nonparametric inference under competing risks and selection-biased sampling

Journal of Multivariate Analysis 99 (2008) 589 –605www.elsevier.com/locate/jmva

Nonparametric inference under competing risks andselection-biased sampling

Jean-Yves Dauxoisa,∗, Agathe Guillouxb

aDépartement de Mathématiques, UMR CNRS 6623, Université de Franche Comté, 16, route de Gray, 25030 Besançon

Cedex, FrancebLSTA, Université de Paris VI, 175 rue de Chevaleret, 75013 Paris, France

Received 6 July 2006Available online 20 February 2007

Abstract

The aim of this paper is to carry out statistical inference in a competing risks setup when only selection-biased observation of the data of interest is available. We introduce estimators of the cumulative incidencefunctions and study their joint large sample behavior.© 2007 Elsevier Inc. All rights reserved.

AMS 2000 subject classification: 62G05; 62N01; 62N02; 60F17; 62G15; 62G20

Keywords: Competing risks; Selection bias; Right-censored data; Cumulative incidence function; Nonparametricinference; Weak convergence of processes; Skorohod space; Martingale; Counting process

1. Introduction

Consider a homogeneous population of individuals who are subjected to K competing causesof death. Recall that the modeling of this situation using latent random variables (r.v.), assumesthat there exist absolutely continuous positive r.v. XIn

k , for k = 1, . . . , K , which denotes the ageat the time of death in the hypothetical situation where the kth cause is the only possible cause.These r.v.’s are often dependent. Each death is due to a single cause and, when a member of thispopulation dies, the age at death T In and the underlying cause of death �In are recorded. Thismeans that, in the competing risks setup, we only observe the lifetime T In = ∧jX

Inj (where x∧y

denotes the minimum between x and y) and �In which is equal to k when the underlying cause ofdeath is the kth one. Thus, the r.v.’s XIn

k , for k = 1, . . . , K , are unobservable and called “latent

∗ Corresponding author. Fax: +333 81 66 66 23.E-mail addresses: [email protected] (J.-Y. Dauxois), [email protected] (A. Guilloux).

0047-259X/$ - see front matter © 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.jmva.2007.02.001

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590 J.-Y. Dauxois, A. Guilloux / Journal of Multivariate Analysis 99 (2008) 589 –605

survival times”. Let G be the survivor function of T In. The distribution of the pair (T In, �In) isspecified by the cumulative incidence functions (CIFs)

Gk(t) = P(T In� t, �In = k),

for t �0 and k = 1, . . . , K . The importance of the CIFs is well recognized in demography,epidemiology and survival analysis literature. In the competing risks setup, they are the primarymeasures of interest. For examples of the use of the CIF see Lin [34], Pepe and Mori [41], andGaynor et al. [21]. Estimation and testing in this setup have been studied by several authors. Wemention, among others, Babu et al. [8], Benichou and Gail [9], Aly et al. [3], Dauxois and Kirmani[17]. Good reviews of this topic can be found in the books of Crowder [16] and Andersen et al.[4].

The aim of this paper is to carry out statistical inference in this competing risks frameworkwhen only selection-biased observation of the pair (T In, �In) is available. Indeed it may happen, inpractice, that drawing a direct sample from (T In, �In) is impossible and that one can only observea sample from (T , �) with distribution defined as follows. Let us write Fk(t) = P(T � t, � = k),with k = 1, . . . , K , and F the CIFs and the survival function associated to (T , �), respectively.The distribution of the observable random pair (T , �), called the selection-biased random pair,is supposed to be related to the distribution of the initial random pair through the followingequations:

Fk(t) = −∫ t

0 w(x) dGk(x)∫ ∞0 w(x) dG(x)

, (1)

for k = 1, . . . , K , and

F(t) = 1 − F (t) =∫ t

0 w(x) dG(x)∫ ∞0 w(x) dG(x)

, (2)

for all t �0. The so-called bias function w is supposed to be known and such that all the integralsexist. For the sake of simplicity, we shall refer to Gk , for k = 1, . . . , K , and G as the “unbiased”CIFs and the survivor function, while the ones related to the pair (T , �) will be called “biased”.

Outside the competing risks setup, selection-biased sampling (SBS) is now a relatively well-known phenomenon which was first noted by Fisher [20]. It appears when instead of observing ar.v. of interest XIn (with p.d.f. g(x)) one observes the r.v. X with p.d.f. proportional to w(x)g(x),where w is the bias function. For example, this has been encountered in industrial applications(see [15]), in biostatistics (see [12,13]) and in econometrics (see [27]).

Statistical inference under SBS has been considered by Cox [15], Rao [42] and Patil and Rao[40], among others. More recently, some interesting results on nonparametric density estimationunder selection bias have been obtained by Gill et al. [24], Lo [36], Wu and Mao [57], Wu [56],Sun and Woodroofe [45], Lloyd and Jones [35], Lee and Berger [32] and Efromovich [18].

