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

Abstract

The mean residual life (MRL) function for a lifetime random variable 𝑇⁰ 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.

1. Introduction

The MRL function at time 𝑡 is defined to be the expected remaining lifetime of a system given survival up to time 𝑡. 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 𝑡. 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 𝜒² 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 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 [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.

2. Notations and Estimators

2.1. Data and Model Setup Section

Let 𝐴⁰ and 𝑇⁰ be two independent nonnegative random variables denoting the left-truncated time and the lifetime of a system, respectively. Assume that the truncation probability 𝛼=Pr(𝐴⁰≤𝑇⁰)>0. Let 𝐴 and 𝑇 denote the observed truncation time and lifetime; that is, the pair (𝐴,𝑇) is a realization from the conditional distribution of (𝐴⁰,𝑇⁰) given that 𝐴⁰≤𝑇⁰. Under the stationarity assumption, 𝐴 is uniformly distributed on [0,𝑇], so 𝑇 has a length-biased distribution. Coupled with the censoring time 𝐶, which is applied to the residual lifetime 𝑉=𝑇−𝐴 instead of 𝑇 and independent of (𝐴,𝑇), the observed data is composed of i.i.d. copies of (𝐴,𝑌,𝛿), where 𝑌=min(𝑇,𝐴+𝐶), 𝛿=𝐼(𝑉≤𝐶). Now we suppose {(𝑎_𝑖,𝑦_𝑖,𝛿_𝑖),𝑖=1,2,…,𝑛} to be a sequence of i.i.d. random samples of (𝐴,𝑌,𝛿) which one observes.

Denote by 𝑓(𝑡) and 𝑆(𝑡) the marginal density function and survival function of 𝑇⁰ and by 𝑔(𝑡) and 𝐺(𝑡) the corresponding density function and survival function of the length-biased random variable 𝑇. Let 𝜏=inf{𝑡∶𝑆(𝑡)=0}, 𝜏∈𝑅⁺∪{+∞}. Under length-biased sampling, 𝐴 and 𝑇 are stochastically dependent and (𝐴,𝑇) has the joint density function (see [13]) 𝑓_𝐴,𝑇()=𝑓(𝑡)𝑎,𝑡𝜇,0≤𝑎<𝑡<𝜏, (1) and the density function 𝑓 of 𝑇⁰ is related to the length-biased density function 𝑔 of 𝑇, as follows: 𝑔(𝑡)=𝑡𝑓(𝑡)𝜇,0≤𝑡<𝜏, (2) where 𝜇=[∫^𝜏₀𝑢⁻¹𝑔(𝑢)𝑑𝑢)]⁻¹ is finite.

According to the definition of the MRL function, the mean residual life of 𝑇⁰ at time 0≤𝑡<𝜏 is expressed as 𝑚(𝑡)=𝐸𝑇⁰−𝑡∣𝑇⁰=∫>𝑡^𝜏𝑡𝑆(𝑢)𝑑𝑢𝑆(𝑡), (3) and 𝑚(𝑡)=0 when 𝑡≥𝜏.

In Huang and Qin [8], the nonparametric maximum likelihood estimator of 𝑆(𝑡) has been given: 𝑆(𝑡)=_{𝑢∈[0,𝑡]}1−𝑑Λ(𝑢), (4) where Λ is the estimated cumulative hazard function Λ(𝑡)=^𝑡₀𝑑𝑁(𝑢)𝑅(𝑢), (5) with 𝑁(𝑢)=𝑛⁻¹∑^𝑛_𝑖=1𝛿_𝑖𝐼(𝑦_𝑖≤𝑢), 𝑅(𝑢)=𝑛⁻¹∑^𝑛_𝑖=1𝐼(𝑦_𝑖≥𝑢)−𝑆_𝐴(𝑢), 𝑆_𝐴(𝑢)=∏_{𝑣∈[0,𝑢]}{1−(𝑑𝑄(𝑣)/𝐾(𝑣))}, 𝑄(𝑣)=𝑛⁻¹∑^𝑛_𝑖=1𝐼{𝐼(𝑎_𝑖≤𝑣)+𝛿_𝑖𝐼(𝑦_𝑖−𝑎_𝑖≤𝑣)}, and 𝐾(𝑣)=𝑛⁻¹∑^𝑛_𝑖=1{𝐼(𝑎_𝑖≥𝑣)+𝐼(𝑦_𝑖−𝑎_𝑖≥𝑣)}.

