Extreme value statistics for censored data with heavy tails under competing risks

0.66 MB·

by  Julien Worms

Extreme value statistics for censored data with heavy tails
under competing risks
Julien Worms (1) & Rym Worms1 (2)
(1) Universit´e Paris-Saclay / Universit´e de Versailles-Saint-Quentin-En-Yvelines
Laboratoire de Math´ematiques de Versailles (CNRS UMR 8100),
F-78035 Versailles Cedex, France,
e-mail : [email protected]
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1 (2) Universit´e Paris-Est
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Laboratoire d’Analyse et de Math´ematiques Appliqu´ees
2
(CNRS UMR 8050),
n
a UPEMLV, UPEC, F-94010, Cr´eteil, France,
J
e-mail : [email protected]

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1Correspondingauthor
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Extreme value statistics for censored data with heavy tails
under competing risks
Abstract
This paper addresses the problem of estimating, in the presence
ofrandomcensoringaswellascompetingrisks,theextremevalue
index of the (sub)-distribution function associated to one partic-
ular cause, in the heavy-tail case. Asymptotic normality of the
proposed estimator (which has the form of an Aalen-Johansen
integral, and is the first estimator proposed in this context) is es-
tablished. A small simulation study exhibits its performances for
finite samples. Estimation of extreme quantiles of the cumulative
incidence function is also addressed.
AMS Classification. Primary 62G32 ; Secondary 62N02
Keywords and phrases. Extreme value index. Tail inference. Random censoring. Competing Risks. Aalen-
Johansen estimator.
2
1. Introduction
The study of duration data (lifetime, failure time, re-employment time...) subject to random censoring
isamajortopicofthedomainofstatistics, whichfindsapplicationsinmanyareas(inthesequelwewill, for
convenience, talk about lifetimes to refer to these observed durations, but without restricting our scope to
lifetimedataanalysis). Ingeneral,theinterestliesinobtaininginformationsaboutthecentralcharacteristics
of the underlying lifetime distribution (mean lifetime or survival probabilities for instance), often with the
objective of comparing results between different conditions under which the lifetime data are acquired. In
this work, we will address the problem of inferring about the (upper) tail of the lifetime distribution, for
data subject both to random (right) censoring and competing risks.
Suppose indeed that we are interested in the lifetimes of n individuals or items, which are subject to K
different causes of death or failure, and to random censorship (from the right) as well. We are particularly
interested in one of these causes (this main cause will be considered as cause number k thereafter, where
k P t1,...,Ku), and we suppose that all causes are exclusive and are likely to be dependent on the others.
Thecensoringtimeisassumedtobeindependentofthedifferentcausesofdeathorfailureandoftheobserved
lifetimeitself. However, sincetheothercauses(differentfromthek-thcauseofinterest)generallycannotbe
considered as independent of the main cause, in no way they can be included in the censoring mechanism.
This prevents us from relying on the basic independent censoring statistical framework, and we are thus in
the presence of what is called a competing risks framework (see Moeschberger and Klein (1995)).
Forinstance,ifapatientissufferingfromaveryseriousdiseaseandstartssometreatment,thenthefinal
outcome of the treatment can be death due to the main disease, or death due to other causes (nosocomial
infectionforinstance). Andcensoringcanoccurduetolossoffollowuporendoftheclinicalstudy. Another
example, in a reliability experiment, is that the failure of some mechanical system can be due to the failure
ofaparticularsubpart,orcomponent,ofthesystem: sinceseparatingthedifferentcomponentsforstudying
the reliability of only one of them is generally not possible, accounting for these different competing causes
of failure is necessary. Another field where competing risks often arise are labor economics, for instance in
re-employment studies (see Fermanian (2003) for practical examples).