A particular case, which is of specific interest, is when the bias function w is the identity,i.e. w(x) = x, and is called length-biased (LBS) or size-biased sampling. It has been notedby McFadden [38] for lengths of intervals in a stationary point process, by Blumenthal [11]in industrial life testing and by Cox [16] for estimating the distribution of fiber lengths in afabric. Feinlieb and Zelen [19] recognized LBS in screening for chronic diseases and Simon [44]noted its relevance in etiologic studies. Probability proportional to size sampling also occurs inenvironmetrics. For instance, this has been underlined by Muttlak and McDonald [39] in samplesof shrub widths when shrubs are selected using the line-intercept method. See also Gove [25,26].

J.-Y. Dauxois, A. Guilloux / Journal of Multivariate Analysis 99 (2008) 589 –605 591

Vardi [50] was the first to consider nonparametric estimation of the survivor function (i.e. whenonly one cause is acting) in the presence of LBS. We refer to Vardi [50–52], Gill et al. [24],Vardi and Zhang [53] and Wang [54] for further theoretical developments. de Uña-Àlvarez [46]studied the NPMLE of the survivor function under random right censoring. He also derived amoment based approach in de Uña-Àlvarez [47]. Moreover, Asgharian et al. [5] and Asgharianand Wolfson [7] obtained the unconditional NPMLE of the survivor function and its asymptoticproperties when the data are length-biased and under right censoring.

As far as we know, no statistical inference has been carried out under competing risks togetherwith SBS. The only result available can be found in the interesting and very broad paper ofHuang and Wang [28]. Estimation of CIFs under length bias (i.e. w(x) = x) is considered in thispaper, among other topics. But this is not in the general frame of SBS. Multivariate pointwise weakconvergence of their estimators is given, under the restriction of compactly supported distributionsand without random right-censoring.

Selection-biased observation of competing risks data can often be encountered in practice.In particular, it should be the case in the situation considered by Wang [55] if the gap timebetween the first and the second event (denoted by Z in her paper) was due to several competingrisks. Also, we show in the Appendix that SBS and competing risks can be concomitant insurvival analysis when the distribution of (T In, �In) has to be estimated from a cross-sectionalsample. Indeed, we consider a population where the time of births are distributed accordingto a nonhomogeneous Poisson process. We suppose that, in this population, a cross-sectionalobservation is driven at time t0. Then, instead of observing a random sample from (T In, �In),one observes a random sample from (T , �) with distribution equal to the conditional distributionof (T In, �In), given that the individual is alive at t0. Mathematical developments based on pointprocess theory prove that the distributions of (T In, �In) and (T , �) are such that Eqs. (1) and (2)are fulfilled (seeAppendix for details). This generalizes to any bias function the results obtained byLund [37] and Van Es et al. [49] with w(x) = x, which corresponds to the homogeneous Poissonprocess case.

The objective of the present paper is firstly to develop estimation of the CIFs under SBS,allowing the possibility of random right censoring and, secondly, to state a joint weak convergencetheorem for the processes associated to these estimators (see our Theorem 2).A preliminary step isto extend to [0, ∞] the well-known theorem of Aalen [1] about the functional weak convergenceof the CIF estimators. Note that this extended convergence (see our Theorem 1) may be ofindependent interest in classical survival analysis.

2. Estimation

2.1. Notation

As stated in the previous section, let us suppose that we are interested in estimating the CIFsGk , for k = 1, . . . , K , and the survivor function G of a lifetime with competing risks (T In, �In)

but under the restriction that only selection-biased observation of this random pair is available.That is, we can just observe a sample from (T , �) with distribution defined in (1) and (2).

Now, in order to handle the most general situation, and even if it is not needed or justified for allpractical situations where SBS occurs (see below), let us furthermore suppose that each individualof our selection-biased sample is subjected to independent random right censoring. Let C denotethis censoring r.v., suppose that it is independent from the r.v. T and let H be its survivor function.


Hence, instead of the (Ti, �i), we observe for i = 1, . . . , n,{

T ∗i = Ti ∧ Ci,

�∗i = �iI ({Ti �Ci}),

where I (A) denotes the indicator function of the set A. Let S denote the survivor function ofT ∗ which, by independence, is equal to F H . This type of censoring has been introduced in theLBS setup (i.e. w(x) = x) by de Uña-Àlvarez [46], in particular. It is clear from Asgharian[5] that this kind of censoring does not correspond to a “loss to follow up” or “end of study”mechanism. Indeed, independence between lifetime and censoring time in the initial populationis lost under LBS. Note, however, that in case of follow-up studies, there are several ways totake into account independent censoring before SBS. One possible approach is provided by themodel of censoring introduced by Asgharian et al. [6]. Another one is given by our strategy whereindependent censoring can be taken as another cause of death before SBS.

