An Efficient Estimation of the Mean Residual Life Function with Length-Biased Right-Censored Data

Themean residual life (MRL) function for a lifetime randomvariableT is one of the basic parameters of interest in survival analysis. In this paper, we propose a new estimator of theMRL function with length-biased right-censored data and evaluate its performance through a small Monte Carlo simulation study. The results of the simulations show that the proposed estimator outperforms the existing one referred to in Data and Model Setup Section in terms of Monte Carlo bias and mean square error, especially when the censoring rate is heavy. We also show that the proposed estimator converges in distribution under some conditions.


Introduction
The MRL function at time  is defined to be the expected remaining lifetime of a system given survival up to time .It can be used to model various lifetime data in many areas of science.For example, a life insurance company may be interested in the life expectancy of a person, or an engineering firm may wish to estimate the expected remaining lifetime of a system, given survival up to time .In literature, there has been a lot of work on the inference of the MLR function.Based on the i.i.d copies of the life time and the closed relationship between the MRL function and survival function, nonparametric estimators and NA based confidence intervals have been constructed (see [1,2]).For right-censored data, consistent estimator of the MRL function with its asymptotic normal distribution has been given by Yang [3] and Kumazawa [4].EL approach has been developed to construct the confidence interval of the MRL function by Zhao and Qin [5], where, for censored data, the corresponding log-EL ratio was defined and the limiting distribution was proved to be a scaled  2 distribution.A class of transformed MRL function for fitting survival data under right censoring was proposed and efficiency and robustness of estimators have been studied by Sun and Zhang [6].A more recent work on estimation of the MRL with left-truncated and right-censored data has been constructed by Zhao et al. [7], and they showed that the proposed estimator converges weakly to a Gaussian process.
This paper considers survival data arising from lengthbiased sampling, which is a special case of left truncation.Following the terminology in the literature (see [8,9]), lengthbiased data are defined for left-truncated and right-censored data under the stationarity assumption, which means that the probability of the survival time being sampled is proportional to its length or the survival times are left-truncated by uniformly distributed random truncation times.Statistical inferences of length-biased data have been considered by many authors.Vardi [9] considered the nonparametric maximum likelihood estimator of the survival distribution.Gupta and Keating [10] obtained the relations for reliability measures of the length-biased distribution and some characterization results.Concerning the effect of the length bias on the sampling distribution of the covariates, Bergeron et al. [11] presented a joint likelihood approach and studied the large-sample behavior of the resulting maximum likelihood estimators.To estimate the survival distribution, Luo and Tsai [12] proposed a pseudopartial likelihood approach, and Huang and Qin [8] proposed a new nonparametric estimator based on the product-limit estimator.Their estimators were proved to be uniformly consistent and converge weakly to Gaussian processes.
In this paper we propose a new estimator of the MRL function with length-biased and right-censored data.We compare the performance of our estimator with the existing one referred to in Section 2.1 through a small Monte Carlo simulation study and the results of the simulations show that the proposed estimator outperforms the existing one in terms of Monte Carlo bias and mean square error, especially when the censoring rate is heavy.The research is organized as follows.In Section 2, we introduce the model and the proposed estimator of the MRL function.The large sample property of our proposed estimator is stated in Section 3. The performance of the estimators is illustrated in Section 4.

Notations and Estimators
2.1.Data and Model Setup Section.Let  0 and  0 be two independent nonnegative random variables denoting the lefttruncated time and the lifetime of a system, respectively.Assume that the truncation probability  = Pr( 0 ≤  0 ) > 0. Let  and  denote the observed truncation time and lifetime; that is, the pair (, ) is a realization from the conditional distribution of ( 0 ,  0 ) given that  0 ≤  0 .Under the stationarity assumption,  is uniformly distributed on [0, ], so  has a length-biased distribution.Coupled with the censoring time , which is applied to the residual lifetime  =  −  instead of  and independent of (, ), the observed data is composed of i.i.d.copies of (, , ), where  = min(, +),  = ( ≤ ).Now we suppose {(  ,   ,   ),  = 1, 2, . . ., } to be a sequence of i.i.d.random samples of (, , ) which one observes.
Denote by () and () the marginal density function and survival function of  0 and by () and () the corresponding density function and survival function of the length-biased random variable .Let  = inf{ : () = 0},  ∈  + ∪ {+∞}.Under length-biased sampling,  and  are stochastically dependent and (, ) has the joint density function (see [13]) and the density function  of  0 is related to the length-biased density function  of , as follows: where According to the definition of the MRL function, the mean residual life of  0 at time 0 ≤  <  is expressed as and () = 0 when  ≥ .
In Huang and Qin [8], the nonparametric maximum likelihood estimator of () has been given: where Λ is the estimated cumulative hazard function with 3) and ( 4), when 0 ≤  ≤  () , the estimator of () can be naturally derived: where Let   (⋅) be the survival function of  and   = inf{ :   () = 0}.Assume the following conditions hold: (i)  is a nonnegative constant satisfying  < sup{ : ()  () > 0}; (ii)  and   are absolutely continuous on [0, ]; In the next subsection, we shall state our proposed estimator of (), and the simulation study in Section 4 shows that it performs better in several terms than the one in (6).

The Proposed Estimator.
According to the joint density function of (, ), we can derive that where   (⋅) is the density function of the censoring time .
Applying (7) we shall use the method of moments to derive an estimator for the MRL function ().To proceed we note that () can be written as An alternative estimator of the MLR function (), for 0 ≤  ≤  () , can be derived as follows: where Ŝ () is the product-limit estimation of   (), and it is expressed as where () = ∑  =1 (  −   ≥ ).If  >  () , m() = 0.In the next section we shall establish that m() converges in distribution under some conditions by using the Hadamard derivative method.

The Asymptotic Properties
Firstly, let us introduce some notations and lemmas. Denote and assume that where  * (⋅, ⋅) is the joint distribution function of (, ).Lemma 1 gives the Hadamard derivative of  at ((), ()).
Using the same method, let us give the proof of Lemma 2 (ii).Under the regular conditions (i-iv) and ( 12), because (( > )/  ( − )) = (), by the central limit theorem, we can derive where   → denotes convergence in distribution.N m is Gaussian as a linear form on two Gaussian variables and it is easy to check that N m = 0, since  1 and  2 are centered.It is also easy to calculate the variance of N m and it can be expressed as Var (N m) = The proof is completed.

Simulations
To compare the performance of our proposed estimator with the one in (6), we conducted a series of Monte Carlo simulations with  = 100 and 200.In these simulations, the underlying survival time  0 has a uniform (0, 1) distribution.