One way of formalising this is to say that we observe a sample of n independent couples pZ ,ξ q
i i 1ďiďn
where "
0 if δ “0,
Z “minpX ,C q, δ “I , ξ “ i
i i i i XiďCi i C if δ “1.
i i
The i.i.d. samples pX q and pC q , of respective continuous distribution functions F and G, represent
i iďn i iďn
the lifetimes and censoring times of the individuals, and are supposed to be independent. For convenience,
we will suppose in this work that they are non-negative. The variables pC q form a discrete sample with
i iďn
valuesint1,...,Ku,andrepresentthecausesoffailureordeathofthenindividualsoritems. Itisimportant
to note that these causes are observed only when the data is uncensored (i.e. when δ “ 1), therefore we
i
only observe the ξ ’s, not the complete C ’s.
i i
One way of considering the failure times X is to write
i
X “minpX ,...,X q,
i i,1 i,K
where the variable X is a (rather artificial) variable representing the imaginary latent lifetime of the i-th
i,k
individualwhenthelatterisonlyaffectedbythek-thcause(theothercausesbeingabsent). Thisviewpoint
may be interesting in its own right, but we will not keep on considering it in the sequel, one reason being
that such variables X ,...,X cannot be realistically considered as independent, and their respective
i,1 i,K
distributions are of no practical use or interpretability (as explained and demonstrated in the competing
risks literature, these distributions are in fact not statistically identifiable, see Tsiatis (1975) for example).
The object of interest is the probability that a subject dies or fails after some given time t, due to the
k-th cause, for high values of t. This quantity, denoted by
Fspkqptq“PrX ąt, C “ks,
is related to the so-called cumulative incidence function Fpkq defined by
Fpkqptq“PrX ďt, C “ks.
NotethatFspkqptqisnotequalto1´Fpkşqptq,buttoPpC “kq´Fpkqptq,becauseFpkqisonlyasub-distribution
function. However we have Fspkqptq “ 8dFpkqpuq. In the sequel, the notation S¯p.q “ Sp8q´Sp.q will be
t
3
used, for any non-decreasing function S.
s
Inthispaper,weareinterestedininvestigatingthebehaviourofFpkqptqforlargevaluesoft. Thisamounts
to statistically study extreme values in a context of censored data under competing risks, and will lead us
s
to consider some extreme value index γ related to Fpkq, which will be defined in a few lines. Equivalently,
k
the object of interest is the high quantile xpkq “pFspkqq´ppq“inftxPR; Fspkqpxqěpu when p is close to 0,
p
which can be interpreted as follows (in the context of lifetimes of individuals or failure times of systems) :
in the presence of the other competing causes, a given individual (or item) will die (or fail), due to cause k
after such a time xpkq, only with small probability p. A nonparametric inference for quantiles of fixed (and
p
therefore not extreme) order, in the competing risk setting, has been already proposed in Peng and Fine
(2007).
Onewayofaddressingthisproblemcouldbethroughaparametricpointofview(seeCrowder (2001)for
further methods in the competing risk setting), however, the non-parametric approach is the most common
choice of people faced with data presenting censorship or competing risks. Of course, the standard Kaplan-
Meier method for survival analysis does not yield valid results for a particular risk if failures from other
causes are treated as censoring times, because the other causes cannot always be considered independent of
the particular cause of interest.
The commonly used nonparametric estimator of the cumulative incidence function Fpkq is the so-called
Aalen-Johansen estimator (see Aalen and Johansen (1978), or Geffray (2009) equation p7q) defined by
Fnpkqptq“ ÿ nδGsiICpiZ“´kq,
Ziďt n i
s s s
where G denotes the standard Kaplan-Meier estimator of G (and G pt´q denotes lim G psq), so that we
n n sÒt n
s
can introduce the following estimator for Fpkq :
Fsnpkqptq“ ÿ nδGsiICpiZ“´kq.
Ziąt n i
But if the value t considered is so high that only very few (if any) observations Z (such that C “k) exceed
i i
t, then this purely nonparametric approach will lead to very unstable estimations Fspkqptq of Fspkqptq. This is
n
whyasemiparametricapproachisdesirable,andtheonewewillconsiderhereistheoneinspiredbyclassical
extreme value theory.
