A Central Limit Theorem for a Nonparametric Maximum Conditional Hazard Rate Estimator in Presence of Right Censoring

In this paper, we consider a non-parametric kernel type estimator of the time where a hazard rate function is maximum in the presence of covariate and right censoring. Via a strong representation of the estimator, we establish weak convergence and asymptotic normality results.


Introduction
The estimation of the hazard rate and the related topics are important subjects in statistics because of the variety of their applications.Those subjects may be considered in several manners according to the data and there are many techniques used in the literature to estimate the hazard functions.In this paper we focus on the investigation of the maximum hazard rate with covariate.More precisely let T be a life time, Z a covariate and C a right censoring variable independent of T conditionally on Z. Assume that T , Z, and C are continuous and denote by F(t|z) (resp.G(t|z)) the conditional distribution function of T (resp.C) given Z = z, f (t|z) the conditional probability density of T and f (z) the marginal density function of Z. Define X = min(T, C) and δ = I(T ≤ C) where I(A) is the indicator function of a Borel set A.
The conditional hazard rate function λ(t|z) of T given Z = z is defined by This function is very useful in statistical applications such as in survival analysis, medical follow up, industrial reliability studies or in earthquake studies.In this setting, knowing how to estimate the maximum of the instantaneous risk allows to predict the maximum risk when a new seismic series occurs and the knowledge of the maximum may arise when exploring relationship between responses and potential covariates.Denote by θ the time in an interval [a z , b z ] of R + corresponding to the maximum of the conditional hazard rate function, that is, θ(z) = Argmax a z ≤t≤b z λ(t|z). (1) Non-parametric estimation of the hazard rate function was first introduced in the statistical literature by Watson andLeadbetter (1964a, 1964b).The topic was developed by other authors among which Singpurwalla and Wong (1983), Tanner and Wong (1983) and Gneyou (1991).The conditional case was considered later by Van Keilegom and Veraverbeke (1997) and (2001).
The problem of estimating the maximum of a conditional hazard rate function is somewhat similar to the problem of estimating the conditional mode of a random variable.The methods employed here are inspired by the methods used to the treatment of the latter problem which has received much attention during the last twenty five years.For the references, see e.g.Collomb et al. (1987), Samanta and Thavaneswaran (1990), Ould-Saïd (1993), Quinteladel-Río and Vieu (1997), Louani and Ould-Saïd (1999), Ferraty et al. (2005), Dabo-Niang and Laksaci (2007) and Ezzahrioui and Ould Saïd (2008) for uncensored models.In censored data case, see e.g.Ould Saïd and Cai (2005) and Khardani et al. (2010Khardani et al. ( , 2011)).
Concerning the maximum hazard rate estimation, Quintela-del-Río ( 2006) considered a non-parametric estimator under dependence conditions in uncensored case.Gneyou (2012) considered a kernel-type estimator in the model of right censored data with covariate and establish strong uniform consistency results.The aim of this paper is to extend these results to the weak convergence.The paper is organized as follow.In Section 2 we recall the definitions of the non-parametric estimator of the conditional hazard rate function λ(t|z) and the corresponding estimator θ n (z) of its maximum value θ(z) as given in Gneyou (2012) and state the assumptions under which the results will be obtained.In Section 3 we establish a basic almost sure asymptotic representation for the estimator θ n (z) which leads to some main results such as weak convergence and asymptotic normality.Detailed proofs are given in the appendix.

Definitions and Assumptions
Let (T i , Z i , C i ) n i=1 be a sample of size n of the random variables (T, Z, C).As it is often the case in clinical trials or industrial life tests, the lifetimes T 1 , T 2 , • • • are not completely observable due to the presence of right-censored variables.In presence of covariates Z i and right-censoring

Denote by
the conditional cumulative hazard function of T given Z = z and define H(t|z )dF(s|z) the conditional sub-distribution function of the uncensored observation (X, δ = 1) given Z = z.Since it is assumed that T and C are independent conditionally on Z, Λ(t|z) can be written in the form Hence, non-parametric kernel-type estimators of the functions Λ(t|z), λ(t|z) and θ(z) are respectively given by where , for all x ∈ R d , h > 0, s ∈ R and a > 0. Note that H n (t|z) and H 1n (t|z) are kernel estimators of H(t|z) and H 1 (t|z) respectively obtained by regression.
Let τ z = sup{t ∈ R + /F(t|z) < 1}.In applications, τ z is typically not know in advance, but may be chosen such that the size of the observed risk set is sufficiently large.
For later use, introduce the Fourier transforms and for a general conditional (sub-distribution) function L(t|z), t ∈ R + , z ∈ R d , denote by L z (t) the function t → L(t|z); L z (t), L z (t), its first and second derivatives with respect (w.r.) to t and L (i, j) (t|z) = ∂ i+ j L(t|z) ∂t i ∂z j its derivative of order i + j w.r. to t and z, for all (i, j) ∈ N 2 , whenever all those derivatives exist.For a sequence of conditionals

