NEW CHARATERIZATIONS OF THE NO-AGING PROPERTY AND THE /-ISOTROPIC MODEL Stephen E. Chick Dept of Industrial and Operations Engineering University of Michigan Ann Arbor, MI 48109-2117 Max B. Mendel Dept of Industrial Engineering and Operations Research University of California Berkeley, CA 94720 Technical Report 96-4 March 1996

NEW CHARACTERIZATIONS OF THE NO-AGING PROPERTY AND THE f1-ISOTROPIC MODEL Stephen E. Chick* Max B. Mendelt March 21, 1996 Abstract The Ino-aging property and the El-isotropic model it implies have been introduced to overcome certain shortcomings of the exponential model. However, its definition is abstract and not very useful for practitioners. This ipaper presents several additional characterizations of the no-aging property. Included are (1) characterizations that appropriately generalize the lmemoryless property and the constant-failure-rate property of the exponential, (2) behavioral characterizations based on fair bets, and (3) geometric characterizations of the survival and density function and differential-geometric characterizations based on tensor methods. 1 Introduction In this paper, we characterize the no-aging property and the joint probability models it, implies. Tlie no-aging property was proposed by Barlow and Mendel ill [2]. They consider a finite, exchangeable population of lifetimes and argue that the correct, prol)abilistic Iloldel for aging is-given by a Schur-concave joint survival function (for tlhe theory of Schur concavity, see e.g. Marshall and Olkin [12]) No(-alging is desc(ril(dl iy.i Schtiir-co(.istailt survival function and these funlctions are also called /'l-isot.rolic''Their (lefinition of no-aging is particularly apl)pealing for a Bayesianl whio wishel, II'us the unconditional distribution of th llilifetimes, rather thllai i (l strilbltl io (ondlitiolial on somte pIarameter. *DeIt.. of llldustrial alnd ()lperantl;lngmilteerilg. 130 1.O.E. Building, 1205 Beal Avenle, Thle University o(f Michigan, Alnn Art (i. M1 4i1()9-'211 7. tDept. of Itdustrial Engineering andl ()lO atilns Research, 4131 Etcheverry Hall, The University of California, Berkeley, CA 94720. 1

The problem with the no-aging property is that its definition in terms of Schur-constancy can be far removed from the type of assessments a practitioner is comfortable in making. This paper compiles eight characterizations, five of them new, of the noaging property and the 1f-isotropic model in an attempt to make these concepts m6re useful to practitioners. The characterizations are compiled in the theorem in Section 2. We have opted for characterizations that have a direct behavioral interpretation or an appealing geometric interpretation. The behavioral characterizations are particularly relevant to a Bayesian decision maker. They are stated ill terms of fair bets. The geometric characterizations concern symmetries and invariances of the survival function and the density function. We also give several lifferential-geometric characterizations. The use of differential geometry is iot very common in probability (see, however, Santalo [13]). However, these (haracterizations are very compact and provide perhaps the best pictures. Iwo of the characterizations that ill the spirit of the classical constantI'ailure-rate and memnoryless properties of the exponential. These two properties hlave playe(d a central role ill characterizing the exponential model; see, e.g., [4-10, 15]. These are usefull for seeilng how the no-aging property and the'lisotropic mrod(el departs fronm the classical no-aging and the exponential model. 2 Characterizations \Ve state the various characterizations in the form of a theorem. An item-byitem discussion precedes the proof. We first introduce the main notation and we define the empirical failure freq(lency. Consider a population = (..... AN) of lifetimes, i.e., a seq(lelnce of 1IOIi-llt'gatiVe' real-valued random variables. Let.\ >.r mean that,Xi > l,. for i = 1.... Write the joint survival function. + ( [0-. ]), (1 P(o)b(= ~, >.i), (1) 2

and the density finction f:IR -R IR as (_-1)NaNF(.) ax.. aoxN wlhen it, exists. It will be convenlient, though, to use the notation F and f also for the marginal survival and density functions of any subsequence of' F. This will be indicated by the arguments. Finally, let V denote the gradient operator and call -V log F the hazard gradient, (see also Marsliall [11]) For the differential-geometric characterizations, let 9/9xi be the infinitesimal vector in the i-th coordinate. The exterior derivative is denoted by d, the Lie derivative by ~(), the anti-symmetric product for tensors is denoted A, and tensor saturation is denoted by. (See e.g. Burke [3]). Definition: For a population X = (X1,.,XN) of lifetimes, the failure frequency is: ~- N AN(X) N zi=1 Xi We use the following definition of tle no-aging property originally proposed 1>,! Ilr low andl NMeletlll [2] Definition: A population X = (A 1.-., aN) Iha(i t1 i no-aging property if F l.s fI J.nll.ctzo of tilh failrf J( ( q'l( t1 cy AX (a0lon(. The following theoremI shows the e(lqlivalellce of eight characterizations of thle lo-aging 1)rol)erty.'I'll (letillitioil aplpears as Char. 3. The equivalence of C(lar. 3 and Cliar. 4 is eleeniltary. aln( the equivalence of Char. 3 and Char. 5 is (tiue to Barlow and Mendel [2]. Thle otlier characterizations are new. Char. 1 caln be viewed as a specializationo of Spizzichinlo's characterization of the Schurconcave class [14]. 3

