IMPROVED TRUNCATION FOR SPRT WITH NORMAL OBSERVATIONS Technical Report 83-20 Damodar Y. Golhar Stephen M. Pollock Industrial and Operations Engineering Department The University of Michigan, Ann Arbor

IMPROVED TRUNCATION FOR SPRT WITH NORMAL OBSERVATIONS Damodar Y. Golhar Stephen M. Pollock Industrial and Operations Engineering Department The University of Michigan, Ann Arbor Abstract: A desirable truncation of the SPRT would produce actual error probabilities less than or equal to desired error probabilities. We present such a truncation, as a function of desired error probabilities and distribution parameters, when the observations are IID Normal. This truncation is shown to be superior to others in the literature, particularly to Johnson's working rule. Key words: SPRT, truncation point, discrimination factor, error probabilities.

Introduction: Suppose we wish to test HO: W = wO against H1: W = wl, by sequentially observing independent random variables X1, X2,..., Xn with density fi(xi/wj), j=O, 1. Wald (7) proposed the SPRT, truncated at point m, as follows: Define the log-likelihood ratio at time i fi(xi/wi) Zi = In -------, fi(xi/wo) f and the log-likelihood ratio up to time n n Zn i - Zi i=l Then, reject H0 if Z > b for n = 1,2,...,m accept H0 if Z < a for n = 1,2,...,m and take one more observation if a < Zn < b for n = 1,2,...,m-1. If the experiment has not stopped at or before m observations then reject H0 if b > Zm > and accept H0 if a < Z < 0.... (1) Then stopping bounds a and b are given by the approximation a In ----- and b I1 n (2) i ~1-Q<~ S ~I d J where ~od and ld are the desired error probabilities of type I and type II respectively. It is our objective to choose (in advance) the smallest 1

truncation point m such that the actual error probabilities da and 3a' achieved by the test, are less than or equal to the desired values ed and Pd. In addition, we restrict ourselves to using Wald's bounds (relation (2)), since they are easily computable and well understood. We thus do not consider nonconstant bound techniques such as those in (1,2). Wald (7) suggested setting m "large enough" such that the effect of truncation on the actual error probabilities would be minimal. (This tends to produce a conservative test, as we shall see). In this spirit, Ghosh (4) and Johnson (6) give a working rule for choosing a non-integer truncation point mj* when the observations are independent and identically distributed Normal random variables: set mj* = 3 SupW E[N/W], where E[N/W] is the expected number of observations for the untruncated SPRT when W is the true state of nature. The integer point mj** is then obtained by rounding up mj* to the next higher integer. It will be shown later that the mj** thus chosen tends to be conservative. Aroian and Robison (3) showed that for small m, actual error probabilities can be computed to any desired degree of accuracy by using numerical integration. Their method, however, becomes tedious for large m as discussed by Golhar (5). We present here a more efficient method, which produces (for IID Normal random variables, and for the symmetric case, i.e., when id = 1d) a simple relationship between m* (the non-integer truncation point), id = d and the distribution parameters. We also show that the value of m* obtained through such a relationship is superior to the one obtained by Johnson's working rule, in the sense that the resulting truncated SPRT gives a smaller 2

E(N) for any desired error probability. Computing the non-integer truncation point m*: To find the operating characteristic function Lj = L(wj), we n-l need the probability density of Z given kl a<Z<b. Let n-1 Pj(z,n) = Prob. 1(Zn < z) f^ (a < Z < b)W=wj } and let pj(z,n) be k=l k the derivative of Pj(z,n) with respect to z. Then successive convolutions are required to calculate Fj(z,n) namely pj(z,n) = fl(z/wj) b pj(z,n) = f pj(u,n-l) f(z-u/wj) du for n > 1 a Using these relationships we can calculate the operating characteristic function m-1 a 0 Lj (m) = pj(2,n) d z+ pj(z,m) d...(3) The following procedure is used to obtain m* when fi(Xi/wj) is Normal with mean wj (j=0,l) and variance r2: i) Given (dd = d' compute Wald's stopping bounds a and b from equation (2). ii) For a given value of discrimination factor d = (w0-wi)/use relationship (3) to compute o a(m) (=a(m)) = Ll(m) for different values of m. iii) Find m* by interpolation so that Oa (m) = od. Results: Values of c were varied between.01 and.2, and values of d between.5 and 2. For each o, figure 1 shows ln(m*) vs. ln(d). An essentially linear relationship between ln(m*) and 3