First note that in this paper, we will only consider situations where the underlying distributions F and
G of the variables X and C are supposed to present power-like tails (also commonly named heavy tails),
and we will focus on the evaluation of the order of this tail. Our working hypothesis will be thus that the
s s
different functions Fpkq (for k “1,...,K) as well as G“1´G belong to the Fr´echet maximum domain of
attraction. In other words, we assume that they are (see Definition 1 in the Appendix) regularly varying at
infinity, with respective negative indices ´1{γ ,...,´1{γ and ´1{γ
1 K C
@1ďk ďK, @xą0, lim Fspkqptxq{Fspkqptq“x´1{γk and lim Gsptxq{Gsptq“x´1{γC. (1)
tÑ`8 tÑ`8
ř
Consequently,Fs “1´F “ K FspkqandHs “FsGs(thesurvivalfunctionofZ)areregularlyvarying(at`8)
k“1
with respective indices ´1{γ and ´1{γ, where γ “ maxpγ ,...,γ q and γ satisfies γ´1 “ γ ´1`γ ´1
F F 1 K F C
(these relations are constantly used in this paper).
The estimation of γ has been already studied in the literature, as it corresponds to the random (right)
F
censoring framework, without competing risks. We can cite Beirlant et al. (2007) and Einmahl et al.
(2008),wheretheauthorsproposetouseconsistentestimatorsofγ dividedbytheproportionofnon-censored
observations in the tail, or Worms and Worms (2014), where two Hill-type estimators are proposed for γ ,
F
based on survival analysis techniques. However, our target here is γ (for a fixed k “ 1,...,K) and the
k
point is that there seems to be no way to deduce an estimator of γ from an estimator of γ . Note that the
k F
useful trick used in Beirlant et al. (2007) and Einmahl et al. (2008) to construct an estimator of γ does
F
not seem to be extendable to this competing risks setting. To the best of our knowledge, our present paper
is the first one addressing the problem of estimating the cause-specific extreme value index γ .
k
4
Considering assumption (1), it is simple to check that, for a given k, we have
ż
1 `8
tÑlim`8Fspkqptq t logpu{tqdFpkqptq “ γk.
It is therefore most natural to propose the following (Hill-type) estimator of γ , for some given threshold
k
value t (assumptions on this threshold are detailed in the next section) :
n
ż ˆ ˙
γp “ φp puqdFpkqpuq where φp puq “ 1 log u I ,
n,k n n n Fsnpkqptnq tn uątn
which can be also written as
γpn,k “ nFsnpk1qptnqiÿ“n1loGsgnpZpZi{i´tnqqIξi“kIZiątn “ nFsnpk1qptnqZpiÿqątn lGosgnppZZppiiq´{t1nqqqδpiqICpiq“k,
where Z ď ... ď Z are the ordered random variables associated to Z ,...,Z , and δ and C are
p1q pnq 1 n piq piq
the censoring indicator and cause number which correspond to the order statistic Z . It is clear that this
piq
estimator is a generalisation of one of the estimators proposed in Worms and Worms (2014), in which the
situation K “ 1 (with only one cause of failure/death) was considered. The asymptotic result we prove in
the present work is then valid in the situation studied in the latter, where only consistency was proved and
a random threshold was used.
Our paper is organized as follows: in Section 2, we state the asymptotic normality result of the proposed
estimator, and of a corresponding estimator of an extreme quantile of the cumulative incidence function.
Section 5 is devoted to the proofs. In Section 3, we present some simulations in order to illustrate finite
sample behaviour of our estimator. Some technical aspects of the proofs are postponed to the Appendix.