Theorem: The following are equivalent: 1. F(xi + h I X > x) = F(xj + h I X > x) for all x, and all h > 0. 2. The components of-V log F(x) are identically-equalfunctions of AN alone. 3. F is a function of AN alone (definition of the no-aging property). 4. f is a function of AN alone. 5. F(x) is arn 1-isotropic survvlal function, i.e, for all n, 0 < n < N. (1 r,...,)= Ji l - N l P(dAN). Lft Wi, = 3/Ox.i - 9/Olj and W thl wuedge product of any (N - 1) line arly ilndpendeint W, j. Lft p b( a probability N-form7 field on iR, and I)IXN th.( conditzioltl prob)(bil7ty (N - 1)-formn field on the sirmple:.p X {AN(X) AN } (. -d log F = -g(AN)dAN, with g: R+ -- +. 7. t'w,,(p) = 0 for al i, j. Y. P)IIN I W i.s a function of AN alone. Remarks: ('lhar. is thle appropriate generalization of the memoryless property. From Char. 1 it follows that the lifetimes are exchangeable. However, Char. 1 does o70 illlly that thlle',\'s are indle)ependellt.. If we assume independence in (ddtlll on t.o ('liar. 1. 1. w fiiil t hiat F(.r,, + hl > I-) F(h/), Vi, whicli is the memoryless prop)erty t hat chlaracterizes the exlponleitial. Char. 1 has a behiavioral ililterprletatioll: i'(liven two similar comlllp lellts which have not vet, failed, one wolll(l btet the same armoltnt. for the same return on the event that either component fails during an additional increment of usage, regardless of tlheir ages.' 4

Char.2 is the appropriate generalization of the constant-failure-rate property. If we assume independence in addition to Char. 2, we find that: - log F(xi) = constant, Vi, dxi which characterizes i.i.d. exponential lifetimes (see Marshall [11]). Char. 6 addresses several problems with the the use of the failure gradient (see remarks below). Char.3 is the definition of the no-aging property and the 41-isotropic model. This is Schur-constancy of F: F can be moved anywhere along a simplex {AN(X) = AN} without changing its value. (See further Barlow and Mendel [2]). (Char. 3 is physically covariant under changes in the scales or units in which lifetime is measured. Tllese are here the admissible coordinate transfortltiatloisl Mattlemlatically, a scale change is represented by a smooth ionltotolne increasing function Y of lifetime: (I.....A\') (Y'(Xi),..., Y(XN)). Thle new,y valules cail siriplyl lbe stubstitltted inl the expression for F. ('lCar.4 states that, the density is uniforni on simplexes {AN(X) = AN}. This utliforimity, however, depends on the scale used for X. f is 7no physically covariant, as changes in scale involve a Jacobian factor. See Char. 7 for a llphysically covarianlt version of Char. 4. ('ltar.5 gives a de Finetti-type represenltation for the family of fl-isotropic survival functions. For anly N, t. lis family strictly includes the i.i.d. exponentials, although the intergra;t(l represctts depetndlent variables. When N -0oo, it converges to a llixtlre of i.id. expotnentials with rate A,, the limiting failitre freql te cy: F(J. ) = P(dA,). rl= (See also Barlow attd Mettdel [2] ) 5