Applying (3) and (4), when 0≤𝑡≤𝑦_(𝑛), the estimator of 𝑚(𝑡) can be naturally derived: 𝑚(𝑡)=∫^𝑦(𝑛)_𝑡𝑆(𝑢)𝑑𝑢𝑆(𝑡), (6) where 𝑦_(𝑛)=max(𝑦₁,𝑦₂,…,𝑦_𝑛). If 𝑡>𝑦_(𝑛), 𝑚(𝑡)=0.

Let 𝑆_𝐶(⋅) be the survival function of 𝐶 and 𝜏_𝐶=inf{𝑡∶𝑆_𝐶(𝑡)=0}. Assume the following conditions hold:(i)𝐿 is a nonnegative constant satisfying 𝐿<sup{𝑡∶𝑆(𝑡)𝑆_𝐶(𝑡)>0};(ii)𝑆 and 𝑆_𝐶 are absolutely continuous on [0,𝐿];(iii)∫^𝜏_𝑡𝑔(𝑢)𝑑𝑢<∞, ∫^𝜏_𝑡𝑢⁻¹𝑔(𝑢)𝑑𝑢<∞, 𝑡∈[0,𝜏);(iv)either 𝜏_𝐶<𝜏 with 𝑆_𝐶(𝜏⁻_𝐶)>0 or 𝜏_𝐶≥𝜏.

As a straightforward result of the functional delta method, when 𝑛→∞, √𝑛(𝑚(𝑡)−𝑚(𝑡)) converges in distribution to a zero mean normal variable with the variance function 𝜎²(𝑡)=𝑆⁻⁴(𝑡)[𝑆²(𝑡)𝐸(∫^𝜏_𝑡𝑍(𝑢)𝑑𝑢)²−2𝑆(𝑡)∫^𝜏_𝑡𝑍(𝑢)𝑑𝑢𝐸𝑍(𝑡)∫^𝜏_𝑡𝑍(𝑢)𝑑𝑢+(∫^𝜏_𝑡𝑆(𝑢)𝑑𝑢)²𝐸𝑍²(𝑡)] for every 𝑡∈[0,𝐿] under Assumptions (i–iv).

In the next subsection, we shall state our proposed estimator of 𝑚(𝑡), and the simulation study in Section 4 shows that it performs better in several terms than the one in (6).

2.2. The Proposed Estimator

According to the joint density function of (𝐴,𝑇), we can derive that 𝐸𝐼(𝑌≥𝑡)𝛿𝑆_𝐶(𝑌−𝐴)=^𝜏_𝑡𝑓(𝑢)𝜇^𝑢₀^∞_𝑢−𝑎𝑓_𝐶(𝑣)1𝑑𝑣𝑆_𝐶(𝑢−𝑎)𝑑𝑎𝑑𝑢=^𝜏_𝑡𝑔(𝑢)𝐸𝐼()𝛿𝑑𝑢,𝑌≥𝑡𝑌𝑆_𝐶()=𝑌−𝐴_𝜏𝑡𝑓(𝑢)𝜇^𝑢₀^∞_𝑢−𝑎𝑓_𝐶(𝑣)1𝑑𝑣𝑢1𝑆_𝐶()=𝑢−𝑎𝑑𝑎𝑑𝑢_𝜏𝑡𝑢⁻¹𝑔(𝑢)𝑑𝑢, (7) where 𝑓_𝐶(⋅) is the density function of the censoring time 𝐶.