2.2. Estimators

For the sake of simplicity of notations and mathematical developments, we consider in thesequel only the case where two causes of death are acting, i.e. K = 2. However, all the definitionsand results below can be extended to the general case (K causes of death) in a straightforwardmanner.

First, it has to be noted that, in the survival analysis area, there are some links between cross-sectional observation and delayed entries. The two main approaches for nonparametric inferencein the setup of delayed entries (in case of a single risk, i.e. when K = 1) are known as: lefttruncation and left filtering (see [30] or [4]). They are both based on an invariant property fulfilledby the intensity under left truncation or filtering, when the time of entry is known. Thus, straightnonparametric estimation in the case of delayed entries can be carried out using Nelson–Aalentype estimators of the intensities (see again [4,30]). But our problem is rather different. Indeed, ourscope of application is larger than the one of cross-sectional observation in survival analysis. Moreimportantly, we do not assume the truncation time to be available. Hence, straight nonparametricestimation of the (cause specific) hazard rate is, more or less, hopeless in our case and anotherway has to be found.

Thus, our main idea in order to estimate the unbiased functions is to use the inverse formula

Gk(t) = −

∫ t

01

w(x)dFk(x)

∫ ∞0

1

w(x)dF (x)

, (3)

for k = 1, 2, and

G(t) = 1 − G(t) =

∫ t

01

w(x)dF (x)

∫ ∞0

1

w(x)dF (x)

, (4)

which are easily obtained from Eqs. (1) and (2). Indeed, estimation of the biased functions (Fk ,for k = 1, 2, and F ) on the basis of SBS can be driven using Aalen [1] and Aalen & Johansen’s


[2] method. In this order, let Nk , for k = 1, 2, be the counting process associated to the kth causeof death and Y the “at-risk” process, i.e.

Nk(t) =n∑

i=1

I ({T ∗i � t, �∗

i = k}), k = 1, 2,

Y (t) =n∑

i=1

I ({T ∗i � t}).

Let us also denote by N· = N1 + N2 the aggregated counting process and T ∗(1) � · · · �T ∗

(n) the

ordered statistics. Then the Kaplan–Meier estimator of the survivor function F is given by

F(t) = �]0,t]

(1 − d�

)=

∏

i:T ∗(i)

� t

(1 −

�N·(T ∗(i))

Y (T ∗(i))

),

where � is the product-integral (see [23]) and � is the Nelson–Aalen estimator of the biasedcumulative hazard rate function � (see [4]). Recall also that the Aalen–Johansen estimators ofthe CIFs Fk are given by

Fk(t) =∫ t

0

F(x−) d�k(x) =∫ t

0

F(x−)J (x)dNk(x)

Y (x), (5)

where �k is the Neslon–Aalen estimator of the biased cause specific cumulative hazard ratefunction and J (t) = I ({Y (t) > 0}). See also Andersen et al. [4, p. 298] and following.

From this and relations (3) and (4), plug-in estimators of the Gk , for k = 1, 2, and G are easilyobtained and defined, for all t > 0, by

Gk(t) = −

∫ t

01

w(x)dFk(x)

∫ ∞0

1

w(x)dF(x)

(6)

for k = 1, 2 and

G(t) =

∫ t

01

w(x)dF(x)

∫ ∞0

1

w(x)dF(x)

. (7)

These estimators can be seen as a generalization of many estimators introduced earlier in theliterature.

First, in the case of LBS and K = 1 (i.e. a single cause of death is acting), only Eq. (7) withw(x) = x has to be considered. In this case, our estimator of the unbiased survivor function isequal to the one introduced by de Uña-Álvarez [46]. But due to a different type of censoring, thisestimator is different from the one of Asgharian et al. [6]. However, when there is no censoring,or equivalently H ≡ 1, our estimator and the one of Asgharian et al. [6] turn out to be equal.

Secondly, let us consider the competing risks setup under LBS and without censoring. It is wellknown that the Kaplan–Meier estimator of F is equal to the empirical survival function

F(t) = 1 − Fn(t),


where Fn(t) = 1n

∑ni=1 I ({Ti � t}). In this case, estimators of Eqs. (6) and (7) can be simplified

and we get, for k = 1, 2:

Gk(t) =1

n

∫ t

01

xdNk(x)

∫ ∞0

1

xdFn(x)

=∑n

i=11Ti

I ({�i = k})I ({Ti � t})∑n

i=11Ti

.

These estimators are just those given by Huang and Wang [28, p. 1409].

