110 H. A. D A D

NBdae, A. (1971). ‘’ The distribution of the identified minimum of e normsl pair determinee the distribution of the pair.” Tcolrrrmnckics. 13, 201-2.

Neyman, J. (1976). ‘‘ Aseeeslng the chain : energy crisis, pollution end health.” Int. SkAvt. Rev., 43, 263-67.

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Todhunter, I. (1866). A Hietwy of the Mathemata Theory of ProbaWity. Reprinted 1931.

T&th, A. (1976). “ A nortidentiliability aspect of the problem of competing rkh” Pm. Nut. A d . Soi., 72.20-22.

Stechert end CO., New York ; 1949, Chelsea, New Pork.

INVITED DISCUSSION OF THE PAPER BY PROFESSOR DAVID

Dr. J. S. Maritz (CIYLRO Bwi& oj i?€crthemaliCs and Lstatistics, Yelbourne) : My own contact with the subject of Professor David’s talk has been through its overlap with the analysis of survival data, especially in studies of cancer survival. Professor David’s paper ha8 introduced me to several ideas which I had not previonsly considered, and has made me reaJize that I may have taken a rather

In survival d y s i a it hss seemed reasonable to me to compound all the censoring agencies and thus to deal with two random variables, X, the survival time and 8=min(P,,P, . . .‘.), the censoring time. One of Professor David’s references (Gail [1976]) pointa out that the L L m W a l method ” is widely used in these circumstancee. The “product limit method” of Kaplan and Meier (J.A.S.A. [1958]) may be regmded as a refinement of the actuarial method and produces a maximum likelihood estimate of the survivorship function. It depends, 5 t ~ I understand it, on X and 8 being independent. On the other hand, the actuarial method does not appear to depend explicitly on such independence and is a step by step, that is, grouped, reconstruction of an approximation to the complement of the cdf of X ; for certain X and B distributions, and X and 8 independent this reconstruction is exact. Now, it seems to me that a general study of the goodness of this approximation may be worthwhile, and I wonder whether Professor David knows of such studies.

Professor David has dealt mainly with estimation. In survival studies one may be concerned with the comparison of two or more survivorship curves; they would typically correspond to different treatments. An hypothesis testing approach, which I do not neces- sarily wish to espouse, can, in this Bind of situation, be simpler than an estimation approach, e8peciaUy if one uses test procedures that rely on permutation arguments. The only assumption that has to be made to test a null hypothesis of no differential treatment effect would seem t o be that the censoring agencies act in the same way on the Merent patient groups.

It is a great pleasure to propose a vote of thanks to Professor David for a fascjnating review of an interesting, and obvionaly important, topic.

IlaI”0W View Of S & d d y B k .

TBE HNIBBS LEcTuB;E FOB 1976 111

Professor A. M. Hasofer (University of N.8.W.) : First of all I would like to thank Professor David for a very lucid and interesting lecture.

The theory of competing risks, as presented by Professor David, is a particular aspect of the generd theory of distribution of extreme values of severd random variables. The characteristic feature of his approach is the fact that on the one hand the random variables are neither identically distributed nor independent, and on the other hand attention is focussed on which of the random variables causes the extremum to be attained.

My own interest in the theory of extreme values lies in the asymptotic distribution of the extreme values, suitably normalized. It might well be relevant, in the context of Prof. David’s paper, to ask whether the lifetime distribution of a complex system, which could be represented by a large number of components in series, would tend to a particular form.

The ,answer, in the case of independent, identically distributed component life-times, is well known.

In view of the fact that the lifetimes am non-negative, there are only two possible asymptotic laws,

. (a) 1 -exp( -xk) (Weibull) (b) 1 -exp[ -exp(m)] (Gumbel). If we introduce the generalization that the component life times

are no longer independent and/or identicdy distributed, the previous asymptotic results need no longer hold. To make the discussion more general, I shall consider the distribution of the maximum (which is only notationally different from that of the minimum) and eliminate the boundedness of the component variables. Let M,=max ( X i , i=1, . . . , n). Then if the X, are independent, identically distributed, the aspptot ic distribution of X,,, if it exists, must be one of the three laws (Gnedenko (1943))

+,(.I =exp i: -exp( -41, + A 4 =exp [ -.3 +.3(4 =exp [ -( -#I,

x 2 0, r c l 0.

One generalization which has been studied in much detail is that of the random stationary sequence.