Applying (7) we shall use the method of moments to derive an estimator for the MRL function 𝑚(𝑡). To proceed we note that 𝑚(𝑡) can be written as 𝑚(𝑡)=𝐸𝑇⁰−𝑡∣𝑇⁰=∫>𝑡^𝜏𝑡𝑢𝑓(𝑢)𝑑𝑢∫^𝜏_𝑡𝑓(𝑢)=∫𝑑𝑢−𝑡_𝜏𝑡𝑔(𝑢)𝑑𝑢∫^𝜏_𝑡𝑢⁻¹𝑔(𝑢)𝑑𝑢−𝑡. (8)

An alternative estimator of the MLR function 𝑚(𝑡), for 0≤𝑡≤𝑦_(𝑛), can be derived as follows: 𝑚(𝑡)=(1/𝑛)∑^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖/𝑆_𝐶𝑦_𝑖−𝑎_𝑖(1/𝑛)∑^𝑛_𝑖=1𝑦_𝑖⁻¹𝐼𝑦_𝑖>𝑡𝛿_𝑖/𝑆_𝐶𝑦_𝑖−𝑎_𝑖−𝑡, (9) where 𝑆_𝐶(𝑡) is the product-limit estimation of 𝑆_𝐶(𝑡), and it is expressed as 𝑆_𝐶(𝑡)=^𝑛_𝑖=11−1−𝛿_𝑖𝐼𝑦_𝑖−𝑎_𝑖≤𝑡𝑅𝑦_𝑖−𝑎_𝑖, (10) where 𝑅(𝑡)=∑^𝑛_𝑗=1𝐼(𝑦_𝑗−𝑎_𝑗≥𝑡). If 𝑡>𝑦_(𝑛), 𝑚(𝑡)=0.

In the next section we shall establish that 𝑚(𝑡) 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 𝑀(𝑡)=^𝜏_𝑡𝑔(𝑢)𝑁(𝑡)=𝑑𝑢,_𝜏𝑡𝑢⁻¹𝑔(𝑢)𝑀(𝑡)=()∑𝑑𝑢,1/𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖,𝑁(𝑡)=(1/𝑛)∑^𝑛_𝑖=1𝑦_𝑖⁻¹𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖,𝛾(𝑀(𝑡)(𝑡(𝑡)=𝑀(𝑡),𝑁))=𝑚𝑁(𝑡)𝛾𝑀(𝑡),𝑁(𝑡)=𝑚(𝑡)=𝑀(𝑡)−𝑡,𝑁(𝑡)−𝑡, (11) and assume that 1𝑆_𝐶()𝐹𝑡−𝑎^∗()1𝑑𝑎,𝑑𝑡<∞,𝑡²𝑆_𝐶()𝐹𝑡−𝑎^∗(𝑑𝑎,𝑑𝑡)<∞, (12) where 𝐹^∗(⋅,⋅) is the joint distribution function of (𝐴,𝑇).

Lemma 1 gives the Hadamard derivative of 𝛾 at (𝑀(𝑡),𝑁(𝑡)).

Lemma 1. For 𝑡∈[0,𝐿], the function 𝛾 is Hadamard-differentiable at (𝑀(𝑡),𝑁(𝑡)), where its derivative is equal to 𝛾^′_{(𝑀(𝑡),𝑁(𝑡))}ℎ₁(⋅),ℎ₂(⋅)=ℎ₁(𝑡)𝑁(𝑡)−𝑀(𝑡)ℎ₂(𝑡)𝑁²(𝑡). (13)