3. Large sample behavior

Our estimators Gk , for k = 1, 2, and G introduced in Eqs. (6) and (7) are functions of thebiased ones Fk , for k = 1, 2, and F . Thus, the large sample study of the associated processeswill straightforwardly be based on the results available for the last ones. Unfortunately, the weakconvergence result proved by Aalen [1], which can also be found in Andersen et al. [4, p. 319]in a more general framework, is only established on intervals of the form [0, �] where � < ∞.However, as it is clear from Eqs. (6) and (7), results on the whole line [0, ∞] will be needed tohandle the denominators of our estimators. Hence, our first step is to extend this asymptotic resulton the whole line, i.e. in the Skorohod space of càdlàg functions D

3[0, ∞]. This is the aim ofTheorem 1, which may be of independent interest in classical survival analysis, since it generalizesto the CIFs estimators the result of Gill [22] on the product limit estimator. For example, it allowsconstruction of a confidence band for F and the CIFs Fk on the largest possible interval and notonly on bounded intervals of the form [0, �], with � < +∞.

Let Assumption A be defined as

Assumption A :∫ ∞

0

dF(x)

H (x−)< ∞.

Theorem 1. Under Assumption A, we have the following weak convergence result in D3[0, ∞],

as n goes to ∞:

Z =

⎛⎜⎝

Z0

Z1

Z2

⎞⎟⎠ ≡

√n

⎛⎜⎝

F − F

F1 − F1

F2 − F2

⎞⎟⎠ D−→ Z∞ =

⎛⎜⎝

Z∞0

Z∞1

Z∞2

⎞⎟⎠ ,

where (Z∞0 , Z∞

1 , Z∞2 )′ is a mean-zero Gaussian process defined by

Z∞0 = FU0,

Z∞k (·) =

∫ ·

0(Fk(·) − Fk(u)) dU0(u) +

∫ ·

0F (u) dUk, for k = 1, 2


and U1 and U2 are square integrable, orthogonal and mean zero Gaussian local martingales with

covariance functions given, for k = 1, 2, by

〈Uk(s), Uk(t)〉 =∫ s∧t

0

dFk(u)

F (u)S(u−)

and U0 = −(U1 + U2).

Note that, since we supposed that the probability measures associated to F and Fk (k = 1, 2)are absolutely continuous with respect to the Lebesgue measure on R+, the Gaussian martingalesZ∞

k (k = 0, 1, 2) have a.s. continuous paths.

Proof. From Aalen [1] (see also [4, p. 319]), we know that the process Z is weakly convergentin D

3[0, �], for � < +∞, to a process Z defined by

Z(·) =

⎛⎝

Z0(·)Z1(·)Z2(·)

⎞⎠ =

⎛⎝

F (·)U0(·)∫ ·0 F1(u) dU0(u) − F1(·)U0(·) +

∫ ·0 F (u) dU1(u)∫ ·

0 F2(u) dU0(u) − F2(·)U0(·) +∫ ·

0 F (u) dU2(u)

⎞⎠ .

Under Assumption A, the processes U1 and U2 are orthogonal mean-zero square integrableGaussian martingales with variance function given by, for k = 1, 2 and 0�s, t < � < ∞,

〈Uk(s), Uk(t)〉 =∫ s∧t

0

dFk(u)

F (u)S(u−),

U0 = −(U1 +U2), Fk(t) = P(T � t, � = k) and F = F1 + F2. Hence, U0 is a mean-zero squareintegrable Gaussian martingale with variance function:

〈U0(s), U0(t)〉 =∫ s∧t

0

dF(u)

F (u)S(u−).

Thus, the first step in the proof of Theorem 1 is to show that the process Z can be extended to [0, ∞].But, underAssumption A, using a result from Gill [22, Remark 2.2], the process Z0 = FU0 can beextended to [0, ∞]. As a matter of fact we have a.s. Z0(t) → 0, as t → ∞. The same conclusioncan be drawn for the −FkU0 parts of the processes Zk , for k = 1, 2. Moreover, under AssumptionA, the remaining terms,

∫ ·0 Fk(u) dU0(u)+

∫ ·0 F (u) dUk(u), are square integrable martingales on

[0, ∞[ with variance functions⟨∫ s

0Fk(u) dU0(u) +

∫ s

0F (u) dUk(u),

∫ t

0Fk(u) dU0(u) +

∫ t

0F (u) dUk(u)

⟩

=∫ s∧t

0

F 2k (u) dF (u)

F (u)S(u−)+

∫ s∧t

0

F 2(u) dFk(u)

F (u)S(u−)− 2

∫ s∧t

0

Fk(u)F (u) dFk(u)

F (u)S(u−),

for k = 1, 2. Assumption A guarantees that these “stochastic integral pieces” of the Zk , fork = 1, 2, can also be extended to [0, ∞] by simply taking their limits. Hence, the extension Z∞

of Z to [0, ∞]3 is well defined.Let us now consider the process Z

T ∗(n) , defined as the process Z stopped at time T ∗

(n). The second

step in our proof of Theorem 1 is to establish its weak convergence to Z∞ in D3[0, ∞]. Following

