# Rank Estimation for Mean Residual Life Transformation Model

Authors: Xiaoping Chen
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper proposes a new and important class of mean residual life regression model, which is called the mean residual life transformation model.  The link function is assumed to be unknown and increasing in its second argument, but it is permitted to be not differentiable. The mean residual life transformation model encompasses the proportional mean residual life model, the additive mean residual life model, and so on. Under maximum rank correlation estimation, we present the estimation procedures, whose asymptotic and finite sample properties are established. The consistent variance can be estimated by a resampling method via perturbing theU-statistics objective function repeatedly which avoids the usual sandwich choice. Monte Carlo simulations reveal good finite sample performance and the estimators are illustrated with the Oscar data set.


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## Body

## 1. Introduction

The mean residual life function at timet is the remaining life expectancy for those given survival up to t. It is defined as(1)mt=ET−t|T>t=∫t∞SuduSt,where T is the time from the onset to the failure event of interest and St=PT>t is the survival function of T. The mean residual life function is often of interest and obviously important in many fields of application, such as industrial reliability study, actuarial research, and survival analysis. For example, when T is the patient’s remaining life from cancer to death, mt is the remaining life expectancy for those given survival up to t. The patients and physicians both care about how long cancer patients can survive. When t=300 is the minimum score of Graduate Record Examination foreign language test for a school graduate program, mt is the average excess score for those more than 300 points. From model (1), m0=ET, and it is easy to induce that the conditional survival function of T given covariate X:(2)St|X=m0|Xmt|Xexp−∫0t1mu|Xdu.How to describe the effect of the factors on the remaining residual life expectancy has aroused much interest (Song et al., Wang et al., Lin et al., and therein [1–3]). For example, they are interested in exploring the relation between the remaining survival time of cancer patients and age, smoking, and various treatments. All kinds of semiparametric statistical models have been widely used for assessing such relation. For instance, in order to concern with the correlation between mt|X and X′β0, Oakes and Dasu [4] studied the proportional mean residual life model:(3)mt|X=m0texpX′β0,where m0t is the unknown baseline hazard function, covariate X is p-dimensional vector, and β0 is the true parameter. The proportional mean residual life model can compare the ratio when the covariate is a binary indicator. For example,(4)mt|X=1mt|X=0=expβ0,where X=1, combined treatment, and X=0, no treatment. For instance, β0=log3 means that the combined treatment would be 3 times the mean remaining life as compared to no treatment.In model (3) without censoring, Maguluri and Zhang [5] introduced the estimation method for β0. The regression methods for model (3) with right censoring data have been studied by Chen and Cheng [6]. Chen and Cheng [6] proposed the estimating equations which can be used to estimate m0· and β0:(5)∑im0tdNit−ZitdΛit;β,m0=0;∑i∫Xim0tdNit−ZitexpXi′β0dt+dm0t=0.Sun and Zhao [7] presented a class of mean residual life regression model:(6)mt|X=m0tgX′β0,where g·≥0 is a prespecified link function and is assumed to be continuous almost everywhere and twice differentiable. In model (6) with censoring data, Sun and Zhao [7] proposed the estimating equations which can be used to estimate m0· and β0:(7)∑im0tdNit−ZitgXi′β0dt+dm0t=0,∑i∫g1Xi′βgXi′βXim^0t,βdNit−Zit1gXi′βdt+dm^0t,β=0,where g1x=dgx/dx. They also developed estimation procedures and a goodness-of-fit test for model (6) with right censoring data.Just like the additive hazards regression model being an attractive alternative to the Cox proportional hazards model, Chen [8] considered the additive mean residual life model:(8)mt|X=m0t+X′β0.Chen [8] adapted estimating equations for estimating parameter β0. The model (8) explains the relative difference for mean residual life:(9)mt|X=1−mt|X=0=β0.In this paper, we consider a new and important class of mean residual life regression model given by(10)mt|X=ht,X′β0,where link function h·,·>0 is assumed to be an unknown and increasing function in its second argument. The function h·,· is permitted to be not differentiable. Model (10) is called the mean residual life transformation model. Obviously, the mean residual life transformation model includes the proportional mean residual life model (3) and the additive mean residual life model (8) and partially extends model (6). Model (10) is robust, but traditional methods like the counting process approach cannot be used to estimate the unknown regression parameter β0 because the estimating equations depend on the differentiation of link function.In this paper, under maximum rank correlation estimation, we propose the smoothed objective function to estimate unknown regression parameterβ0 for model (10) with complete data and right censoring data. We motivate this paper in three ways. First, the consistencies and asymptotic normalities of such approximated estimators for the general mean residual life model have not been established yet in the literature. Second, the smoothed objective functions make it feasible to adapt the rank-based approach, without loss of asymptotic efficiency. Third, the censoring variable is correlated, not being independent with a covariate.The rest of this article is organized as follows. In the next section, we give estimation procedures for model (10) with complete data. Section 3 considers the extension for right censoring data. A numerical study of Monte Carlo simulations and an application to the Oscar data are conducted in Section 4 to investigate the finite sample properties. Section 5 discusses future research.