2. Assumptions and Statement of the results
The central limit theorem which is going to be proved has the rate ?v where v “nFspkqpt qGspt q and
n n n n n
t is a threshold tending to 8 with the following constraint
n
v nÝÑÑ8`8 such that n´η0v nÝÑÑ8`8 for some η ą0. (2)
n n 0
s s
Ifwenotelk theslowlyvaryingfunctionassociatedtoFpkq (i.e. suchthatFpkqpxq“x´1{γklkpxqincondition
(1)), the second order condition we consider is the classical SR2 condition for l (see Bingham, Goldie and
k
Teugels (1987)),
l ptxq
@xą0, k ´1 tÑ„8 h pxq gptq p@xą1q, (3)
l ptq ρk
k
where g is a positive measurable function, slowly varying with index ρ ď 0, and h pxq “ xρk´1 when
k ρk ρk
ρ ă0, or h pxq“logx when ρ “0.
k ρk k
Theorem 1. Under assumptions p1q, p2q and p3q, if there exists λ ě 0 such that ?v gpt q nÝÑÑ8 λ, and if
n n
γ ăγ then we have
k C
?
v pγp ´γ q ÝdÑ N pλm,σ2q as nÑ8
n n,k k
where $
m“& 1´γγk2kρk if ρk ă0, and σ2 “ γk2 `p1`r2q´2cr˘,
% γ2 if ρ “0, p1´rq3
k k
s s
with c“lim Fpkqpxq{FpxqPr0,1s and r “γ {γ Ps0,1r.
xÑ8 k C
Remark 1. Note that when γ ăγ , then c“0, and, when γ “γ and c“1 (for instance when there is
k F k F
only one cause of failure/death), then σ2 reduces to γ2{p1´rq.
F
Proposition 1. Under assumptions p1q and p2q, we have
γp ÝPÑ γ as nÑ8.
n,k k
Remark 2. The condition γ ăγ (weak censoring) is not necessary for the consistency of γp .
k C n,k
5
Now, concerningtheestimationofanextremequantilexpkq (oforderp tendingto0)associatedtoFspkq,
pn n
we propose the usual Weissman-type estimator (in this heavy tailed context), associated to the threshold t
n
used in the estimation of γ ,
k ˜ ¸
Fspkqpt q γpn,k
xˆpkq “t n n ,
pn,tn n p
n
` ˘
where p is assumed to satisfy the constraint p “o Fspkqpt q . Remind that by definition Fspkqpxpkqq“p ,
n n n pn s n
and thus the definition of this estimator is based on the fact that, by the assumed regular variation of Fpkq,
the ratio Fspkqpxppknqq{Fspkqptnq is close to pxppknq{tnq´1{γk.
s
Corollary 1. UndertheassumptionsofTheorem1,ifinadditionρ ă0(in(3))andd “Fpkqpt q{p Ñ8
k n n n
satisfies the condition
?
v {logpd q nÝÑÑ8 8, (4)
n n
then (with λ, m and σ2 being defined in the statement of Theorem 1)
˜ ¸
?v xˆpkq
n pn,tn ´1 ÝdÑ N pλm,σ2q as nÑ8.
logpdnq xppknq
3. Simulations
In this section, a small simulation study is conducted in order to illustrate the finite-sample behaviour
of our new estimator in some simple cases, and discuss the main issues associated with the competing risks
setting.
For simplicity, we focus on the situation with two competing risks (K “ 2), also called causes below,
and our aim is the extreme value index γ associated to the first cause. Data are generated from one of the
1
following two models : for c , c non-negative constants satisfying c `c “ 1, we consider the following
1 2 1 2
s
(sub-)distribution for each cause-specific function Fpkq (k Pt1,2u) :
s
´ Fr´echet : Fpkqptq“ck expp´t´1{γkq, for tě0 ;
s
´ Burr : Fpkqptq“ck p1`tτk{βq´1{pγkτkq, for tě1, where τk ą0, β ą0.
The lifetime X, of survival function F¯ “F¯p1q`F¯p2q, is generated by the inversion method (with numerical
s
computation of F´1). Censoring times are then generated from a Fr´echet or a Burr distribution :
Gsptq“expp´t´1{γCqptě0q or Gsptq“p1`tτC{βq´1{pγCτCq ptě1q.