Gill [22] in his proof of Theorem 2.1, this convergence will be obtained using Theorem 4.2 of


Billingsley [10]. From what has already been proved, the hypothesis which remains to check isthe “tightness at ∞” of Z

T ∗(n) . That is, we have to show that for all ε > 0,

lim�↑∞

lim supn→∞

P

⎛⎝ sup

�� t �T ∗(n)

‖Z(t) − Z(�)‖3 > ε

⎞⎠ = 0,

where ‖Z(t) − Z(�)‖3 = |Z0(t) − Z0(�)| + |Z1(t) − Z1(�)| + |Z2(t) − Z2(�)|. This will befulfilled if, for all k = 0, 1, 2 and for all ε > 0, we have

lim�↑∞

lim supn→∞

P

⎛⎝ sup

�� t �T ∗(n)

|Zk(t) − Zk(�)| >ε

3

⎞⎠ = 0. (8)

The case k = 0 has been proved by Gill [22], under Assumption A. In order to deal with theZk’s, for k = 1, 2, note that we have for t �T ∗

(n):

Zk(t) =√

n(Fk(t) − Fk(t)) =√

n

(∫ t

0

F(u−) d�k(u) −∫ t

0F (u) d�k(u)

)

=√

n

∫ t

0

F(u−)d(�k − �k)(u) +√

n

∫ t

0(F(u−) − F (u)) d�k(u),

from Eq. (5). Hence, for �� t :

Zk(t) − Zk(�) =√

n

∫ t

�

F(u−)d(�k − �k)(u)

+√

n

∫ t

�(F(u−) − F (u)) d�k(u). (9)

Let us first investigate the behavior of the first term on the right-hand side of this equality. Theprocess

√n

∫ ·

�

F(u−)d(�k − �k)(u)

is a local square integrable martingale on [�, �], where � < +∞, with predictable variation processgiven by

⟨√n

∫ ·

�

F(u−)d(�k − �k)(u)

⟩=

∫ ·

�n

F2(u−)J (u)

Y (u)d�k(u).

By the inequality of Lenglart [33] and Lemmas 2.6 and 2.7 of Gill [22], we have, for any � < ∞,� > 0 and all � ∈ (0, 1):

P

⎛⎝ sup

�� t ��∧T ∗(n)

∣∣∣∣√

n

∫ t

�

F(u−)d(�k − �k)(u)

∣∣∣∣ >ε

6

⎞⎠

�36�

ε2 + P

⎛⎝

∫ �∧T ∗(n)

�n

F2(u−)J (u)

Y (u)d�k(u) > �

⎞⎠

�36�

ε2 + � + e(1/�)e−1/� + P

(∫ �

��−3 F 2(u)

S(u−)d�k(u) > �

).


The last term of the inequality may be rewritten as

P

(∫ �

��−3 F 2(u)

S(u−)d�k(u) > �

)= P

(∫ �

�

�−3

H (u−)dFk(u) > �

).

Hence, letting � ↑ ∞ and taking � =∫ ∞�

�−3

H (u−)dFk(u), which is possible under Assumption A,

we obtain

P

⎛⎝ sup

�� t �T ∗(n)

∣∣∣∣√

n

∫ t

�

F(u−)d(�k − �k)(u)

∣∣∣∣ >ε

6

⎞⎠

�

36∫ ∞�

�−3

H (u−)dFk(u)

ε2 + � + e(1/�)e−1/�.

This states that

lim�↑∞

lim supn→∞

P

⎛⎝ sup

�� t �T ∗(n)

∣∣∣∣√

n

∫ t

�

F(u−)d(�k − �k)(u)

∣∣∣∣ >ε

6

⎞⎠ = 0.

From this and Eq. (9), the result of Eq. (8) will be guaranteed if we prove that

lim�↑∞

lim supn→∞

P

⎛⎝ sup

�� t �T ∗(n)

∣∣∣∣√

n

∫ t

�(F(u−) − F (u)) d�k(u)

∣∣∣∣ >ε

6

⎞⎠ = 0.