Char.6 improves upon Char. 2 in two important ways. (see Figure 1.) First, Char. 6 is physically covariant. There are no explicit references to X. Second, the gradient, operator is replaced with the exterior-derivative operator d, representing the differential or total derivative of log F. Unlike the gradient operator, the exterior derivative does not depend on a metric or inner product. This dependence on a metric is a not-so-well-known defect of Marshall's [11] hazard gradient: a gradient give the dir:ction of greatest increase, but without a natural metric there is no natural notion of direction. Our definition makes -d log F a 1-form rather than a vector, so a name such as "hazard 1-form" or "hazard differential" would be more appropriate.1 Char.7 states that the joint density is uniform on simplexes as in Char. 4, but now using differential N-forms, thereby eliminating the references to X. The Wij's are the vectors that leave AN invariant. By requiring that the Lie derivative of p vanislles, we obtain the invariant density. (See Figure 2.) ('liarr.i states that the conditional denlsity oll a simplex is uniform with respect, to ti lle lturl tiotilo of vle o r-area oni the sirmplex. W gives a unit of liyper-area onl tlie simplex. P) AI,. W is a scalar function that gives the (infinlitesinlal) tlamollunlt of ptroblali lity )er uinit of hyper-area. If this is constalnt on tlhe si illex, t lie c11 (ol(litionlal (ldelsity is uniform. Tliis is showij ii Figture 3. This chlaracterizat ioi alsco 1;i. a;L itllt'rlpretationI in termsls of fair bets:'Given tllhe failure fre(lquenclly, onl~ w\ould b[et the same amount for the same return oti any oltconle for \ Proofs: We show how Chars. 1 thiroughi.5;are e'(livallent, to the definition in Char. 3. Cliars. 6 and 7 are showii to )be et(quival'int to their respective coordinate-based 1Force is a 1-f(ori, so "force of Inlit aitl " I., ap)l)pr()priate from tiis persI)ective. 6

X2 {AAn (X) = AN } d ^ JlogF' 1") dANF (:') l og F( ) dA.A(J")%-' Figure 1: The hazard differelltials at, x and.T'; they are pictured by two infinitesimally-spaced level lines of the hazard function -log F with the arrowhead flagging the positive direction (see also Burke [3]). Also pictured are the differentials of AN at.' and 5". -g is the factor with which dAN is multiplied to obtain the hlazard different.ial and this factor is constant on the simplex. A2 A { (X)= AN} p(-) /)a- _ —- o I.,.1.r1.. 0.''l ~i. x Figure 2: The probability N-forll (. = 2) at two locations pictured by infinitesilial sqiuares colntaining ati inlfiniit.esilial unit of probability. Pulling p(') back via', 1 arnd subtracting it froit Ip)() shlows that the Lie derivative vanishes. 7

X1.T.~~~'- O - T-1,2 A 42 J = W -''2,= -a. PlA' 2 oxo A.?\.{' ( = AA x2 Figure 3: Here N 3. The simplex contains the bivector W and the conditional probability 2-forms PA,N on the simplex. PlAX J W is the number of times the 2-forn fits into the bivector (in the figure this is approximately 3 times) and thlis lnumber is constant over the sinpllex as re(luired.

characterizations. 1.:>3. Assume 3, F = g(AN) for some g. Then Prob(X, > xi + h,X >2 ) = Prob(X >.' 3 +/,.Y > J) = g(N/(h + >N1 X,)). 1 follows after dividing both sides of the first equality by F(Z). Assume 1, multiply both sides by F(x), collect terms, and substitute - hej for x. Thus Prob(X > 5-hej+hei) = Prob(X > x) for all 0 < h < xj. Both F and AN are the same at Z - hej + hei and x. One can similarly shift lifetime between any pairs i, j while still conserving F and AN. As a consequence, F is a function of AN alone. 2.=>3. Assume 2. Then (0/a9x, -/9cxj )F = 0, or Vwi, (log F) = 0. This implies that log F and hence F is a function of AN alone. If not, there would be point where V'w,, log F: 0. Assume 3. This implies log F is a function of AN alone. Then W (... dP (I AXN AN ) Vw,,(logF) = (- ) dAN ax.i Oxj = 0. 4.<>3. Because tlle Lit' irackets of the coordinate vectors in the X system all va.lisiil, Wte liavc (-1).. -, (,_,.":-1)( P - w, (f) T ( F)= TW ( - 7 F) = VW,, C(f) - If F is a fullct.ionl of A, alone, thellll V W, (F) = O. From the above equality it follows thalt Vw, (.f) = U. wllich iniplies 4. To prove 4 - 3 revert, this argulmerit. 5.,=3. Assume 3. Then F(-IA,A) is constant on the simplex. Integrate out.r,.+i, J.N Tills is a Diriclhlet integral and gives the result in 5 (see also, e.g., Barlow and Mendel [2]) That. 5 implies 3 can be read off directly from the expression. 6.>2. Assume 6. Use the; operator for the Euclidean metric tensor (See e.g. Abraham and Marsden [1]). Ix.\ coor(dinates: dlogF 0 1 logF = f(-d logF) = - g(N)t (dN). (J'l ()i' dO: x ON 9