## 2. Estimation

Without loss of generality, we suppose the survival timeT is a positive random variable and T is absolutely continuous with a density ft with respect to the Lebesgue measure. In the mean residual life transformation model (10), the link function h·,· is unknown; thus, the approach of estimation equations cannot estimate parameter β0 for model (10). Under mean residual life transformation model (10) with complete data, we present the following smoothed objective function to estimate parameter β0:(11)βn=argmaxβ∈ℬ1nn−1∑i≠jITi>Tj×1+exp−Xi′β−Xj′βσn−1,where ℬ is a parameter subset of ℝp, σn is a strictly positive and decreasing number satisfying limn⟶∞σn=0, and I is the indicator function.To obtain the consistency ofβn, the sufficient and necessary condition of the following proposition is important.We list four regularity assumptions for rank estimation below. All assumptions are ordinary, which are also used in other literature.Assumption 1.
(Covariates):a the support of the regressor X=X1,X2,…,Xp is not contained in a proper linear subspace of ℝp. bX’s first component X1 has an everywhere positive Lebesgue density, conditional on X2,…,Xp.Assumption 2.
(Normalization): the parameter spaceℬ is a compact subset of β=β1,β2,…,βp∈ℝp:β1=1.
To identify the parameterβ0, Assumptions 1 and 2 are sufficient conditions. To obtain the asymptotic normality, we list Assumptions 3 and 4, which were presented by Sherman [9].Assumption 3.
LetN be a neighborhood of true parameter β0.(i)
All mixed second partial derivatives ofτz,· exist on N, for any z.(ii)
There exists an integrable functionMz. For any z and β∈N, Mz satisfies(12)∇2τz,β−∇2τz,β0≤Mzβ−β0,where cij=∑i,jcij21/2.(iii)
E∇1τz,β02<∞.(iv)
E∇2τz,β0<∞, where ∇2τz,β=∑i1,i2∂2/∂βi1∂βi2τz,β.(v)
Thep−1∗p−1 dimensional matrix E∇2τz,β0 is negative definite.Assumption 4.
LetN be a neighborhood of true parameter β0. Then,(i)
All mixed second partial derivatives ofτcz,· exist on N, for any z.(ii)
There exists an integrable functionMcz. For any z and β∈N, Mcz satisfies(13)∇2τcz,β−∇2τcz,β0≤Mczβ−β0.(iii)
E∇1τcz,β02<∞.(iv)
E∇2τcz,β0<∞.(v)
Thep−1∗p−1 dimensional matrix E∇2τcz,β0 is negative definite.Proposition 1.
Under mean residual life transformation model (10) with complete data,(14)PTi>Tj|Xi,Xj>PTi<Tj|Xi,Xj,iffXi′β0>Xj′β0.
It is known that the interest parameterβ0 is not unique, that is, not identifiable. To avoid identifiability problem and without loss of generality, the first component of β0 is assumed to be 1, that is, β0=1,θ0′′. Recall that Z=T,X denotes an observation from the distribution P on the set S⊆ℝ⊗ℝp. For each z in S, define(15)τz,β=EIt>TIx′β>X′β+EIT>tIX′β>x′β.
Then consistency and asymptotic normality of βn are given by the following Theorem 1.Proof .
of Proposition1.(16)PTi>Tj|Xi,Xj=∫0∞PTi>tjdFjtj=∫0∞SitjdFjtj=∫0∞Sitjfjtjdtj=∫0∞SitjλjtjSjtjdtj,where the fourth equality holds since the hazard function λjt=fjt/Sjt. Then,(17)PTi>Tj|Xi,Xj−PTj>Ti|Xi,Xj=∫0∞SitjλjtjSjtjdtj−∫0∞SjtiλitiSitidti=∫0∞SitSjtλjt−λitdt.
Because(18)St|X=m0|Xmt|Xexp−∫0t1mu|Xdu,the derivative of St|X is(19)dSt|XdX′β0=1m2t|Xdmt|XdX′β0mX′Xexp−∫0t1mu|Xdu+mt|X⋅dm0|XdX′β0exp−∫0t1mu|Xdu+m0|Xexp−∫0t1mu|Xdu−∫0t−1m2u|Xdmt|XdX′β0du.
Becausemt|X is an increasing function in X′β0, that is, dmt|X/dX′β0≥0, then(20)dSt|XdX′β0≥0.
Thus,St|X is an increasing function of X′β0. Because Sit=exp−Λit, where Λit=∫0tλiudu is a cumulative hazard function. Hence, λit is a decreasing function of Xi′β0, that is,(21)λjt>λitif and only ifXi′β0>Xj′β0.
Therefore,(22)PTi>Tj|Xi,Xj>PTj>Ti|Xi,Xj,if and only ifXi′β0>Xj′β0.Theorem 1.
Under Assumptions1 and 2,(23)βn⟶pβ0,where ⟶p denotes convergence in probability. Under Assumptions 1-3,(24)nβn−β0⟶D0W,where ⟶D denotes convergence in distribution and W has a normal distribution N0,V−1ΔV−1 with(25)Δ=E∇1τ·,β0∇1τ·,β0′,2V=E∇2τ·,β0.
From theorem1, we find that the asymptotic variance of βn is V−1ΔV−1, which is equivalent to nonsmooth. Also, the smooth estimation βn has another advantage, which avoids the situation that the estimation of β0 involves maximizing indicator function. The asymptotic variance of βn can be estimated by a resampling method, which was studied by Cai et al. [10]. For each z in S, let the mean and variance of the positive random variable U both be 1. Define(26)βn∗=argmaxβ∈ℬ1nn−1∑i≠jUiUjIti>tj×1+exp−xi′β−xj′βσn−1.Theorem 2.
Under Assumptions1–3,(27)nβn∗−βn⟶D0W∗,where W∗ follows N0,V−1ΔV−1 and V and Δ are given in Theorem 1.
From theorem2, we find that the asymptotic variance of βn can be approximated by the sample variance of βn∗. The proof of Theorem 2 is omitted because we can use similar proof as Cai et al. [10].