But, this follows from arguments given by Gill in the proof of his Theorem 2.1. Indeed, we havefor all t �T ∗

(n),

√n

∫ t

�(F(u−) − F (u)) d�k(u) =

∫ t

�

Z0(u−)

F (u)dFk(u) = −

∫ t

�

Z0(u)

F (u)dFk(u),

by continuity of Fk , and Fk is nonincreasing.We are now in a position to apply Theorem 4.2. of Billingsley [10], which gives the following

weak convergence in D3[0, ∞]:

ZT ∗

(n) =

⎛⎝

Z0

Z1

Z2

⎞⎠

T ∗(n)

=√

n

⎛⎝

F − F

F1 − F1

F2 − F2

⎞⎠

T ∗(n)

D→ Z∞. (10)

Our last step is now to prove that this weak convergence still holds without stopping Z at timeT ∗

(n). Indeed, by Lemma 3 of Ying [58] and Assumption A, we have

√nF (T ∗

(n))P→ 0,


as n → ∞. This leads to the following inequality:√

n supt∈[0,∞)

‖ZT ∗(n)(t) − Z(t)‖3 �

√n

(F (T ∗

(n)) + F1(T∗(n)) + F2(T

∗(n))

)

= 2√

nF (T(n))P→ 0,

as n goes to ∞. With this remark and (10), we get the desired result. �

This theorem enables us to deal with the asymptotic behavior of the unbiased estimators.

Theorem 2. Under Assumption A, the following weak convergence result holds in D3[0, ∞], as

n goes to ∞:

√n

⎛⎝

G − G

G1 − G1

G2 − G2

⎞⎠ D→

⎛⎝

L0L1L2

⎞⎠ , (11)

where the limiting process is Gaussian with mean zero and defined by

L0(·) =

∫ ·0

1

w(x)dZ∞

0 (x)

∫ ∞0

1

w(x)dF (x)

− G(·)

∫ ∞0

1

w(x)dZ∞

0 (x)

∫ ∞0

1

w(x)dF (x)

and, for k = 1, 2,

Lk(·) = Gk(·)

∫ ∞0

1

w(x)dZ∞

0 (x)

∫ ∞0

1

w(x)dF (x)

−

∫ t

01

w(x)dZ∞

k (x)

∫ ∞0

1

w(x)dF (x)

.

Moreover, the estimators G, G1 and G2 are asymptotically efficient.

Proof. We denote by D�[0, ∞] the space of càdlàg functions of bounded variation such that, iff ∈ D�[0, ∞], then

∫ ∞0

1w(x)

df (x) < ∞. In addition, let �t be defined as

�t : D2�[0, ∞] → D[0, ∞]

(f, g) →

∫ t

01

w(x)df (x)

∫ ∞0

1

w(x)dg(x)

. (12)

We have

√n

⎛⎝

G − G

G1 − G1

G2 − G2

⎞⎠ =

√n

⎛⎜⎝

�t (F, F) − �t (F , F )

�t (F1, F ) − �t (F1,F)

�t (F2, F ) − �t (F2,F)

⎞⎟⎠ .

Thus, the weak convergence in Theorem 2 may be obtained by using the functional delta method inits version of Theorem 3.9.4 of Van der Vaart and Wellner [48]. The Hadamard differentiability of


�t is implied by some elementary algebraic developments and its differential is equal to

D�(f,g)t (h1, h2) =

∫ t

01

w(x)dh1(x)

∫ ∞0

1

w(x)dg(x)

−

∫ t

01

w(x)df (x)

∫ ∞0

1

w(x)dg(x)

∫ ∞0

1

w(x)dh2(x)

∫ ∞0

1

w(x)dg(x)

.

Note that the integration by parts formula (see, e.g. [48, p. 382]), yields, for any u ∈ D�[0, ∞]:∫ b

a

1

w(u)du = u(b)

w(b)− u(a)

w(a)+

∫ b

a

u(x−)w′(x)

w(x)2 dx.

Hence D�(f,g)t (h1, h2) is defined even if h1 or (and) h2 are not of bounded variation. In our case,

integrals defined via integration by parts coincide with stochastic integrals, see Revuz and Yor[43, p. 113 and following].

So, under Assumption A and with the result of Theorem 1, the functional delta method yieldsthe following weak convergence result, as n goes to ∞, in D

3[0, ∞]:

√n

⎛⎝

G − G

G1 − G1

G2 − G2

⎞⎠ D→

⎛⎝

L0L1L2

⎞⎠ ,

where

L0(·) =

∫ ·0

1

w(x)dZ∞

0 (x)

∫ ∞0

1

w(x)dF (x)

− G(·)

∫ ∞0

1

w(x)dZ∞

0 (x)

∫ ∞0

1

w(x)dF (x)

and, for k = 1, 2

Lk(·) = Gk(·)

∫ ∞0

1

w(x)dZ∞

0 (x)

∫ ∞0

1

w(x)dF (x)

−

∫ t

01

w(x)dZ∞

k (x)

∫ ∞0

1

w(x)dF (x)

.