Lemma 2. Assume the conditions (i–iv) and (12) hold, for every 𝑡∈[0,𝐿], as 𝑛→∞. Then(i)𝑍_1𝑛(𝑡)=√𝑛(𝑀(𝑡)−𝑀(𝑡)) converges in distribution to a zero mean normal variable 𝑍₁ with the variance function 𝐸(𝐼(𝑌≥𝑡)𝛿/𝑆²_𝐶(𝑌−𝐴))−𝑀²(𝑡);(ii)𝑍_2𝑛(𝑡)=√𝑛(𝑁(𝑡)−𝑁(𝑡)) converges in distribution to a zero mean normal variable 𝑍₂ with the variance function 𝐸(𝐼(𝑌>𝑡)𝛿/𝑌²𝑆²_𝐶(𝑌−𝐴))−𝑁²(𝑡).

Proof. Firstly, let us give the proof of the result in Lemma 2(i). Under the regular conditions (i–iv) and (12), since 𝐸(𝐼(𝑌>𝑡)𝛿/𝑆_𝐶(𝑌−𝐴))=𝑀(𝑡), by the central limit theorem, we can derive √𝑛1𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖−𝑀(𝑡)_ℒ𝐼()𝛿⟶𝑁0,𝐸𝑌>𝑡𝑆²_𝐶(𝑌−𝐴)−𝑀²(𝑡), (14) where −−^ℒ→ denotes weak convergence.
Since 1𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖(𝑡=1−𝑀)𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖−1𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖+1𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖(𝑡≤1−𝑀)𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖−1𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖+1𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖(𝑡≤1−𝑀)𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖1𝑆_𝐶𝑦_𝑖−𝑎_𝑖−1𝑆_𝐶𝑦_𝑖−𝑎_𝑖+1𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖−𝑀(𝑡), (15) and recall that 𝑆_𝐶(⋅) is the product-limit estimator of 𝑆_𝐶(⋅), so on the basis of (14) and the consistency of 𝑆_𝐶(⋅) (see [14]), we can obtain 1𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖(𝑡𝑍−𝑀)⟶0,𝑛⟶∞,Var_1𝑛𝐼𝑦=𝐸^𝑖>𝑡𝛿_𝑖𝑆²_𝐶𝑦_𝑖−𝑎_𝑖−𝑀²(𝑡)𝐼()𝛿⟶𝐸𝑌>𝑡𝑆²_𝐶(𝑌−𝐴)−𝑀²(𝑡),𝑛⟶∞. (16)
Since 𝐼(𝑦_𝑖>𝑡)𝛿_𝑖/𝑆_𝐶(𝑦_𝑖−𝑎_𝑖), 𝑖=1,2,…,𝑛, 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 𝐸(𝐼(𝑌>𝑡)𝛿/𝑌𝑆_𝐶(𝑌−𝐴))=𝑁(𝑡), by the central limit theorem, we can derive √𝑛1𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑦_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖−𝑁(𝑡)_ℒ𝐼()𝛿⟶𝑁0,𝐸𝑌>𝑡𝑌²𝑆²_𝐶(𝑌−𝐴)−𝑁²(𝑡). (17)
Since 1𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑦_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖(𝑡=1−𝑁)𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑦_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖−1𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑦_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖+1𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑦_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖(𝑡≤1−𝑁)𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑦_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖−1𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑦_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖+1𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑦_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖(𝑡≤1−𝑁)𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑦_𝑖⁻¹1𝑆_𝐶𝑦_𝑖−𝑎_𝑖−1𝑆_𝐶𝑦_𝑖−𝑎_𝑖+1𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑦_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖−𝑁(𝑡), (18) so on the basis of (17) and the consistency of 𝑆_𝐶(⋅), we can derive 1𝑛^𝑛_𝑖=1𝐼𝑦_𝑖>𝑡𝛿_𝑖𝑦_𝑖𝑆_𝐶𝑦_𝑖−𝑎_𝑖(𝑡𝑍−𝑁)⟶0,𝑛⟶∞,Var_2𝑛𝐼𝑦=𝐸^𝑖>𝑡𝛿_𝑖𝑦²_𝑖𝑆²_𝐶𝑦_𝑖−𝑎_𝑖−𝑁²(𝑡)𝐼()𝛿⟶𝐸𝑌>𝑡𝑌²𝑆²_𝐶(𝑌−𝐴)−𝑁²(𝑡),𝑛⟶∞. (19)
Since 𝐼(𝑦_𝑖>𝑡)𝛿_𝑖/𝑦_𝑖𝑆_𝐶(𝑦_𝑖−𝑎_𝑖), 𝑖=1,2,…,𝑛, 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.