## 3. Extension

For right censoring data, we observe the dataZi=Vi,δi,Xi, where δi=ITi≤Ci, Vi=TiIδi=1+CiIδi=0. Ci are right censoring random variables, i=1,…,n.Under model (10) with right censoring data, we present the following smoothed objective function to estimate parameter β0:(28)β˜=argmaxβ∈ℬG˜nβ,where(29)G˜nβ=1nn−1∑i≠jδjIVi>Vj×1+exp−Xi′β−Xj′βσn−1.ℬ is a parameter subset of ℝp, and σn is a strictly positive and decreasing number satisfying limn⟶∞σn=0. The objective function (28) is inspired by Khan and Tamer [11] and Song et al. [12].For obtaining the consistency ofβ˜, the identification (i.e., uniquely maximum) is an important condition. As Khan and Tamer [11] proposed, define(30)V1i=δiVi+1−δi+∞,V0i=Vi.Proposition 2.
Under mean residual life transformation model (10) with censoring data,(31)PV0i>V1j|Xi,Xj>PV1i<V0j|Xi,Xj,iffXi′β0>Xj′β0.
Recall thatZ=V,δ,X denotes an observation from the distribution P on the set S⊆ℝ⊗ℝ⊗ℝp. For each z in S, define(32)τcz,β=EδIv>VIx′β>X′β+EδIV>vIX′β>x′β.
Then consistency and asymptotic normality of β˜ are given by Theorem 3.Proof .
of Proposition2.
First of all, we calculate the conditional expectation ofV0j>V1i given Xi,Xj,Ci,Cj, which can be decomposed as two expressions:(33)PV0j>V1i|Xi=xi,Xj=xj,Ci=ci,Cj=cj=PV0j>V1i,δi=1|xi,xj,ci,cj+PV0j>V1i,δi=0|xi,xj,ci,cj=PV0j>V1i,δi=1|xi,xj,ci,cj=PV0j>V1i,δi=1,δj=1|xi,xj,ci,cj+PV0j>V1i,δi=1,δj=0|xi,xj,ci,cj=defI1+I2.
The two above expressions are calculated as follows:(34)I1=PTj>Ti,Ti≤Ci,Tj≤Cj|xi,xj,ci,cj=∫0ci∧cj∫ticjdFjtjdFiti=∫0ci∧cjFjcj−FjtidFiti=FjcjFici∧cj−∫0ci∧cjFjtidFiti,I2=PCj>Ti,Ti≤Ci,Tj>Cj|xi,xj,ci,cj=PTi<Ci∧Cj,Tj>Cj|xi,xj,ci,cj=Fici∧cj1−Fjcj.Thus,(35)I1+I2=Fici∧cj−∫0ci∧cjFjtidFiti=∫0ci∧cj1−FjtidFiti=∫0ci∧cjSjtidFiti.
Similarly, the conditional expectation ofV0i>V1j given Xi,Xj,Ci,Cj is(36)PV0i>V1j|xi,xj,ci,cj=∫0cj∧ciSitjdFjtj.
The rest of the proof of Proposition2 is the same as Proposition 1.Theorem 3.
Under Assumptions1 and 2,(37)β˜⟶pβ0,where ⟶p denotes convergence in probability. Under Assumptions 1, 2, and 4,(38)nβ˜−β0⟶D0W˜,where ⟶D denotes convergence in distribution and W has a normal distribution N0,Vc−1ΔcVc−1 with(39)Δc=E∇1τc·,β0∇1τc·,β0′,2Vc=E∇2τc·,β0.
From Theorem3, we find that the asymptotic variance of β˜ is Vc−1ΔcVc−1, which is equivalent to nonsmooth. Also, the smooth estimation β˜ has another advantage, which avoids the situation that the estimation of β0 involves maximizing indicator function. The asymptotic variance of β˜ can be estimated by a resampling method, which was studied by Cai et al. [10]. For each z in S, let the mean and variance of the positive random variable U both be 1. Define(40)β˜∗=argmaxβ∈ℬG˜n∗β,where(41)G˜n∗β=1nn−1∑i≠jUiUjδjIvi>vj×1+exp−xi′β−xj′βσn−1.Proof .