vn U) CM Cr 1 -'I. I cn U oc — i 0c IQ C.= =- I.75 0 -0.50 -0.25 0.00 In(d) 0.25 0.50 0.75 Figure 1 - The relationship between ln(m*) and ln(d) for IID Normal random variables.

ln(d) is seen. Using a least squares procedure, a constant slope of -2.09 is obtained for all fixed o( between.01 and.2. This suggests that m* and d have the following simple relationship: ln(m*) 2 ln[k(o)] - 2.09 ln(d), (4) k(d) k (et) or, m* --- 2.09 d where ln[k(o)] is the intercept at d=l, and depends upon the value of o(. To obtain k(oc), m* was plotted against for d=l as shown in figure 2. The curve is well fit by the equation: k(oc) S -79 + 72 (C)-.079 (5) Using equation (5), relationship (4) can be written: ln(m*) ln[-79 + 72 (a)-.079] - 2.09 ln(d) (6) Thus there is a useful and simple linear relationship (6) between m*, o and the discrimination factor d. We can obtain the smallest integer m** by rounding up the solution to (6). To show that the truncated SPRT using m** thus chosen gives a smaller E(N) than the one obtained by using Johnson's working rule it suffices to demonstrate that m* < mj* for different d and o(. The solid lines in figure 3 show (linear) relationships between ln(m*) vs. ln(d) for o(=.01,.1, and.2. The dotted lines correspond to ln(mj*) vs. ln(d) for the same values of x. As can be seen, mj* is 15% to 50% higher than m*. Thus, Johnson's rule gives a very conservative truncation point, giving a value of E(N/W) greater than that obtained by using m*. Conclusion: We have established a simple relationship between a 4

C) *- e o c * 0C Li.o00 0.04 Figure 2 0.08 0.12 0.16 0.20 - The relationship between m and a when d=1 for IID Normal random variables.

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useful truncation point m**, o< and d for tests using Wald's bounds. We have also shown that the SPRT using this truncation point gives a smaller E(N/W) than using a truncation point obtained from Johnson's working rule, yet still gives actual error probabilities within desired limits. In this paper we only analyzed the symmetric case (od = /d) and Normal observations. Work is in progress on the nonsymmetric case, and when the observations are exponentially distributed. 5

References: 1. Anderson, T.W. "A Modification of The Sequential Probability Ratio Test to Reduce The Sample Size", Ann. Math. Stat., 1960, Vol. 31, pp. 165-196. 2. Armitage, P., "Restricted Sequential Procedures", Biometrika, 1957, Vol.44, pp. 9-26. 3. Aroian, L.A. and Robison, D.E., "Direct Methods For Exact Truncated Sequential Tests of The Mean of a Normal Distribution", Technometrics, 1969, Vol. 11, no. 4, pp. 661-675. 4. Ghosh, B.K., Sequential Tests of Statistical Hypotheses, John Wiley and Sons Inc., New York, 1966. 5. Golhar, D.Y., "Sequential Analysis: Non-Stationary Processes and Truncation", a Ph.D. dissertation submitted to industrial and operations engineering department at the University of Michigan, Ann Arbor, 1983. 6. Johnson, N.L., "Sequential Analysis: A Survey", Journal of Royal Stat. Society, series A, 1961, Vol. 124, pp. 372-411. 7. Wald, A., Sequential Analysis, Dover Publications, Inc., New York, 1947. 6