14) (M(lr) z r2 —, Al'...A. K., (P, )),(PA ) where the summation is over all partitions P,. 7 of r excluding those with unit parts and ~Ar F ^)-1 — (72 1 1 r >. 2 ( — r -rn n n * Their result erroneously contains a (7T-1)l in the numerator.

as defined by Abdel-Aty (1954)o It may be noted that Ar - d r-1 in Dwyer's (1962) notation. Wishart's (1952) formulae for M(pl ), M(pl ), M(p14) have also been generalized by Dwyer (1962) for M(plr), He shows, using (4,3), that M(plr) = EN (kpk) - K EN (k) I'C',I -i(;)*1Z(~'~ ~~~r' )r, Z (l) >P 1 (r,', I.., ra )r'-i ~' 1 Kp+I,r,'. or, (5.15) r l Kp Kr o or - Kpr,.r, where the second summation in the first term is over all non-unitary partitions r',OO.,ort of r' - r-I and the summation in the second term is over all non-unitary partitions r,.o.,rr of r. Although (5o15) appears formidable, it is easy to apply for small ro For example, M(pl ) 4Kp4 + 6^3 Kp,2 2 + 42,2 K+1,3 3 K K4 - 4] - 3~< [ K22 K 22 which agrees with Wishart (1952, p.9). Also, with the help of (5.6), we find M(pqlr) = - Kq nI). (ra' o*..r ))*o k. +K pKp (r,)...*r)i,..^'' K<,' (5.16) where the summae over the non-unitary partitions of r', the third summation in the first term is over pairs of part partitions of third summation in the first tere is over pairs ofyipart partitions of

p and q and p,p + q. o + I means that the Pi are added to the qj in all possible ways and the units of I are then added in all possible ways. As a special case of (5o16), we obtain (5.17) M(pql) = Ch (n)(p,..-P )(q, q).)..pq q, ql - KK "p1l where the first summation is over pairs of g-part partitions of p and q. For q = 2, M(p2l) ot, o)Kp~a p+1(l)K +c(2)Z(p1p9)K p 3 K -K K M(p21) - r(O )Kp2 t ll p)K+3 +2)2PlP2+1)Kp P 3 2 P+lJ (5-18) x Kpi,2+ Kp -+ 1K +2 2(p)p ^ ^ -K K K K (5.18) [ pf+l,12 3 n P+K3 nl 1 (y2) KPp2+21 KpK3. K 1K J We note that (5.18) checks with Wishart's (1952) result when the latter is appropriately expanded. Also, by putting r ~ 2 in (5o16), M(pql2) = 1Z 1 f (n)(p,..p) (qg. qr) Kp^ p oqj + tdZ2 ~ n)(p. ")(q~ ~q ) (5.19) K O Pp+ q 0 qf'2 "Kp I Kq 2 +. Kq2 KK; i02 KPp2 +a(Kp42j4KpK K2 where the first summations in the first two terms are over pairs of P-part partitions of p and q. For q a 2 in (5.19), M(p212) IKp4- KpK4+KP22 - 2 K2 + 2Kp 1,3 + n P4 ~` n-l2 t(PP2) Kpb4 -2 Z(plp2 ) K231 t n 2 (PlP2) Kp~ 2,pj+2] (5.20) 4-lK^ 22 - KpK22 - Kp2K2 2 KpK + 1 p+2,2+ PlP2, * Includes K since i{+z = o+,]+, = Z}+{ I I K~.,,...~+