of Theorem3.
We consider the nonsmoothly objective function ofβ:(42)β˜non=argmaxβ∈ℬ1nn−1∑i≠jIV0i>V1jIXi′β>Xj′β=argmaxβ∈ℬ1nn−1∑i≠jδjITi>TjIXi′β>Xj′β.Define(43)Gnβ=1nn−1∑i≠jδjITi>TjIXi′β>Xj′β;Gβ=EGnβ.Then,(44)Gnβ−Gβ=1nn−1∑i≠jδjITi>TjIXi′β>Xj′β−Eδ2IT1>T2IX1′β>X2′β≡1nn−1∑i≠jfzi,zj,β,that is,(45)fzi,zj,β=δjITi>TjIXi′β>Xj′β−Eδ2IT1>T2IX1′β>X2′β.
First of all, we prove the consistency ofβ˜non.
Identification (Gβ has unique maximum at β0) follows from Proposition 2 and Assumption 1. We omit the details, which follow from identical steps used in proving Theorem 2.1 (Khan and Tamer [11]).
Obviously,Gnβ−Gβ is a degenerated U-statistic of order 2. Because the objective function is composed of an indicator, f·,·,β,β∈ℬ is Euclidean with an envelope. We can derive from Corollary 7 of Sherman [9] that(46)supβ∈ℬGnβ−Gβ=Op1n.
Note that the conclusion is stronger thansupβ∈ℬGnβ−Gβ=op1.
By Assumption1, it is easy to obtain that PX1′β=X2′β=0, where X1′≠X2′. Then,(47)limβ˜non⟶βIX1′β˜non>X2′β˜non−IX1′β>X2′β=0,with probability one. Thus, continuity (Gβ is continuous about β) follows directly from the dominated convergence theorem.
Because of Assumption2 (compact parameter space), identification, uniform convergence, and continuity, we obtain β˜non⟶pβ0.
In order to obtainβ˜⟶pβ0, it suffices to show that supβ∈ℬG˜nβ−Gnβ⟶p0. For any ϵ>0,(48)G˜nβ−Gnβ=1nn−1∑i≠jδjITi>TjIXi′β>Xj′β−1+exp−Xi′β−Xj′βσn−1≤1nn−1∑i≠jIXi′β−Xj′β≥εδjITi>TjIXi′β>Xj′β−1+exp−Xi′β−Xj′βσn−1+1nn−1∑i≠jIXi′β−Xj′β<εδjITi>TjIXi′β>Xj′β−1+exp−Xi′β−Xj′βσn−1.
Note that(49)IXi′β>Xj′β−1+exp−Xi′β−Xj′βσn−1IXi′β−Xj′β≥ε≤exp−Xi′β−Xj′βσnIXi′β−Xj′β≥ε≤exp−εσn.
Under Theorem 7 of Nolan and Pollard [13], it is easy to obtain that(50)1nn−1∑i≠jIXi′β−Xj′β<ε⟶EIX1′β−X2′β<εa.s.
According to Assumption1, EIX1′β−X2′β<ϵ⟶0 as ϵ⟶0. Hence,(51)supβ∈ℬG˜nβ−Gnβ⟶p0.Define(52)τsz,β=EδIt>T1+exp−x′β−X′βσn−1+EδIT>t1+exp−X′β−x′βσn−1.
BecauseG˜nβ is an U-statistic of order 2, it is easy to obtain that by the Hoeffding decomposition:(53)G˜nβ−G˜nβ0=Γnβ+Pnf1n·,β+Un2f2n·,·,β.
Similarly to the proof of Theorem4 of Sherman [9], we obtain(54)Γnβ=12β−β0′E▽2τsz,β02β−β0+opβ−β02;Pnf1n·,β=1nβ−β0′nPn∇1τs·,β0+opβ−β02;Un2f2n·,·,β=op1n,uniformly inop1neighborhoods ofβ0.
According to Assumption4, we obtain(55)∇1τsz,β=∇1τcz,β+Opσn;,∇2τsz,β=∇2τcz,β+Opσn.
Hence, we obtain(56)G˜nβ−G˜nβ0=12β−β0′Vcβ−β0+1nβ−β0′Wn+opβ−β02+op1n,
uniformly inop1 neighborhoods of β0, where Wn⟶dN0,Δc, Vc=E∇2τcz,β0/2,(57)Δc=E∇1τcz,β0∇1τcz,β0′.
Then, by Theorem2 of Sherman [9],(58)nβ˜−β0⟶D0W˜,where ⟶D denotes convergence in distribution and W has a normal distribution N0,Vc−1ΔcVc−1.Theorem 4.
Under Assumptions1, 2, and 4,(59)nβ˜∗−β˜⟶D0Wc∗,where Wc∗ follows N0,Vc−1ΔcVc−1 and Vc and Δc are given in Theorem 3.


