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Mathematical Problems in Engineering
Volume 2014 (2014), Article ID 937397, 5 pages
An Efficient Estimation of the Mean Residual Life Function with Length-Biased Right-Censored Data
1School of Mathematics, Shandong University, Jinan, Shandong 250100, China
2College of Science, China University of Petroleum, Qingdao 266580, China
Received 20 March 2014; Accepted 26 May 2014; Published 9 June 2014
Academic Editor: Jian Guo Zhou
Copyright © 2014 Hongping Wu and Yihui Luan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The mean residual life (MRL) function for a lifetime random variable is one of the basic parameters of interest in survival analysis. In this paper, we propose a new estimator of the MRL 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.
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  and Kumazawa . EL approach has been developed to construct the confidence interval of the MRL function by Zhao and Qin , where, for censored data, the corresponding log-EL ratio was defined and the limiting distribution was proved to be a scaled 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 . A more recent work on estimation of the MRL with left-truncated and right-censored data has been constructed by Zhao et al. , and they showed that the proposed estimator converges weakly to a Gaussian process.
This paper considers survival data arising from length-biased sampling, which is a special case of left truncation. Following the terminology in the literature (see [8, 9]), length-biased 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  considered the nonparametric maximum likelihood estimator of the survival distribution. Gupta and Keating  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.  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  proposed a pseudopartial likelihood approach, and Huang and Qin  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.
2. Notations and Estimators
2.1. Data and Model Setup Section
Let and be two independent nonnegative random variables denoting the left-truncated time and the lifetime of a system, respectively. Assume that the truncation probability . Let and denote the observed truncation time and lifetime; that is, the pair is a realization from the conditional distribution of given that . Under the stationarity assumption, is uniformly distributed on , 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 , . Now we suppose 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 and by and the corresponding density function and survival function of the length-biased random variable . Let , . Under length-biased sampling, and are stochastically dependent and has the joint density function (see ) and the density function of is related to the length-biased density function of , as follows: where is finite.
According to the definition of the MRL function, the mean residual life of at time is expressed as and when .
In Huang and Qin , the nonparametric maximum likelihood estimator of has been given: where is the estimated cumulative hazard function with , , , , and .
Let be the survival function of and . Assume the following conditions hold:(i) is a nonnegative constant satisfying ;(ii) and are absolutely continuous on ;(iii), , ;(iv)either with or .
As a straightforward result of the functional delta method, when , converges in distribution to a zero mean normal variable with the variance function for every under Assumptions (i–iv).
2.2. 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 , can be derived as follows: where is the product-limit estimation of , and it is expressed as where . If , .
In the next section we shall establish that converges in distribution under some conditions by using the Hadamard derivative method.
3. 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 .
Lemma 1. For , the function is Hadamard-differentiable at , where its derivative is equal to
Lemma 2. Assume the conditions (i–iv) and (12) hold, for every , as . Then(i) converges in distribution to a zero mean normal variable with the variance function ;(ii) converges in distribution to a zero mean normal variable with the variance function .
Proof. Firstly, let us give the proof of the result in Lemma 2(i). Under the regular conditions (i–iv) and (12), since , by the central limit theorem, we can derive
where denotes weak convergence.
Since and recall that is the product-limit estimator of , so on the basis of (14) and the consistency of (see ), we can obtain
Since , , are i.i.d, it is easy to derive the result of Lemma 2 (i) by the central limit theorem.
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
Since so on the basis of (17) and the consistency of , we can derive
Since , , are i.i.d, it is easy to derive the result of Lemma 2 (ii) by the central limit theorem. The proof of Lemma 2 is completed.
Theorem 3. Assume the conditions (i–iv) and (12) hold, for every , as , then converges in distribution to a zero mean normal variable with the variance function .
Proof. Under the conditions (i–iv) and (12), Lemma 2 allows a straightforward application of Lemma 1. So we can derive where denotes convergence in distribution. is Gaussian as a linear form on two Gaussian variables and it is easy to check that , since and are centered. It is also easy to calculate the variance of and it can be expressed as The proof is completed.
To compare the performance of our proposed estimator with the one in (6), we conducted a series of Monte Carlo simulations with and . In these simulations, the underlying survival time has a uniform distribution. Then the length-biased density function of is , , and the left-truncated time is uniformly distributed on . We simulated the censoring time from either a uniform , a uniform or a uniform distribution, corresponding to a censoring rate of , , and .
Table 1 summarizes the Monte Carlo bias, standard deviation, and mean square error for each estimator at and based on replications. The survival probabilities at the selected time-points are , , , and . The corresponding true values of at the selected time-points are , , , and , respectively.
The outcomes show that the proposed estimator works very well. Through the numerical results in Table 1, we derive the following conclusions:(1) outperforms obviously in terms of Monte Carlo bias and mean square error for every values of when or ;(2)the two estimators are very close to each other in terms of the Monte Carlo standard deviation when the censoring rate is small;(3)when the censoring rate is heavy, outperforms obviously in all terms that we referred to in the simulations.
Through the conclusions given above, it is appropriate to explore our proposed method in real applications.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported by the National Science Foundation of China Grants 11371227 and 10921101.
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