Semiparametric Estimator of Mean Conditional Residual Life Function under Informative Random Censoring from Both Sides

In this paper we study estimator of mean residual life function in fixed design regression model when life times are subjected to informative random censoring from both sides. We prove an asymptotic normality of estimators.


Introduction
In survival data analysis, response random variable (r.v.) Z, the survival time of a individual (in medical study) or failure time of a machine (in industrial study) that usually can be influenced by r.v.X, is often called prognostic factor (or covariate).X represents e.g. the dose of a drug for individual or some environmental conditions of a machine (temperature, pressure,…).Moreover, in such practical situations it often occurs that not all of survival times 1 , , n Z Z  of n identical objects are complete observed, that they can be censored by other r.v.-s.In this article we consider a regression model in which the response r.v.-s are subjected to random censoring from both sides.
We first introduce some notations.Let the support of covariate is the interval [0,1] and we describe our regression results in the situation of fixed design points , where ( ) Hence the observed data is consist of n vectors: Assume that components of vectors ( ) In sample x x P L Y ≤ = , then we obtain the interval random censoring model.The main problem in considered fixed design regression model is consist on estimation the conditional d.f.F x of lifetimes and its functionals from the samples ( ) n S under nuisance d.f.-sK x and G x .The first product-limit type estimators for F x in the case of no censoring from the left (that is ( ) x K t ≡ ) proposed by Beran [1] and has been investigated by many authors (see, for example [2] [3]).In this article supposing that the random censoring from both sides is informative we use twice power type estimator of F x from [4] [5] for estimation the mean conditional residual life function.Suppose that d.f.-sK x and G x are expressed from F x by following parametric relationships for all 0 t ≥ : ( , where x θ and x β are positive unknown nuisance parameters, depending on the covariate value x.Informative model (1.1) include the well-known conditional proportional hazards model (PHM) of Koziol-Green, which follows under absence of left random censorship (that is 0 x β ≡ ).Estimation of F x in conditional PHM is con- sidered in [6].Model (1.1) one can considered as an extended two sided conditional PHM.In the case of no covariates, model (1.1) first is proposed in [7] [8].It is not difficult to verify that from (1.1) one can obtain following expression of d.f.F x : ( ) ( ) ( ) where Then estimator of F x one can constructed by natural plugging method as follows: , , ( ) π y is a known probability density function (kernel), and { } n h is a sequence of positive constants tending to 0 as n → ∞ , called the bandwidth sequence.Note that in the case of no censoring from the left the estimator (1.3) is coincides with estimator in conditional Koziol-Green model in [6].Note also that a class of power type estimators for conditional d.f.-s for several models authors have considered in book [9].Estimator (1.3) was presented in [4] and its asymptotic properties have been investigated in [5].Now we demonstrate some of these results.

Asymptotic Results for Estimator of Conditional Distribution Function
For asymptotic properties of estimator (1.3) we need some notations.For the design points 1 , , n x x  and kernel π we denote ( ) ( ) = are lower and upper bounds of support of d.f.F x .Then by (1.1): and In [4] authors have proved the following property of two sided conditional PHM (1.1).Theorem 2.1 [5].For a given covariate x, the model (1.1) holds if and only if r.v.x ξ and the vector χ χ χ are independent.This characterization of submodel (1.1) plays an important role for investigation the properties of estimator (1.3).
Let's introduce some conditions: Let's also denote: Note that existence of all these derivatives follows from conditions (C3) and (C4).Now we state some asymptotic results for estimator (1.3), which have proved in [5].
Corollary.Under the conditions of Theorem 2.3, and as n → ∞ , for
0, ; , , It is necessary to note that Theorems 2.1-2.4 are extended the corresponding theorems in conditional PHM of Koziol-Green from [6].
In the next Section 3 we use these theorems for investigation the properties of the estimator of mean conditional residual life function.

Asymptotic Normality of Estimator of Mean Conditional Residual Life Function
The conditional residual lifetime distribution defined as ( ) ( ) i.e. the d.f. of residual lifetime, conditional on survival upon a given time t and at a given value of the covariate x.Then for 0 One of main characteristics of d.f.(3.1) is its mean, i.e. mean conditional residual life function We estimate functional ( ) x t µ by plugging in estimator (1.3) instead of F x in (3.2).But from section 2 we know that estimator (1.3) have consistent properties in some interval [ ] . Therefore, we will consider the following truncated version of (3.2): Now we estimate (3.3) by statistics We have following asymptotic normality result.
) π is a probability density function with compact support [ ]

2 .
Then we see that all these re- mainder terms uniformly on [ ] A) and (B) of theorem follows from corresponding statements of the theorem 2.4 by standard arguments.Theorem 3.1 is proved.