## 4. Numerical Examples

### 4.1. Monte Carlo Results

This section aims to conduct the finite sample properties of parameter estimationβn for model (10) with complete data and β˜ for model (10) with right censoring data by a number of simulation studies. Simulation results are reported for n=100 and 400 observations, with 500 replications. We choose the smoothed parameter σn=on−1/2. Weighted random variables U follow a standard exponential distribution with 500 replications. The empirical bias (BIAS), empirical standard deviation (SD), an average of the estimated standard errors (SE), and empirical coverage probability of 95% confidence interval (CP) are reported in Tables 1–4.Table 1
Simulation results ofβn and βcom for the additive model.
βSizem0tMethodBIASSDSECPn=100t+1βn0.10941.58641.509192.8t+1βcom0.24741.45221.314585.2n=1001βn0.08610.62310.657697.01βcom0.17620.56140.418381.2n=400t+1βn0.06340.47360.458193.0t+1βcom0.15040.39670.384293.2n=4001βn0.02810.16510.187296.41βcom0.07940.11470.171190.4Table 2
Simulation results ofβ˜ and βrc for the proportional model with right censoring data.
SizeβRateMethodBIASSDSECPn=10010%β˜0.01830.18500.194696.230%β˜0.02740.22390.211893.860%β˜0.03820.27080.243893.4n=10010%βrc−0.01510.12520.114588.230%βrc−0.06710.13100.122586.260%βrc−0.23220.17970.135961.4n=40010%β˜−0.00290.10370.101794.630%β˜0.00930.12180.128395.460%β˜0.02080.16410.169195.8n=40010%βrc−0.00850.05770.053591.630%βrc−0.04040.07770.057483.660%βrc−0.15650.11470.067355.0Table 3
Simulation results for model (62).
SizeβRateBIASSDSECPn=10010%0.01010.08720.118296.830%0.01470.12870.146396.460%0.02680.13960.153897.2n=40010%0.00320.07190.068294.430%0.00940.08010.083496.260%0.01380.12460.136995.4Table 4
Results for Oscar data.
MethodCovariateCoef.SE^p value95% CIβ˜Num.1.00170.21262.4628e−006(0.5849, 1.4184)β˜Id.0.31020.26030.2334(−0.2000, 0.8204)βrcNum.1.37910.18376.0366e−14(1.0190, 1.7391)βrcId.0.59280.37910.1178(−0.1502, 1.3358)Our design for model (10) is specified by(60)mt|X=ht,X1+X2β,where β=1 and ht,X1+X2β is assumed to be an increasing function in X1+X2β. The following subsections consider three functional forms for link function h·,· and different covariates X=X1,X2.


#### 4.1.1. Complete Data

First, we want to evaluate the performance of parameter estimationβn for model (10) with complete data. The first link function h·,· and different covariates X=X1,X2 which we consider are as follows:(61)Additive modelmt|X=ht,X1+X2β=m0t+X1+X2β,where β=1, m0t=t+1 or 1, X1 is generated from a uniform U0,1, X2 follows a chi-square distribution with 1 degree of freedom, and X1 is independent of X2. According to the conditional distribution function Ft|X=1−St|X, equation (2) and model (53), then T=F−1U.Letβcom denote the parameter estimation for additive mean residual life model (53) with complete data proposed by Chen [8]. Table 1 shows the simulation results of βn and βcom for the additive model (53) with complete data. The biases and standard errors of parameter estimation βn and βcom decrease as sample size from 100 to 400. The biases and standard errors of parameter estimation βn perform well as sample size n=400. The average of estimated standard errors of βn are close to empirical standard deviations, and the empirical coverage probabilities are approximately 95%. The biases of parameter estimation βn are smaller than βcom. The standard errors of βn are larger than βcom; it is probably because that mean residual life transformation model (11) is more general than the additive model (8). The results of Table 1 demonstrate that the parameter estimation βn for the mean residual life transformation model (10) with complete data is robust and reasonable.