Assumptions
The following assumptions are needed throughout the proofs of the main results: (F1.) (i) the r.v.Z takes values in a compact subset Δ of R d and the variables T and C are independent conditionally on Z; (ii) the marginal density function f of Z is a continuous function with bounded derivative at each z ∈ Δ. (K1.)K is a symmetric Kernel of bounded variation on R d with compact support satisfying (i) ) N is a symmetric Kernel of bounded variation on R vanishing outside the interval [−M, +M] for some M > 0 and satisfying (iv) N is two times derivable, the derivative N is of bounded variation and satisfies R N 2 (u)du < +∞.

Main Results
Gneyou (2012) proved the uniform convergence of the estimators λ n (t|z) and θ n (z).In what follows we establish weak convergence and asymptotic normality of the estimator θ n (z).For that, we need to consider the process l z (t, X, δ) is a centred random process which plays a major role in our investigations.The following theorem establishes a strong representation of the maximum hazard rate estimator θ n (z).We apply it to derive a weak convergence leading to the asymptotic normality of the estimator θ n .
Theorem 1 Under the assumptions (F1)-( F5), (K1)-( K2) and (H1)-(H2), we have, for all z ∈ Δ where θ * n is between θ and θ n , with l z i (t) as in ( 7) and sup The proof of Theorem 1 is given in the next section.Let D[0, τ z ] be the standard Skorohod space on [0, τ z ].We have Theorem 2 If the assumptions of Theorem 1 hold then as n → +∞, for all z ∈ Δ, the process converges weakly in D[0, τ z ] to a Gaussian process with covariance function given by .
As a consequence of this theorem in conjunction with the following proposition, we obtain the asymptotic normality of the maximum conditional hazard rate estimator θ n .
Proposition 1 Assume that assumptions (H3)(iii) and (HK) hold.Then for all z ∈ Δ λ z n (t) converges in probability to λ z (t) uniformly in t.
The proof of this proposition is postponed to the next section.

Appendix: Proofs of the Results
The following lemmas are needed to prove the main results.
Lemma 1 Let : R d → R be a function continuous at z. Then A. Under assumptions (F1)(ii), (K1)(i) and (H1)(i), B. If is a function twice continuously differentiable at z then, under assumptions (K1) and (H1)(i), we have The proof of this lemma is analogue to the proofs of Lemmas 4 and 5 in Bordes and Gneyou (2011a), hence we omit it.
and if the assumptions (K2) and (H2) − (ii) are satisfied then, where Proof.By Fubini's Theorem, it is easily seen that where with It is easy to check that El * z (t, X i , δ i , s) = 0 and hence El z (t, X i , δ i , s) = Q z (t, s).Thus the first part of the lemma is proved.For the second part, we have by ( 16), Then the expectation under the last integral equals By Fubini's theorem, we check that the eight last expectations equal respectively to zero while the first one EAA equals to d * (t ∧ t |z) where Thus integrating by parts under the assumption (K2), we check that where M is the upper boundary of the support of the kernel N. Developing the function λ * (t − a n v|z) by Taylor's theorem in order one at a neighbourhood of t yields which ends the proof of the Lemma 2.
Proof of Theorem 1.Notice that by definition of θ and θ n , λ z (θ) = λ z n (θ n ) = 0, λ z (θ) ≤ 0 and λ z n (θ n ) ≤ 0. Hence by Taylor's expansion of order one in a neighbourhood of θ , we have where θ * n is between θ and θ n .It follows from ( 22) that By Assumption (K2) (i) it is easily seen that Hence by the derivation theorem and an integration by parts under the assumption (K2) we have where A n (t|z) is a process which can be written in the form and R n (t|z) is a remainder term which vanishes almost surely under assumptions (H1) and (H2) (see in Gneyou (2012)).It follows that where By Assumptions (F4), (F5), ( K1) and (H1), we show that sup , Theorem 1 follows from the representation ( 28) and (30).