It is useful to have on record as a special case of (5o16), in order to obtain moment formulae through weight' 8, M(p21r) - r ) ~, o,_. +I,'..K I + n,' (t'' 1,I I, - }' * ) " z' -1- Kz I''f-Io,,- - K i (I)J Z ( r/)o -l.'. / 1 T< (5.21) + Kp K S -.,'' f v. Thus, M(221r)' Z, (i) r,') r)' ~f l 22+Ior'o wr K4,. _ ""h 1|,' T o oo r... K..,T~, + K':''(:' - Z&,~,"y>' (5.22) +~ K_ z, This gives M(2212),. |2K42 + 2K33 K + n K42 + K3 K222 + n K4 33K n 4+l K42} 3 (5.23) ^n K222 - 2K2 ^ K4 + o K223 + n o K2 n-l which checks with Wishart's (1952, po9) resulto Most moment formulae through weight 8 can be written with the help of general results obtained aboveo For example, for weight 7, (5~22) gives M(2213) = ) 2K52 + 6K43 + K + -- K 1 K3 + 3, 2K22+ K 2 2 52* 43~ n n-1 52 n-i 43 + 322 n 52 n-l K322J'+{K322 n K43 + n- K322 -2K2 t < Ks + 3(K 3 ) +aK32j +K o< K2 + 22K3K (5.24) + 3 K3 2K3K- 2

Another formula needed for writing all M( o ) through weight 8 is M(p221') - I (I)/,, -, t',, t 2,i' - l zQ>Z''KK,. ~-' —*y. - 1 n^)+, - 2' (5.25) This gives M(p221) = + n-l2 < ) f1+ a / L K+,1,2,2+2 32 + 1 1,4 KP +3, 2 + il Kp+2,3+ 1 Kp+ 1 4 + 2 - 2K n(n- Kp2,3 nn-) Kp+ l, l4 -+ K+l,p+2,2 + Kp+ljp+l,3jn i ) I L.I. n -2 2 (PlP2) K+2,p+12 + 1,2,2 2 t- 1 p32 n p 3, 2 -n 4 1n r 2 - 12 K n2 n p P+5 + nnl) 2 ((P+2), (p+2)) K(p+2) +2,(p+2) +1 l (p+(2)+1. (p +2)+2 + n-l- 2 (PlP2) Kp pt2, 1, 2 4- K+,1,p+2,2+ Kp~+ l, 3 2 + nn(n'l) Z(P1P2) { -l pi3F2 2 Kl:: +4) l "+ K^,p^e.)+2j + n(~l) 2(P1P2) Kp2,p+3 t- K+l p 4. n(n-1)22 (PlP2) K R+3 P+24,P3j + (n4 12 2 (P +1, 2l )P 2 + +K(pll)+1,(pl)+12,;l+K Ptl) l,(p+l lp2+2 32 2(l2)1 K~rtl)il,(p,+l)+2)p K +ltl~l,!e1K p lp p + n S~ PP

-93 p+l), (P41)) (p l,( l - P+) P1tl,( 1+ K+ 9~ (+1 )+ 2, (pC+l)+14 I L+1, (pel)+l, (p+1)-+2} - K K n p5 n-1 K 2 2 n-1 p 32 p2K211,2 -K2KP3 - 2 42K -K2 C KKp+3 - n-l K2 (P1P2) (5.26) Kp+2,p.l + K1 j,. Hence, (24li 48 2 16 5 M(3221) = o n(1 "+ )- +yz)(K422+ 2K332) + (n;)(K + K ) + ( 1 + 24 - n n )(K53 + K44) + n2 K (K5+ n1 32)K3 n7fTn-l))K3 ) 3K(3 8 n-l (5.27) - (2 + -)(K42+ K33)K2 KK2 Recursive Approach to Formulae involving the Mean using Substitution Products As observed by Irwin and Kendall (1944), M(rl) - oKr+1 In terms of substitution products, however, (5.28) M(rl) - LKrKln - Krl K Proceeding further, we find (5.29) M(r12) a [Krln -2K1 L[ KrKil 2r 2 2 - Kr KiJ + 2Kr K and in general, M(rl ) - EN (kr-Kr)( (-1) lf "^s~~t., (t)kst K~