#### 4.1.2. Censoring Data

Second, we want to evaluate the performance of parameter estimationβ˜ for model (10) with right censoring data. The second link function h·,· and different covariates X=X1,X2 which we consider are as follows:(62)proportional modelmt|X=ht,X1+X2β=m0texpX1+X2β,where β=1, m0t=1, X1 follows a binary distribution of 0 or 1 with equal probabilities of 0.5, X2 follows a standard normal distribution, and X1 is independent of X2. Similar to the above section, we calculate survival time T for model (61). The right censoring variable C follows an exponential distribution, where the exponential parameter is selected to yield the censoring rate which is approximately 10%, 30%, and 60%, respectively.Letβrc be the parameter estimation for the proportional mean residual life model (3) with right censoring data being studied by Chen and Cheng [6]. The simulation results of β˜ and βrc for the proportional model with right censoring data are shown in Table 2. The mean biases and standard errors of β˜ and βrc increase when the censoring rate increases or sample size decreases. The average of estimated standard errors of β˜ is close to empirical standard deviations, and the empirical coverage probabilities are approximately 95%. The biases of parameter estimation β˜ are smaller than βrc. The standard errors of β˜ are larger than βrc; it is probably because that mean residual life transformation model (10) is more general than the proportional mean residual life model (3). The results of Table 2 demonstrate that the proposed estimator β˜ is comparable with βrc.


#### 4.1.3. Nondifferential Link Function

Third, we want to evaluate the performance of parameter estimationβ˜ for model (10) with right censoring data and nondifferential link function. The third link function h·,· and different covariates X=X1,X2 which we consider are as follows:(63)mt|X=ht,X1+X2β=m0tqX1+X2β,whereβ=1, m0t=1, qs=log1+esIs<3+esIs≥3, X1 follows a chi-square distribution with 1 degree of freedom, X2 is generated from a uniform U0,4, and X1 is independent of X2.The simulation results for model (62) with right censoring data and nondifferential link function are summarized in Table 3. Table 3 shows that β˜ performs well in finite sample size n = 100 and 400. The results are in accordance with the theory; that is, β˜ is asymptotically unbiased and normal. The mean biases and standard errors of β˜ decrease as sample size n increases. The 95% confidence intervals of β˜ are close to their nominal levels. With the censoring rate increasing, the mean biases and standard errors of β˜ increase. From Table 3, we find that β˜ is suitable for the link function h·,· which is not differentiable. Table 3 demonstrates that β˜ is reasonable for model (10) with right censoring data and nondifferential link function.


### 4.2. Application to Oscar Data

Redelmeier and Singh [14] compiled the Oscar Awards from 1929 to 2000 which list 766 nominees. We treat the data set as right-censored data. Because 327 died before the end of 2001, hence the censoring ratio is about 57.3%. Han et al. [15] elaborated the description of the Oscar data. Whether winning an Oscar Award increases life expectancy is an interesting topic.The number of total films in career, number of times the person was nominated for an Oscar (Num.), and whether the person won an Oscar (Id.) are chosen as covariates. We have intuitive thinking that there is a positive correlation between life expectancy and the number of total films in a career. To identify the parameter, we assume the coefficient of the number of total films in a career is 1. Random variableU follows a standard exponential distribution with 500 replications, and we choose σn=on−1/2. Because the survival time of nominees belongs to censoring data, we exploit the parameter estimator β˜ for mean residual life transformation model (10) and βrc for proportional mean residual life model (3) with censoring data.From Table4, the coefficients of whether the person won an Oscar for β˜ and βrc are positive, and the 95% confidence intervals of whether the person won an Oscar contain zero. So, we can deduce that a performer winning Oscar may not have a longer life span than those without winning.