But the components of }(dAN) depend only on AN and 2 follows. Assume 2. Use the b operator of the Euclidean metric tensor. -log F Xlog F b(V(- logF)) = - dX -... O- dXN. Ox1 OX.N This is -d log F in X coordinates. If the alog,'s depend only on AN, 6 follows. 7.:4. Write the density N-form in X coordinates: p = fdX. The Leibniz property of the Lie derivative gives:.~w,,(fdX,) = Lw, (f). + f wLw _, (dX). But ~'w,,(dX) = 0 and so we have L'w,,(p) = 0 if and only if t'w,., (f) 0, or \7w,,(f) = 0, since for fulnctions the Lie derivative is the directional derivative. 8.-:7. Because ~w,,,(h(.AN)) = 0 for any function h, 8 implies: ~w,,j(pIN I W) = 0 for all i,j. We have: -'w,.,(P|AN J W) ='w,. (W)J P|AX + W I ~Wi,j(P|AN) But L'w,,(W) = 0 and so we have 8 if and only if ~w,,(pIxA) = 0 for all i,j. Now write p = PAN A PlAN; PAN is a marginal probability 1-form for AN- We have ~'w,, (PAN A PlAN) = -' W,, (PAN) A Pl|A + PAN A Z'W,.j (PIAN). But. Lw, (PAN) = 0 and so we have 7 also if and only if L~'w, i(PAXN) = 0 for all i. j Now t le W11,, iave ilitegral mIanifolds which are the level sets of l(nlbd.r N an(d th e eqllivaleltce follows. References [1] R. Abraham andl. E. NMars(lell, F.'oundatlioit of Mc:chailcls, AddisollWesley, Redwood City, CA, 211l ed(., )19S5. [2] R. E. Barlow and M. B. Meid(el, D( Finctti-typc reprTCesentations for life distributions, Journal of t.le Aliericall Statistical Association, 87 (1992), pp. 1116-1122. 10

[3] W. L. Burke, Applied Differential Geometry, Cambridge University Press, New York, 1985. [4] E. B. Fosam and D. N. Shanbhag, Certain characterizations of exponential and geometric distributions, Journal of the Royal Statistical Society, Series B, 56 (1994), pp. 157-160. [5] J. Galambos and C. Hagwood, An unreliable server characterization of the exponential distribution, Journal of Applied Probability, 31 (1994), pp. 274279. [6] J. Galambos andl S. Kotz, Characterizations of probability distributions, Lecture Notes iMi Mathematics, 675 (1978). [7] T. Gather, On a charac trization of the exponential distribution by proper/c(. oJ ol( I/d.s/l//l.st(.i, Stat.istics &' Probabiility Letters, 7 (1989), pp. 93-96. [8] V. J. Hua.ng and.1.M. M Sllhoung, So7r(e characterization of thl exponential (1.1/d jtoi/ tri(c di(st..ribu tio7I (I s a.ri.siny in a queut:ing systertm with at'untreliabl(.s Tr r, Journlal of Applie(l Probability, 30 (1993), pp. 985-990. [9] Z. Khalil and B. Dimit.rov, Thl strvicf( time7 propertie:s of an unlrehlable.s( VT v 7 char(ac1(1f i z tht (~ po ll ntial di.stribution, Advances in Applied Probal)ility, 26 (1994), pp. 172-182. [10] K. S. Lau and ('. R. Rao, Integrated Cauchy functional equation and chracteri'zations of th( exponel.tital law, Sankhya A, 44 (1982), pp. 72-90. [11] A. Marshall, Soimi cormnrltEs on the hazard gradient, Stochastic Processes andl Their Applications, 3 (19)75), pp. 293-300. [12] A. W. Marshall andl I. Olkiui. Inmlquallit.s: Theory of Majorization and Its A pplicalion..s, Acadi'c Prt.ess. New York, 1979. [13] 1. A Saiita1O. l/ it(/r7tl (;t(tnlti/) tt(l1d (;to711.(1'7 Probabzlity, AddisolnWVesley Ptublisllhi(g (o i. I', Hf'adnlg. MA, 1976. [14] F. Spizzichiino, R.l( lIillt it (l1(.I,1n l,) obl( m1s ulnder condiztons oJ agingl, in Bayesian Stattist.ics 1. Bt'rul'ardi. litrgeir, Dawid, and( Smith, eds., Oxford IUi iversity Press, 19 2'. I'p' S();t (11 11

[15] Y. Too and G. D. Lin, Characterizations of uniform and exponential distributions, Statistics &r Probability Letters, 7 (1989), pp. 357-359. 12