-94 ( -I /-2 - 2 (-1)t(t)KrK LK S t._o t (5.30) + (-l)s K Ks Also, from (5o11), M(pql) LKpK K}KPK -K L K (5o 31) -K1 K KK +2KKK1. 1 L p q-f,. And, in general, M(pl) EN (kKp)(kqKq)((l)) ks Kt) 7(-l)t(t) K tA2 ()t(t) K Kt to 1 s-t K K I q. 1 - A-I - 2 (-l) ~t= 0 P% s- t-2 r K, i-t+ (-it=o p t= o''(s)KKK S-t (t p [ s s() K K Kl K 1~ (t pq (5032) + (-1)s+ (s+l) K K Ks p> aL From (5.9), (5.33) M(pP2 P Pp) 2(-)I U CP f1teI Kp.k: r K1% I v- i A4,(P1P2.. Pu) Hence a recursion formula is (5s34) M(P1P2o.Pu) = IZ (-1)I1 K LK lff K Pi 7 l'r C- f -K1 M(P P2.p~) u) This can be successively applied to get expression (5o28) to (5,32)o As an example,we consider M(32) - EN(k3-K3)(k2-K2) - 3KK - KKI] -K2n K3K2* 3K2" 3 K2~ - L3J, 3K2 K1 +nK -KK -KK JKK K5 n- 2 K32 32 23 32 * For recursion, we use M(32) in this extended form, obtained in derivation, and not as [K3K2jn K3K2 o

- 95 Hence, 1M"-(321) 5 n -5 K2 33 + K32 n~ ~~~ — f rr K3 K3 iK2 - -j -K2 L + K3J ( n 3>1) + K3K2K1 - K1M(32) - K6 ri Kr n+ 5i -+ K5 3n+5 nl K+42+ ni+5 K33 K321 n n(n-) -nn-T ) 33 t- -- 321 K2 n- - n K2K n ----- K3K21 K2K31 n KK32 + 2K3K2K1 n-1 and M(3211) -12 + K61 n n n 2 n K K61 n _52 +K 511 = ) n' K52+K43 K421j n n + nn+5 2K4 K33 n+ 5 n(n-l) nn 331J n-1 K421 + K + K3- + K321 j K3 K4 n RK ( Kr 2 r + K31 - K3 K3 1 K32 1 n L + K2 + K211 n -j+Kll] ( nK6 — ni+ + K1n-. K1K3K 32_ K51 nl. K 321 + 3K1n + K21 +K1K2 K4 K31 + i2K3 n -. K3K2K12 - K1 M(321). Estimators of Moments Once the moment formulae are expressed as linear functions of the Ko o,, their estimators can be readily obtained by replacing Ko o by ko o o 0 since kooo is an unbiased estimator of Ko..o For example, from (.5ol (535) M(22) k (-) k4+ n ) k22

Symbolically, since (5.2) M(22) [K2N ] K4 n+l 2 (5.)2) (2) [KJ] - N [KJ n - K2 { + n 1 K22 - K2 n n-1 we have 2 kj N-l The transformation from (5.2) to (5~36) can be described as changing the sign of the expression for M(2 ) and replacing K's by k~s and n by N to get M(2 ). By a similar operation on the variance-covariance formulae (5o3), (5.4), we get (5.37) M(r2) k2 k] (5-38) M(rs) krs [krks] N The use of substitution products allows us to combine the computation and estimation properties in the same expressionO Thus, either of (5,2), (5~36) can be used to represent both formulae if we note the transformation described above. To take a few more examples, we can write from (555), (5.39) MP(r) s l) +([) (akr a-) kr Also, from (5.9), (5.10), (5.40) M(pl...pU) 2 (-1) FT k FkTT kp I ( t kp Pi L IP' NJ- J /-oo k 0 k( k.k.o.. o (5.o1) PI Pi1 Pl

-97 in the earlier notationo It should -be noted that though the N could be dropped in (599), (5o10), since LKrKs..o N KrKs.o, we can not drop it in (54o0), (5a41)o Thus, whereas the finite moment formulae can be written compactly in terms of simple substitution products L[K. K oO 0 N which are really products K.oooK oo0OO giving the results of Table 3, the compact form of the estimation formulae is more complicated as it involves complex substitution products like [k rkso On kp JN which, of course, can be transformed into simple substitution products by using formulae for krksoo Estimators of moments and curmulants not involving the sample mean are A given in Table 80 (Results for M(o o) for weight 8 have been given by Schaeffer and Dwyer (1963) )o We pass over the first column and the beginning rows which have a zero under the "estimator1'columno The formulae for AA the estimators of moment functions when we read M( o o ), K(o.) for M( 0o ), K(o o o) at the head of the columns are available in terms of substitution products [koook Q0 0 No The suffix N has been dropped in Table 80 The relationship between the expectation and estimation formulae is illustrated by the following exampleo For M(522), we have 2 2 2(nK 1 M(522). 5K2]n nI K7K2 KK52K ni K43I 2 (n K51 n+l K22K5 + 2K5K2 2 (in Table 806, the expansion of K5K2 is written using Table 7ol)o Now for the estimator, A ^. - 2 -I2 - 2(n, 40 - t (522) ksk 2 k7k2 n k52 k432 n [~k5ssWn-12k5 2 k2 n] n~l 2 (5'43) k k k2l2kN 2$ 2 n [4k5 i L~25J 2 Lk5~