## 4.1. Monte Carlo Results

This section aims to conduct the finite sample properties of parameter estimationβn for model (10) with complete data and β˜ for model (10) with right censoring data by a number of simulation studies. Simulation results are reported for n=100 and 400 observations, with 500 replications. We choose the smoothed parameter σn=on−1/2. Weighted random variables U follow a standard exponential distribution with 500 replications. The empirical bias (BIAS), empirical standard deviation (SD), an average of the estimated standard errors (SE), and empirical coverage probability of 95% confidence interval (CP) are reported in Tables 1–4.Table 1
Simulation results ofβn and βcom for the additive model.
βSizem0tMethodBIASSDSECPn=100t+1βn0.10941.58641.509192.8t+1βcom0.24741.45221.314585.2n=1001βn0.08610.62310.657697.01βcom0.17620.56140.418381.2n=400t+1βn0.06340.47360.458193.0t+1βcom0.15040.39670.384293.2n=4001βn0.02810.16510.187296.41βcom0.07940.11470.171190.4Table 2
Simulation results ofβ˜ and βrc for the proportional model with right censoring data.
SizeβRateMethodBIASSDSECPn=10010%β˜0.01830.18500.194696.230%β˜0.02740.22390.211893.860%β˜0.03820.27080.243893.4n=10010%βrc−0.01510.12520.114588.230%βrc−0.06710.13100.122586.260%βrc−0.23220.17970.135961.4n=40010%β˜−0.00290.10370.101794.630%β˜0.00930.12180.128395.460%β˜0.02080.16410.169195.8n=40010%βrc−0.00850.05770.053591.630%βrc−0.04040.07770.057483.660%βrc−0.15650.11470.067355.0Table 3
Simulation results for model (62).
SizeβRateBIASSDSECPn=10010%0.01010.08720.118296.830%0.01470.12870.146396.460%0.02680.13960.153897.2n=40010%0.00320.07190.068294.430%0.00940.08010.083496.260%0.01380.12460.136995.4Table 4
Results for Oscar data.
MethodCovariateCoef.SE^p value95% CIβ˜Num.1.00170.21262.4628e−006(0.5849, 1.4184)β˜Id.0.31020.26030.2334(−0.2000, 0.8204)βrcNum.1.37910.18376.0366e−14(1.0190, 1.7391)βrcId.0.59280.37910.1178(−0.1502, 1.3358)Our design for model (10) is specified by(60)mt|X=ht,X1+X2β,where β=1 and ht,X1+X2β is assumed to be an increasing function in X1+X2β. The following subsections consider three functional forms for link function h·,· and different covariates X=X1,X2.


### 4.1.1. Complete Data

First, we want to evaluate the performance of parameter estimationβn for model (10) with complete data. The first link function h·,· and different covariates X=X1,X2 which we consider are as follows:(61)Additive modelmt|X=ht,X1+X2β=m0t+X1+X2β,where β=1, m0t=t+1 or 1, X1 is generated from a uniform U0,1, X2 follows a chi-square distribution with 1 degree of freedom, and X1 is independent of X2. According to the conditional distribution function Ft|X=1−St|X, equation (2) and model (53), then T=F−1U.Letβcom denote the parameter estimation for additive mean residual life model (53) with complete data proposed by Chen [8]. Table 1 shows the simulation results of βn and βcom for the additive model (53) with complete data. The biases and standard errors of parameter estimation βn and βcom decrease as sample size from 100 to 400. The biases and standard errors of parameter estimation βn perform well as sample size n=400. The average of estimated standard errors of βn are close to empirical standard deviations, and the empirical coverage probabilities are approximately 95%. The biases of parameter estimation βn are smaller than βcom. The standard errors of βn are larger than βcom; it is probably because that mean residual life transformation model (11) is more general than the additive model (8). The results of Table 1 demonstrate that the parameter estimation βn for the mean residual life transformation model (10) with complete data is robust and reasonable.


### 4.1.2. Censoring Data

Second, we want to evaluate the performance of parameter estimationβ˜ for model (10) with right censoring data. The second link function h·,· and different covariates X=X1,X2 which we consider are as follows:(62)proportional modelmt|X=ht,X1+X2β=m0texpX1+X2β,where β=1, m0t=1, X1 follows a binary distribution of 0 or 1 with equal probabilities of 0.5, X2 follows a standard normal distribution, and X1 is independent of X2. Similar to the above section, we calculate survival time T for model (61). The right censoring variable C follows an exponential distribution, where the exponential parameter is selected to yield the censoring rate which is approximately 10%, 30%, and 60%, respectively.Letβrc be the parameter estimation for the proportional mean residual life model (3) with right censoring data being studied by Chen and Cheng [6]. The simulation results of β˜ and βrc for the proportional model with right censoring data are shown in Table 2. The mean biases and standard errors of β˜ and βrc increase when the censoring rate increases or sample size decreases. The average of estimated standard errors of β˜ is close to empirical standard deviations, and the empirical coverage probabilities are approximately 95%. The biases of parameter estimation β˜ are smaller than βrc. The standard errors of β˜ are larger than βrc; it is probably because that mean residual life transformation model (10) is more general than the proportional mean residual life model (3). The results of Table 2 demonstrate that the proposed estimator β˜ is comparable with βrc.