On the basis of Lemmas 1 and 2, now we can discuss the asymptotic property of 𝑚(𝑡).

Theorem 3. Assume the conditions (i–iv) and (12) hold, for every 𝑡∈[0,𝐿], as 𝑛→∞, then √𝑛(𝑚(𝑡)−𝑚(𝑡)) converges in distribution to a zero mean normal variable with the variance function (𝐸𝑍²₁(𝑡)/𝑁²(𝑡))−(𝑀(𝑡)𝐸𝑍₁(𝑡)𝑍₂(𝑡)/𝑁³(𝑡))+(𝑀²(𝑡)𝐸𝑍²₂(𝑡)/𝑁⁴(𝑡)).

Proof. Under the conditions (i–iv) and (12), Lemma 2 allows a straightforward application of Lemma 1. So we can derive √𝑛(𝑚(𝑡)−𝑚(𝑡))⟶^𝑑𝒩_𝑚=𝑍₁(𝑡)𝑁(𝑡)−𝑀(𝑡)𝑍₂(𝑡)𝑁²(𝑡), (20) where →−^𝑑 denotes convergence in distribution. 𝒩_𝑚 is Gaussian as a linear form on two Gaussian variables and it is easy to check that 𝐸𝒩_𝑚=0, since 𝑍₁ and 𝑍₂ are centered. It is also easy to calculate the variance of 𝒩_𝑚 and it can be expressed as Var𝒩_𝑚=𝐸𝑍²₁(𝑡)𝑁²(𝑡)−𝑀(𝑡)𝐸𝑍₁(𝑡)𝑍₂(𝑡)𝑁³(𝑡)+𝑀²(𝑡)𝐸𝑍²₂(𝑡)𝑁⁴(𝑡). (21) The proof is completed.

4. Simulations

To compare the performance of our proposed estimator with the one in (6), we conducted a series of Monte Carlo simulations with 𝑛=100 and 200. In these simulations, the underlying survival time 𝑇⁰ has a uniform (0,1) distribution. Then the length-biased density function of 𝑇 is 𝑔(𝑡)=2𝑡, 𝑡∈(0,1), and the left-truncated time 𝐴 is uniformly distributed on [0,𝑇]. We simulated the censoring time 𝐶 from either a uniform (0,1), a uniform (0,5/3) or a uniform (0,5) distribution, corresponding to a censoring rate of 50%, 30%, and 10%.

Table 1 summarizes the Monte Carlo bias, standard deviation, and mean square error for each estimator at 𝑡=0.2,0.4,0.6 and 0.8 based on 1000 replications. The survival probabilities at the selected time-points are 0.8, 0.6, 0.4, and 0.2. The corresponding true values of 𝑚(𝑡) at the selected time-points are 0.4, 0.3, 0.2, and 0.1, respectively.

Table 1 

Simulation results for nonparametric estimators based on 1000 replications.

The outcomes show that the proposed estimator works very well. Through the numerical results in Table 1, we derive the following conclusions:(1)𝑚(𝑡) outperforms 𝑚(𝑡) obviously in terms of Monte Carlo bias and mean square error for every values of 𝑡 when 𝑛=200 or 𝑛=100;(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, 𝑚(𝑡) outperforms 𝑚(𝑡) 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.

Acknowledgment

This research is supported by the National Science Foundation of China Grants 11371227 and 10921101.