-98 2 Here, the first term k5k2 need not be expanded and appears with a multiplier 1 in the';estimator" column, Thus the expectation formulae, with the use of substitution products, give estimation formulaec For weight 4L 8, Schaeffer and Dwyer (1963) have used this fact in constructing a computer program which can be used either for expectation or estimationo Estimators of Moment Functions involving the Mean For estimators of moments and cumulants involving the mean kl, a similar tranformation of substitution products is madeo For example, from (5013), (5 44) ^(lr) - Z (rloo, )... r rvl 5 ) 15 Z rlCX +rC)S r,-1~.. ~ r-l rl~ o off T from (5.29), A 2 2 F r2 i F Fri 1 F21 t5(rl ) k kl 2 Lk[krkl ] N Lkr Lk1J nN +2 L'kri N krkl - -- + k 2 L r2 kr.j n r1- NL 1 2 (55) [lkk2j n r Lkrk2 [ l] N +2 kr kI and from (5.32), ^si't(pg~5 2(ls~)t(s) 5t s) (jpkl, SI k qk tl l (pqls).2 s)C 1 [w -( i 1 ) -) () k lt Lk k l N(-l) (n t) kkkltLkl N (5.46) +(_1) (s+l) kPk kl N

Infinite Populations For the infinite case, we denote the moments by ( o o ) and their estimators by A (o o) and similarly for cumulants use X(F,. ) From the variance-covariance formulae (5 3),(5 4)., we get (5 47),(r2) [r2] Xr (5.48) e(rs) L Xr xs] -n r)s where [IrAsl r lim [KKs (Schaeffer and Dwyer, 1963) Taking estimates and noting that lim [krkj krs5 we obtain the formulae for estimators, (5~49) A(r2) k 2 kr (5~50) A (rs) a krks krs as given by Glasser (1962) and Schaeffer and Dwyer (1963)o The M(.oo) and K(,co) formulae of Table 8 give Fisher's (1923) formulae as N-c>~o For example, from Table 806, MI(522) K.(522) 2 Kg +b) K72 ^ + i K 3 + n3 a 8n l32^ K54- + M(522) K(522) 1 2(n 1 +910) 1 4(n ) 12 2 KK9 n2 n+I K72 + 3~-nin-41 K63 + - n-I., K54 + (n-l Z K522 + - 432(Y-' 333 7 2 1 K5K4 2(nt9) 40. n+l K KK4 - K52K 2 n-l K43K2 n-l KK 22 + (5o51) 2 5K 2 K5K2 Hence, since lim Krs r Xr s /,(522) =A(522) 1 4nll) 2 7X2 + 2 ni —-- 6- + n2 9 n( -7 nJ 2 + n(n-l)'