In this section, we consider (as it is often done in simulation studies) that the threshold t used in the
n
definition of our new estimator γˆ is taken equal to Z (i.e. we consider it as random). One aim of
n,1 pn´knq
this section is to show how our estimator (with random threshold)
1 ÿkn logpZ {Z q
γˆ1 “ nFsnp1qpZpn´knqqi“1 Gspnn´pZi`p1nq´i,npqnq´knq δpn´i`1qICpn´i`1q“1
of γ behaves when the proportion c of cause 1 events varies : we consider c P t1,0.9,0.7,0.5u, the case
1 1 1
c “1 corresponding to the simple censoring framework, without competing risk.
1
Anotheraimistoillustratetheimpactofdependencybetweenthecauses, whenestimatingthetail. The
starting point is that, if cause 2 could be considered independent of cause 1, then we could (and would)
include it in the censoring mechanism and we would be in the simple random censoring setting, without
competing risk. In this case, it would be possible to estimate γ by one of the following two estimators, the
1
first one being proposed in Beirlant et al. (2007) (a Hill estimator weighted with a constant weight), and
thesecondoneinWormsandWorms (2014)(aHillestimatorweightedwithvaryingKaplan-Meierweights):
1 ÿkn 1
γˆpBDFGq “ logpZ {Z q (5)
1 k pˆ pn´i`1q pn´knq
n i“1 1
γˆ1pKMq “ nFs pZ1 q ÿkn δpnGs´i`1pqZICpn´i`1qq“1 logpZpn´i`1q{Zpn´knqq, (6)
n,b pn´knq i“1 n,b pn´i,nq
6
3 3 3 3
Bias 0.0 c1=1 MSE 0.00 c1=1 Bias 0.0 c1=1 MSE 0.00 c1=1
01 ccc111===000...975 00 ccc111===000...975 01 ccc111===000...975 00 ccc111===000...975
−0. 100 300 0.0 100 300 −0. 100 300 0.0 100 300
kn kn kn kn
(a) Fr´echetcase, γ1“0.1, γ2“0.25, γC “0.3 (b) Samecaseas(a)butforBurrdistribution
3 3 3 3
Bias 0.0 c1=1 MSE 0.00 c1=1 Bias 0.0 c1=1 MSE 0.00 c1=1
01 ccc111===000...975 00 ccc111===000...975 01 ccc111===000...975 00 ccc111===000...975
−0. 100 300 0.0 100 300 −0. 100 300 0.0 100 300
kn kn kn kn
(c) Fr´echetcase, γ1“0.1, γ2“0.25, γC “0.2 (d) Samecaseas(c)butforBurrdistribution
3 3 3 3
Bias 0.0 c1=1 MSE 0.00 c1=1 Bias 0.0 c1=1 MSE 0.00 c1=1
01 ccc111===000...975 00 ccc111===000...975 01 ccc111===000...975 00 ccc111===000...975
−0. 100 300 0.0 100 300 −0. 100 300 0.0 100 300
kn kn kn kn
(e) Fr´echetcase, γ1“0.25, γ2“0.1, γC “0.45 (f) Samecaseas(e)butforBurrdistribution
Figure1: ComparisonofbiasandMSE(respectivelyleftandrightineachsubfigure)ofγpn,k fordifferentvaluesofc1;infigures
paq,pcqandpeq,X andC areFr´echetdistributed,butinfigurespbq,pdqandpfqtheyareBurrdistributed.