Leadbetter (1974) has recently obtained sufEcient conditions for M , to converge to one of the three extreme laws. In particular, it turns out that for stationary normal sequences, with E(Xi)=O, E(X?) =1, E[XJ,+,] =rn, either of the conditions r, log n+O,

The last result was m

n-1 X r:<oo implies convergence of M,, to

previously established by Berman (1964). However, when r , does not decay as fast as the Berman conditions

require, other asymptotic laws may exist. For example, X t t a l and Ylvisaker (1975) have established that if r,=y/log n, In] zdl, the limit distribution is a convolution of and a normal distribution. If, moreover, r, is convex for n>O, r,=o(l), (r, log n)-1 is monotone for large n and is o(l) , then the limit distribution for 111, is normal.

112 H. A. DAVID

There has not been, to my howledge, my attempt t o diacnss the case of dependent, non-identidy distributed random variables. However, by gen&zing a, method due to von lldises (1936) and recently revived by de Ham (1970), one can actually obtain sufficient conditiom for X,, to tend to each of the three asymptotic laws. One can mor8omr obtain an algorithm for calculating normalizing constants a,, and b, such that the distribution of (H, -b,,)/a, tends to each of the

Let P,,(s) =P{X,<s, X,l;z, . . . , X,<s } and G(,,(s)=[P,,(411~n.

limitxi 411 42, 9s .

Then clearly G,,(z) ia a distribution function. We a*ssume it ha-s a density function g,,(z) for Stll a. D e h e u, to be the unique solution of the equation G&,) =l-Iln.

Let H,(s) =log n[l -G,(a;)]. Then

Clearly h,,(s) is the, hazard function of G,(a).

We have H,(u,,) =O and therefore

H,(a;) = - hn(z)h* un

Alao, for m y sequence {q,} such that H,(s,,) tends to a positive limit,

lim log[ -logP,,(z,)] = lim log n+ w Ic-*W

= lim H,(a,)= lim - I" h,(z)&. n e w u,, n--cW

Theorem 1. Let o be the ~ m m o n end-point of the F,,(x) i.e. P,,(o)=l for all n a d Pn(s)<l for a< w. If F,,(a)<l fw uZZ 5, Id

lim P, [ u,+- h,,;,,)l n+ w

Proof. Pat q,=z(,.*. Then hn(un)

THE KNIBBS LECTURE FOR 1976 113

from which lim {h,,(u,,)/hn(E)}=l.

Thus lim log[ -log P,(z,)] = -y, as required. n+m

n+m

Theorem 2. Suppose lim sh,(s)=k>O u n i f m l y in n. Then H m

lim P,(u,y) =exp{ -y-k} provided lim u, = +a. n+m : n+m

Proof. Put s,=u,,y. Then

where u,<E<m,. Thus lim H,(s,,) = -k log y as required. n-+ m

Theorem 3. Let w be the ma-point of F,(a) and suppose

lim ~ ~ [ w + ( w - u , ) y ] = e - ( - ~ ) ~ , y s o , prowidea lim %,=a.

Proof. Put s, =w +(w -u,)y. Then

lim (o -a) h,(m) =k > O uniformly i n n. + f 0

Then

n+ m n-+ m

where u,<[<s,,. Then lim H,(an)=-k log (-y-l) as required.

Applications (a) Return to the problem of the asymptotic distribution of the minimum of lifetime. The criterion for the third extreme will now read : If lim sh,,(s) =k, k > O uniformly in n,

w m

2 1 0

1 a where h,(m) =- ax H,,(1;) and H,(m) =log n 1 -{I -Pfl(a)}l/n , then lim P{X,>u,y , . . . , X , > u , y } = e - ~ where

P(X1>un, . . . , X , > u , } = ( l - l / n ) n , provided lim u,=O. w m

n+m

114 H. A. DAVID

(b) Coneider a sequence of independent Pareto v h b l e s {XJ, 8uch that P{X,,Gv} =1 -a,/$, if a21#~ , =O otherwise. Suppose that the sequence {a,,} is bounded and that inf a,,>O. Then it is comparatively eaay to show that the limit law of max X, is I#,($). The normality constant zc, is the eolution of the equation

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Gnedenko, B. V. (1943). ‘I Sur le distribution limite du terme maximum d’um

de h, L. (1970). O n rcgubr vo+iotiorr and i& app&dion to the weak cunvergena

hadbetter, M. R. (1974). “On extreme values in stationary sequencee.”

von Misas, R. (1936). In:

Mittel, Yeah and Ylviaeker, D. (1976). of

sequentxs/’ Ann. Math. Stdut. 36, 602616.

&ne &8toire.” Annula & Math. 44,423463.

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z. WahrLlebid- * mw. W. 28. 289-303.

Msthematisch Centrum, Amsterdam.

“La distribution de le plum grsnde de n vdeum.” Selected p ~ p e r ~ II ( A m . Math. SOC.) 271-294.

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Stocha&c ’ Pr- A M . 3,1-18.