It is easy to prove that the limiting process is Gaussian.Concerning the asymptotic efficiency of our estimators, let us recall that the model defined

by the K-variate counting process (N1, . . . , NK) has the local asymptotic normality property (cf.[29, Theorem X.1.12]). Moreover, the Aalen–Johansen estimators are regular and efficient in thismodel [4, p. 622]. Since our estimators of the unbiased CIFs Gk , for k = 1, . . . , K , are Hadamard-differentiable functions of the Aalen–Johansen estimators, Theorem VIII.3.5 of Andersen et al.[4] shows that they are efficient estimators of the unbiased CIFs. �

Remark 1. This theorem extends Theorem 5.3 of Huang and Wang [28] in four senses. First,we allow the possibility of general SBS (i.e. w(x) may be different from x). Second, we allowindependent right censoring. Third, there is no restriction on the support of the distributions ofthe r.v. XIn

k , for k = 1, . . . , K . Finally, we state a functional weak convergence result in terms ofprocesses.


Remark 2 (One-dimensional case under LBS). In the LBS setup, the case where only one causeof death (possibly censored) is acting has already been studied by Vardi [50], Asgharian et al. [6]and de Uña-Álvarez (2002). In that case, the estimator of the survivor function given in Eq. (7)with w(x) = x, is

G(t) =

∫ t

01

xdF(x)

∫ ∞0

1

xdF(x)

. (13)

The result of Theorem 2 simplifies to the following weak convergence, as n goes to ∞, in D[0, ∞]:

√n

(G − G

) D→ L0, (14)

where L0 is a mean-zero Gaussian process defined in the previous paragraph. In the uncensoredcase, the estimator given in Eq. (13) was considered by Vardi [50]. de Uña-Álvarez [46] studiedits extension to the censored case (as in 13) and showed that it is the NPMLE of the function G.His Theorem 3.2 establishes a point-wise convergence result, which is extended by our functionalweak convergence in (14). Notice that, in the uncensored case, the limiting process Z∞

0 is justB(F) where B is the Brownian bridge. Then, for all s, t ∈ [0, ∞]:

⟨∫ s

0

1

xdZ∞

0 (x),

∫ t

0

1

xdZ∞

0 (x)

⟩=

∫ s∧t

0

1

x2 dF(x) −∫ s

0

1

xdF(x)

∫ t

0

1

xdF(x).

Moreover,(∫ ∞

0

1

xdF (x)

)2

〈L0(s), L0(t)〉

=⟨∫ s

0

1

xdZ∞

0 (x),

∫ t

0

1

xdZ∞

0 (x)

⟩− G(t)

⟨∫ s

0

1

xdZ∞

0 (x),

∫ ∞

0

1

xdZ∞

0 (x)

⟩

−G(s)

⟨∫ t

0

1

xdZ∞

0 (x),

∫ ∞

0

1

xdZ∞

0 (x)

⟩

+G(s)G(t)

⟨∫ ∞

0

1

xdZ∞

0 (x),

∫ ∞

0

1

xdZ∞

0 (x)

⟩.

Finally, when t = s, a straightforward calculation leads to

〈L0(t), L0(t)〉 = 1(∫ ∞

01

xdF (x)

)2

(G2(t)

∫ t

0

dF(x)

x2 − G(t)

∫ ∞

t

dF(x)

x2

). (15)

This is what de Uña-Álvarez [46] stated in his remark at the end of page 118.

Acknowledgment

The authors gratefully acknowledge the helpful suggestions of Prof. Syed N.U.A. Kirmaniduring the preparation of the paper.


Appendix

The objective of this Appendix is to show that, under some assumptions, formulas (1) and (2)appear in survival analysis area when a cross-sectional observation is made in a population withcompeting risks.

Let the population be indexed by {i ∈ I } and (�i)i∈I be the birth times family of theseindividuals. Suppose that each individual i in I is subject to K competing causes of death. As inSections 1 and 2, we denote by XIn

i = (XIn1i , . . . , XIn

Ki) the vector of the latent survival times ofthe individual i and by T In

i = ∧kXInki its lifetime. We say that a random sample is cross-sectionally

selected at t0 if it is not a random sample from I but from the set J = {i ∈ I : (�i, xIni ) ∈ E},

where E = {(�, xIn) : �� t0, �+xInk � t0, ∀k ∈ {1, . . . , K}} and � is the calendar time of birth of

an individual having XIn as vector of latent survival times. In particular, individuals with lifetimeT In

i = ∧kXInki shorter than t0 − �i are excluded from the population J, meaning that the survival

time T Ini is left truncated by the time t0 − �i . Individuals born after time t0 are also excluded

from population J.Thus, the observable r.v. is not T In

i but Ti , a r.v. whose probability distribution is the same asthe conditional distribution of T In

i given (�i, XIni ) ∈ E. We shall refer to Ti as the length-biased

version of T Ini .

Eqs. (1) and (2), or equivalently Eqs. (3) and (4), are justified in this framework by the followingtheorem.