7
5 5
2 2
as 0.03 SE 0.00 as 0.03 SE 0.00
Bi M Bi M
0.01 0000 0.01 0000
− 100 300 0. 100 300 − 100 300 0. 100 300
kn kn kn kn
(a) Fr´echetcase, γ1“0.1,γ2“0.25,γC “0.3,andc1“1 (b) Case(a)butwithc1“0.9
5 5
2 2
as 0.03 SE 0.00 as 0.03 SE 0.00
Bi M Bi M
0.01 0000 0.01 0000
− 100 300 0. 100 300 − 100 300 0. 100 300
kn kn kn kn
(c) Fr´echetcase, γ1“0.1,γ2“0.25,γC “0.2,andc1“1 (d) Case(c)butwithc1“0.9
3 3
Bias 0.03 MSE 0.00 Bias 0.03 MSE 0.00
1 0 1 0
0 0 0 0
−0. 100 300 0.0 100 300 −0. 100 300 0.0 100 300
kn kn kn kn
(e) Fr´echetcase, γ1“0.1,γ2“0.25,γC “0.45,andc1“1 (f) Case(e)butwithc1“0.9
Figure 2: Comparison of bias and MSE (respectively left and right in each subfigure) for γp (plain thick), γˆpBDFGq (plain
n,k 1
thin)andγˆpKMq (dashed),forFr´echetdistributedX andC.
1
ř
Fwsnh,ebrea,nidn EGsqnu,bataioren pb5aqs,epdˆ1o“nkt1nhe kiδ˜“ni1“δpnδ´iiI`C1iq“I1C.pn´Ti`h1eq“se1,tawnodeinstiEmqautaotrisoncopn6qs,idtehretKhaepulannceMnseoierredestliifmetaitmoerss
associated to cause 2 as independent censoring times. Comparing our new estimator with these latter two
estimators, when c ă1, will empirically prove that considering cause 2 as a competing risk independent of
1
cause 1 has a great (negative) impact on the estimation of γ . Note that when c “ 1, the new estimator
1 1
γˆ and γˆpKMq are exactly the same (therefore the thick and dashed lines in sub-figures (a), (c) and (e) of
1 1
Figures 2 and 3 are overlapping, identical).
We address these two aims for each set-up (Fr´echet, or Burr), by generating 2000 datasets of size 500,
with three configurations of the triplet pγ ,γ ,γ q : p0.1,0.25,0.3q (γ ăγ , moderate censoring γ ąγ ),
1 2 C 1 2 C F
p0.1,0.25,0.2q (γ ă γ , heavy censoring γ ă γ ), or p0.25,0.1,0.45q (γ ą γ , moderate censoring γ ă
1 2 C F 1 2 C
γ ). Median bias and mean squared error (MSE) of the different estimators are plotted against different
F
values of k , the number of excesses used. When Burr distributions are simulated, the parameter β is taken
n
equal to 1, and the parameters pτ ,τ ,τ q are taken equal to p12,6,5q in configurations 1 and 2, and to
1 2 C
p6,12,5q in configuration 3.
Figure 1 illustrates the behaviour of our estimator when c varies. In terms of bias and MSE, we can
1
see that the first configuration is a little better than the second one, which is itself much better than the
third one. We observed this phenomenon in many other cases, not reported here : our estimator behaves
best when itis the smallest parameter γ which is estimated, and when the censoring isnot too strong. Our
k
simulations also show that the quality of our estimator (especially in terms of the MSE) diminishes with c .
1
Figures 2 and 3 present the comparison between our new estimator and the ones described in (5) and
8
6 5 6 5
0 0 0 0
as 0. SE 0.0 as 0. SE 0.0
Bi M Bi M
0 0 0 0
0 0 0 0
0. 100 300 0.0 100 300 0. 100 300 0.0 100 300
kn kn kn kn
(a) Burrcase, γ1“0.1,γ2“0.25,γC “0.3,andc1“1 (b) Case(a)butwithc1“0.9
6 5 6 5
0 0 0 0
as 0. SE 0.0 as 0. SE 0.0
Bi M Bi M
0 0 0 0
0 0 0 0
0. 100 300 0.0 100 300 0. 100 300 0.0 100 300
kn kn kn kn
(c) Burrcase, γ1“0.1,γ2“0.25,γC “0.2,andc1“1 (d) Case(c)butwithc1“0.9
as 0.04 SE 0.005 as 0.04 SE 0.005
Bi M Bi M
4 4
0 0 0 0
−0. 100 300 0.00 100 300 −0. 100 300 0.00 100 300
kn kn kn kn
(e) Burrcase, γ1“0.1,γ2“0.25,γC “0.45,andc1“1 (f) Case(e)butwithc1“0.9
Figure 3: Comparison of bias and MSE (respectively left and right in each subfigure) for γp (plain thick), γˆpBDFGq (plain
n,k 1
thin)andγˆpKMq (dashed),forBurrdistributedX andC.