-100 2 - 1( nsgX4 +I n+9)(ntll (n nl +?2X>2n- +98n-139 - 1 J52 40(n+lL 4o } X + 120 3 21 9 24. X 20(3n-4) 20(5nmY) X + g552) 120 X 2 40 )4)32 + 120 )3 (552) 10 5 2 ~- ) 1 X 3, ) 52 nd-l~ n- n'-Z~ 3 which is formula (18) given by Fisher (1923, ppo 210211) and provides a useful checko Also from the formula for M\(522) in Table 806, we can obtain an estimator of K(522), using lim krk ] - k o Thus from (5 43), we have N-^o L N r s (522) kk2 2 2(n+9) k -,40..2 (522) - ksk - - k 72 -- k- k522f (5.53) 1420 4k80 V^ k 120 -. " 5 -Z)' 522 t n 432+ — 1)2 333 using the expansion of k5k2 from Table 7olo The relationship between (5o52) and (5.53) is immediately visible, since E(kr X rs)... Expressions for..(.oo),( o ) can similarly be obtained from all formulae above for M( o.), M(.0.)o Thus, from (5010), (54.1), (5.5/( oo, ~- (-5)X5 4ox [Ox.....)K (".Xu] (i P5h ~i P. Pih P 7 (5~55) ~ P, ) (^...l.. l. 7 PiJ PU l h jk where...k OPil...pi. is the expansion of k o,,k with LPlc Piu'1 hJ n' I PUj n subscript po O1 Pi added to each term, extending the notation of h

-101 Schaeffer and Dwyer (1963)0 Thus, from (5 55), ~(rst) = kkkt krks t] - [kskt r^ - [krkt. s] +[kro stn +[ksort + [kt rsl - krst (5.56) kkkt - krks t ] [kk r] krkto s +2kst r r s n s to n r n nsnu and so?(332) kk - k k 2 - 2. 3 2 k 32 3 332 k2 -(1 9 n+ 6n k222)o 2} k3k2 k6 + n-1 k2 + n- k33+ n - k222) - 2 k5 + n- k32 3 4 2 k332 k3k2 i nk62+ 9 k422 + n 332 + k222 3 2 n-l - k 332 2 kn,jn —2h 2222 (5~57) n 2 5 1+ 2 5,57) -2 i n k53 n-l k332 332 nJ 332 k3 32 ~ Thus the finite moment formulae and their estimators are directly applicable to infinite populations, though a number of terms need collection to provide the infinite formulaeo

SUMMARY The aim of the work was to generalize Fisher's combinatorial technique to write products of generalized k-statistics and to use these to obtain moments of moments when sampling from a finite population~ The basic material was first reviewed, studying the sample symmetric functions and, in particular, the generalized k-statistics were defined in terms of partition coefficients and symmetric means. It was seen how their property of being unbiased estimators of products of parent cumulants for all distributions implies uniqueness and their seminvariance, when no subscript is unity, was established~ Fisher's (1928) combinatorial approach to write products of single subscript k's and to obtain their cumulants was described, Tukey's (1956) algebraic method for products of two generalized k-statistics and its modification to a combinatorial method by Dwyer and Tracy (1962) were discussed with the use of an example. The steps needed to write semi-general formulae for products ke ko.. with the use of array types and distinct units were mentioned. The rules of Dwyer and Tracy (1962) for products of two generalized k-statistics were shown to hold for multiple productso Four additional rules were stated and proved which are useful in determining the coefficients of array types involved in the use of combinatorial method, These coefficients were obtained for some general patterns and tabulated for many commonly occuring patterns, generalizing those given by Fisher (1923), who had a simpler situation in that all rows could be added for any pattern when dealing with single subscript k's, A generalization of the combinatorial method was next used to write semi-general formulae for multiplication of ki3 by products of ko,,'s up

-103 to weight 4, The rule of proper parts was very helpful in reducing the number of array types to be considered as only those contributing nonvanishing coefficients had -to be retained. These formulae were then applied to write products of seminvariant generalized k-statistics of weights 9 and 10 and selected ones of weight 12o References for product formulae of lesser weight were given. Checks for formulae of both types were indicated. Lastly, the product formulae were used in writing moment functions in terms of Ko o, adapted for computation. Formulae not involving the mean k1 for M(...) and K(...), where it differed from M(..), were given for finite sampling through weight 10 and for special cases of weight 12 in tabular form. Methods for obtaining moment formulae involving kl from those not involving it were given and illustrated. Ways to express estimators of all these moments in terms of substitution products (Schaeffer and Dwyer, 1963) were given and estimators of all moment functions tabulated through weight 12 were also incorporated in the tables. Use in checking was made of the fact that as No->o, the finite K( o ) formulae give Fisher's (1928) cumulant formulae for the infinite case,