Theorem 3. Suppose that:

(i) the birth process � =∑

i∈I ε�i, where ε�i

denotes the random measure concentrated on �i ,is a nonhomogeneous Poisson process with intensity � and let the function w be defined as

w(·) =∫ ·

0 �(t0 − u) du;(ii) conditionally on the process �, the vectors XIn

i , for i ∈ I , are independent and identically

distributed with common probability density function gXIn with respect to the Lebesgue

measure on RK+ ;

(iii) ∀k ∈ {1, . . . , K} : E(w(XInk )) < ∞.

Then, for t �0:

Gk(t) = −

∫ t

01

w(x)dFk(x)

∫ ∞0

1

w(x)dF (x)

,

for k = 1, . . . , K , and

G(t) = 1 − G(t) =

∫ t

01

w(x)dF (x)

∫ ∞0

1

w(x)dF (x)

.


Remark 3. If the Poisson process � is supposed to be homogeneous the above equations become:

Gk(t) = −

∫ t

01

xdFk(x)

∫ ∞0

1

xdF (x)

for k = 1, . . . , K, and G(t) =

∫ t

01

xdF (x)

∫ ∞0

1

xdF (x)

.

It has to be noted that, in this case, the above point process approach of Theorem 3 leads to therelations between biased and unbiased CIFs given in Huang and Wang [28].

Proof. We generalize the method introduced by Lund [37] and van Es et al. [49] to the case ofcompeting risks and nonhomogeneous Poisson process.

First, note that � is a point process on R such that, for each Borel set B in R, the r.v. �(B) givesthe number of births falling in B. We assume that �(B) < ∞ a.s. Moreover, let each birth time �i

of this point process be marked by the vector of latent survival times XIni . Thus, we can consider

the Lexis point process

=∑

i∈I

ε(�i ,XIni )

on (R × RK+ , BR ⊗ B

RK+), where BR (resp. B

RK+

) denotes the Borel �-algebra on R (resp. RK+ ).

Its the mean-measure � is given, for each Borel set B in R × RK+ , by

�(B) =∫

B

(�, xIn1 , . . . , xIn

K ) d�

K∏

k=1

dxInk

=∫

B

�(�)gXIn(xIn1 , . . . , xIn

K ) d�

K∏

k=1

dxInk .

We refer to Kingman [31] for the marking theorem exploited here.Now, let (. ∩ E) denote the restriction of the Poisson process to the measurable set E =

{(�, xIn) : �� t0, � + xInk � t0, ∀k ∈ {1, . . . , K}}. By the well-known restriction theorem for

Poisson processes (see [31]), the point process (· ∩ E) is still Poisson on R × RK+ with mean

measure:

�E(B) = �(B ∩ E) =

∫

B∩E

(�, x1, . . . , xK) d�

K∏

k=1

dxk,

for each Borel set B in R × RK+ .

Note that our sampling scheme is equivalent to selecting a random subset E∗ ⊂ E such that(E∗ ∩E) = n is the sample size. By the order statistics property of Poisson processes (see [31])the arrival times of the Poisson process (· ∩ E) are conditionally distributed, given (E) = N ,like (E) independent r.v.’s, with common probability measure

PE(·) = �(· ∩ E)

�(E),

on Borel subsets of R × RK+ .

Now, let X = (X1, . . . , XK) denote the vector of latent survival times (corresponding to the K

causes of death) of an individual in J where, as defined earlier, J = {i ∈ I : (�i, xIni ) ∈ E}. Taking


Bk = {(�, x) : xk � t, xk �xj , ∀j �= k}, it follows from the above discussion that, conditionallyon (E) = N ,

P (T � t, � = k) = PE(Bk) = �(Bk ∩ E)

�(E)

=∫{xk � t,xk �xj ,j �=k}∩E

�(�)gXIn(x1, . . . , xK) d�∏K

j=1 dxj

∫E

�(�)gXIn(x1, . . . , xK) d�∏K

j=1 dxj

=∫ t

0

∫ ∞xk

. . .∫ ∞xk

gXIn(x1, . . . , xK)∫ x1∧...∧xK

0 �(t0 − u) du∏K

j=1 dxj∫ ∞−0 . . .

∫ ∞0 gXIn(x1, . . . , xK)

∫ x1∧...xK

0 �(t0 − u) du∏K

j=1 dxj

.

Hence

Fk(t) = P (T � t, � = k) =∫ t

0 w(xk) dGk(xk)

E(w(XIn1 ∧ . . . XIn

K ))

which may be informally rewritten as

E(w(∧jXInj )) dFk(t) = w(t) dGk(t). (16)

With the notation G(t) = P(T In � t) and F (t) = P(T � t), the same development would haveled us to

E(w(∧jXInj )) dF (t) = w(t) dG(t). (17)

Since we can always write Gk(t) = Gk(t)/G(0) and G(t) = G(t)/G(0), integrating Eqs. (16)and (17) leads to the desired result. �

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