1
9
(6). A general conclusion (confirmed by other simulations not reported here) is that γˆpBDFGq and γˆpKMq
1 1
behave worse in most cases, even for a value of c of 0.9, which is only a slight modification of the situation
1
without competing risk (c “ 1). Therefore, a contamination of the cause 1 distribution by another cause
1
rapidly yield inadequate estimations of γ if dependency between causes is ignored ; this conclusion is true
1
forboth γˆpBDFGq and γˆpKMq, butto agreater extent for γˆpBDFGq . Inthe thirdconfiguration pγ ,γ ,γ q“
1 1 1 1 2 C
p0.25,0.1,0.45q, the improvement provided by γˆ (with respect to γˆpKMq) becomes notable when c drops
1 1 1
below 0.7.
4. Conclusion
In this paper, we consider heavy tailed lifetime data subject to random censoring and competing risks,
and use the Aalen-Johansen estimator of the cumulative incidence function to construct an estimator for
the extreme value index associated to the main cause of interest. To the best of our knowledge, this is the
first estimator proposed in this context. Its asymptotic normality is proved and a small simulation study
exhibiting its finite-sample performance shows that accounting for the dependency of the different causes is
important, but that the bias can be particularly high. Estimating second order tail parameters would then
be interesting in order to reduce this bias. A first step towards this aim could be to study the following
moments
1 ÿn logαpZ {t q
Mnpαq “ nFsnpkqptnqi“1 GsnpZii´qn Iξi“kIZiątn,
which asymptotic behaviour can be derived following the same lines as in the proof of Theorem 1.
5. Proofs
This section is essentially devoted to the proof of the main Theorem 1. Some hints about the proof of
the consistency result contained in Proposition 1 are given in Subsection 5.3, and Corollary 1 is proved in
Subsection 5.4.
We adopt a strategy developed by Stute in Stute (1995) in orşder to prove his Theorem 1.1, a well-
known result which states that a Kaplan-Meier integral of the form φdF can be approximated by a sum
n
of independent terms. This idea is used in Suzukawa (2002) in the context of competing risks. We thus
ş
intend to approximate γp by the integral γr “ φ dFpkq of some deterministic function φ , with respect
n,k n,k n n n
to the Aalen-Johansen estimator, and approximate this integral by the mean γq of independent variables
n,k
U (defined a few lines below). The passage from γp to γr (which amounts to replacing Fspkqpt q by
i,n n,k n,k n n
Fspkqpt q in the denominator of γp ) will imply an additional sum of independent variables V , which will
n n,k i,n
participate to the asymptotic variance of our estimator.
However, a major difference with Stute (1995) or Suzukawa (2002) is that the function we integrate
here, φnpuq “ Fspkq1ptnqlogpu{tnqIuątn, is not only an unbounded function, depending on n, but it also has
a ”sliding” support rt ,`8r, which is therefore always close to the endpoint `8 of the distribution H. In
n
Stute (1995), a crucial point of the proof consists in temporarily considering that the integrated function φ
has a support which is bounded away from the endpoint of H (condition (2.3) there). Considering the kind
of function φ we have to deal with here, we cannot follow the same strategy : dealing with the remainder
n
terms will thus be a particularly challenging part of our work. Finally note that, in order to deal with the
ratio Fspkqpt q{Fspkqpt q (and somehow approximate γp by γr ) we will have to consider simultaneously
n n n n,k n,k
integrals (with respect to Fpkq) of φ and of another function g , defined below, which basically shares the
n n n
same flaws as φ .
n
10