REF RENCES Abdel-Aty, SHo (1954). Tables of generalized k-statistics, Biometrika, 41, 253-260. Barton, DoEo, and David, FoNo (1961)o The central sampling moments of the mean in samples from a finite population. Biometrika, 48, 199-201. Cornish, E.A., and Fisher, R.Ao, (1937). Moments and cumulants in the specification of distributions. Extrait de la Revue de l'Institut International de Statistique, 4, 1-14. Reprinted in Fisher, R.A. (1950). Contributions to Mathematical Statistics, Wiley, New York, Paper 30. Craig, CoCo (1928). An application of Thiele's semi-invariants to the sampling problemo Metron, 7, 3-74 David, F.N., and Barton, DoE. (1962), Combinatorial Chance, Charles Griffin and Co., London. David, F.N., and Kendall, M.Go (1949), Tables of symmetric functions - Part I. Biometrika, 36, 431-449. Dressel, POL. (1940)o Statistical seminvariants and their estimates with particular emphasis on their relation to algebraic invariantso Annals of Mathematical Statistics, 11, 33-570 Dwyer, PoS. (1938). On combined expansions of products of symmetric power sums and of sums of symmetric power products with applications to samplingo Annals of Mathematical Statistics, 9, Part I, 1-47, Part II, 97-132. Dwyer, P.So (1962). Properties of polykays of deviateso (Unpublished paper) Dwyer, P.S., and Tracy, DoSo (1962). A combinatorial method for products of two multiple k-statistics with some general formulae, 0.NoR. Technical Report, Office of Research Adminstration, Ann Arbor. Fisher, R.A. (1928). Moments and product moments of sampling distributionso Proceedings of London Mathematical Society (2), 30, 199-238, Reprinted in Fisher, R.A. (1950.) Contributions to Mathematical Statistics, Wiley, New York, Paper 20. Fisher, R.A., and Wishart, Jo (1931)o The derivation of the pattern formulae of two way partitions from those of simpler patternsc Proceedings of London Mathematical Society 2), 33, 195-208.

- 105 Glasser, G. J (1962). On estimators for variances and covarianceso Biometrika, 49, 259-262, Georgescu, N. Sto (1932)o Further contributions to the sampling problem. Biometrika, 24, 65-107, IIalm.os, P.R. (19)l.6). The Theory of Unbiaased.istirationc Annals of lathematical Statistics, 17, 34-43. Irwin, J0o0, and Kendall, MoG. (1944) Sampling moments of moments for a finite population. Annals of Eugenics, 12, 138-142o Kendall, M.oG (1940a). Some properties of k-statistics. Annals of Eugenics, 10, 106-111, Kendall, M.Go (1940b)o Proof of Fisher's rules for ascertaining the sampling seminvariants of k-statistics, Annals of Eugenics, 10, 215- 222 Kendall, MoGo (1940c)o The derivation of multivariate sampling formulae from univariate formulae by symbolic operation. Annals of Elgenics, 10, 392-402 Kendall, M.Go (1952). Moment-statistics in samples from a finite population. Biometrika, 39, 14-16. Kendall, M.Go, and Stuart, Ao (1958). The Advanced Theory of Statistics, 1 Charles Griffin and Co,, Londono Riordan, J. (1958). An Introduction -to Combinatorial AnalysiSo Wiley, New York. Schaeffer, Eo, and Dwyer, PS,. (1963). Computation with multiple kstatistics. Journal of the American Statistical Association, 58, 120-151 Tukey, J.W. (1950). Some sampling simplified. Journal of the American Statistical Association, 45, 501-519o Tukey, J.W. (1956). Keeping moment-like sampling computations simple, Annals of Mathematical Statistics, 27, 37-54. Wishart, J. (1952). Moment coefficients of the k-statistics in samples from a finite population. Biomoetrika, 39, 1~13. Zia-ud-Din, MVo (1954). Expression of the k-statistics kg and ko in terms of power sums and sample moments, Annals of Mathematical Statistics, 25, 800-803. Zia-ud-Din, M. (1i59). The expression of k-statistic kll in terms of power sums and sample moments. Annals of Mathematical Statistics, 30, 825-828.