Proof of Theorem 2. Recall the notations ζ
where f n is a consistent kernel estimator of the probability density f of the r.v.Z i .
For all n large enough, we have Thus for all 0 ≤ t ≤ τ z we can write where We show the theorem by applying Theorem 2.11.23 of van der Vaart and Wellner (1996) to the class of function Let us calculate first the mean and the covariance functions of the process f n,t : t ∈ T z .It is straightforward that where by Lemma 2, Develop Ψ(s) by Taylor's theorem in the order two in a neighbourhood of z under the assumption (F5).Since Ψ(z) = 0 we get under the assumption (K1) Recalling that an integration by part yields where Since by the assumption (F5) the function t → L (1,0) (t|s) is continuous at t, apply Lemma 3.1 of Bordes and Gneyou (2011 b) with the kernel N and have It follows that Let us check now the covariance function of the process f n,t : t ∈ T z .Set By ( 32) and ( 16) we have for all t, t ∈ T z , where Using the kernel K , it follows from ( 38), ( 21) and Lemma 2 that It remains to check the three conditions of Theorem 2.11.23 of van der Vaart and Wellner (1996).Set where f n,t is the real function defined on with l z (t, y, x) as in ( 7).Let us check the Lindberg conditions (2.11.21) of Van der Vaart and Wellner (1996).
(i) By the assumption (K2) (i), an integration by parts yield Since and the derivatives dH and dH 1 are bounded under the assumption (F5), we have under the assumption (F4) where A 1 and A 2 are absolute constants.Consequently we have for n large enough, sup where m 0 = m N L(R) .Since h n 0 and f is continuous at z by assumption (F1)(ii), we apply Lemma 1 with the kernel K 2 / K 2 L(R d ) and get (ii) Since K is symmetric, bounded and nh d n → +∞ as n → +∞ by assumption (H1)(ii), we have for all ε > 0, (iii) Consider the pseudo-distance ρ defined by For a z ≤ t ≤ t ≤ b z and ρ(t, t ) ≤ ρ n with ρ n → 0, set Then we have and recall that by ( 16) Since l * z (t − a n u, X i , δ i , s) is a centered random variable, the last term of this last equality has mean zero.It follows that After the development of square in the first brackets, we get by similar computations as in (18), Hence we have Since by the assumption (F1)(ii) f is a continuous function with bounded derivative at z, we apply Fubini's theorem and Lemma 1 with the kernel and But for all v, Q z (v, z) = 0 (see in ( 15)), hence II = 0. Recalling that we have after integrating by parts and that f is bounded on Δ, we have by assumption (F4) It remains to check the entropy condition.Let us consider the following brackets [ f n,t i−1 , f n,t i ] with f n,t − (z, y, x, s) = 1 where l z (t, y, x) is defined as in (7).After some computations similar to above ones we obtain for n large enough, Therefore it is straightforward that for n large enough, Finally we have by (42), J [] (δ n , F n , L 2 (P)) = Since for all i = 1, it readily follows that where C is a positive constant.The middle integral equals to P[Z ≤ +∞, δ = 1].Hence it is less than one.As a consequence, we have by Fubini's Theorem Since the function f is continuous at z, we apply again Lemma 1 with the kernel K 2 / R d K 2 (x)dx to have where C is a novel positive constant.Combining now (45), ( 47) and ( 48) we prove that .) There exists a positive constant η such that inf z∈Δ (1 − H(τ z |z)) ≥ η. (F5.)The conditional sub-distribution functions (t, z) → H(t|z) and (t, z) → H 1 (t|z) are of class C 2 and their first and second partial derivatives are continuous in (t, s) ∈ [a z , b z ] × Δ and are uniformly bounded.
sup t∈[a z ,b z ] − λ z (t))2 ] −→ 0 as n → +∞ by assumption (H3)(ii) where A is a constant.This imply the convergence in probability of λ z n (t) to λ z (t) uniformly in t.Proof of Theorem 3. By Theorem 1 it is straightforward that and by the proof of Theorem 2, U n (z) is a linear functional of the empirical process G n given in (43) and hence, is asymptotically Gaussian with asymptotic variance σ 2 z and mean function equal to n(t|z)