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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.

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

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 _{i} are in fact

Assume that components of vectors _{i}’s are observable only when_{x}, K_{x} and G_{x} the conditional distribution functions (d.f.-s) of r.v.-s Z_{x}, L_{x} and Y_{x} respectively, given that

Let H_{x} and N_{x} are conditional d.f.-s of

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 _{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} from [_{x} and G_{x} are expressed from F_{x} by following parametric relationships for all

where _{x} in conditional PHM is considered in [

It is not difficult to verify that from (1.1) one can obtain following expression of d.f. F_{x}:

where

_{x} one can constructed by natural plugging method as follows:

Here

and

are smoothed estimators of

For asymptotic properties of estimator (1.3) we need some notations. For the design points

Let _{x}. Then by (1.1):

In [

Theorem 2.1 [

This characterization of submodel (1.1) plays an important role for investigation the properties of estimator (1.3).

Let’s introduce some conditions:

(C1) As

(C2) π is a probability density function with compact support

(C3)

(C4)

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 [

Theorem 2.2 [

Theorem 2.3 [

where

and as

Corollary. Under the conditions of Theorem 2.3, and as

Theorem 2.4 [

(A) If

(B) If

where

with

It is necessary to note that Theorems 2.1-2.4 are extended the corresponding theorems in conditional PHM of Koziol-Green from [

In the next Section 3 we use these theorems for investigation the properties of the 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

One of main characteristics of d.f. (3.1) is its mean, i.e. mean conditional residual life function

We estimate functional _{x} in (3.2). But from section 2 we know that estimator (1.3) have consistent properties in some interval

Now we estimate (3.3) by statistics

We have following asymptotic normality result.

Theorem 3.1. Assume (C1)-(C3) in

(A) If

(B) If

Here

and

Proof of theorem 3.1. By standard manipulations and Theorem 2.3 we have that

where

For

mainder terms uniformly on

Now statements (A) and (B) of theorem follows from corresponding statements of the theorem 2.4 by standard arguments.

Theorem 3.1 is proved.

This work is supported by Grant F4-01 of Fundamental Research Found of Uzbekistan.