### 4.1.3. Nondifferential Link Function

Third, we want to evaluate the performance of parameter estimationβ˜ for model (10) with right censoring data and nondifferential link function. The third link function h·,· and different covariates X=X1,X2 which we consider are as follows:(63)mt|X=ht,X1+X2β=m0tqX1+X2β,whereβ=1, m0t=1, qs=log1+esIs<3+esIs≥3, X1 follows a chi-square distribution with 1 degree of freedom, X2 is generated from a uniform U0,4, and X1 is independent of X2.The simulation results for model (62) with right censoring data and nondifferential link function are summarized in Table 3. Table 3 shows that β˜ performs well in finite sample size n = 100 and 400. The results are in accordance with the theory; that is, β˜ is asymptotically unbiased and normal. The mean biases and standard errors of β˜ decrease as sample size n increases. The 95% confidence intervals of β˜ are close to their nominal levels. With the censoring rate increasing, the mean biases and standard errors of β˜ increase. From Table 3, we find that β˜ is suitable for the link function h·,· which is not differentiable. Table 3 demonstrates that β˜ is reasonable for model (10) with right censoring data and nondifferential link function.


## 4.1.1. Complete Data

First, we want to evaluate the performance of parameter estimationβn for model (10) with complete data. The first link function h·,· and different covariates X=X1,X2 which we consider are as follows:(61)Additive modelmt|X=ht,X1+X2β=m0t+X1+X2β,where β=1, m0t=t+1 or 1, X1 is generated from a uniform U0,1, X2 follows a chi-square distribution with 1 degree of freedom, and X1 is independent of X2. According to the conditional distribution function Ft|X=1−St|X, equation (2) and model (53), then T=F−1U.Letβcom denote the parameter estimation for additive mean residual life model (53) with complete data proposed by Chen [8]. Table 1 shows the simulation results of βn and βcom for the additive model (53) with complete data. The biases and standard errors of parameter estimation βn and βcom decrease as sample size from 100 to 400. The biases and standard errors of parameter estimation βn perform well as sample size n=400. The average of estimated standard errors of βn are close to empirical standard deviations, and the empirical coverage probabilities are approximately 95%. The biases of parameter estimation βn are smaller than βcom. The standard errors of βn are larger than βcom; it is probably because that mean residual life transformation model (11) is more general than the additive model (8). The results of Table 1 demonstrate that the parameter estimation βn for the mean residual life transformation model (10) with complete data is robust and reasonable.


## 4.1.2. Censoring Data

Second, we want to evaluate the performance of parameter estimationβ˜ for model (10) with right censoring data. The second link function h·,· and different covariates X=X1,X2 which we consider are as follows:(62)proportional modelmt|X=ht,X1+X2β=m0texpX1+X2β,where β=1, m0t=1, X1 follows a binary distribution of 0 or 1 with equal probabilities of 0.5, X2 follows a standard normal distribution, and X1 is independent of X2. Similar to the above section, we calculate survival time T for model (61). The right censoring variable C follows an exponential distribution, where the exponential parameter is selected to yield the censoring rate which is approximately 10%, 30%, and 60%, respectively.Letβrc be the parameter estimation for the proportional mean residual life model (3) with right censoring data being studied by Chen and Cheng [6]. The simulation results of β˜ and βrc for the proportional model with right censoring data are shown in Table 2. The mean biases and standard errors of β˜ and βrc increase when the censoring rate increases or sample size decreases. The average of estimated standard errors of β˜ is close to empirical standard deviations, and the empirical coverage probabilities are approximately 95%. The biases of parameter estimation β˜ are smaller than βrc. The standard errors of β˜ are larger than βrc; it is probably because that mean residual life transformation model (10) is more general than the proportional mean residual life model (3). The results of Table 2 demonstrate that the proposed estimator β˜ is comparable with βrc.


## 4.1.3. Nondifferential Link Function

Third, we want to evaluate the performance of parameter estimationβ˜ for model (10) with right censoring data and nondifferential link function. The third link function h·,· and different covariates X=X1,X2 which we consider are as follows:(63)mt|X=ht,X1+X2β=m0tqX1+X2β,whereβ=1, m0t=1, qs=log1+esIs<3+esIs≥3, X1 follows a chi-square distribution with 1 degree of freedom, X2 is generated from a uniform U0,4, and X1 is independent of X2.The simulation results for model (62) with right censoring data and nondifferential link function are summarized in Table 3. Table 3 shows that β˜ performs well in finite sample size n = 100 and 400. The results are in accordance with the theory; that is, β˜ is asymptotically unbiased and normal. The mean biases and standard errors of β˜ decrease as sample size n increases. The 95% confidence intervals of β˜ are close to their nominal levels. With the censoring rate increasing, the mean biases and standard errors of β˜ increase. From Table 3, we find that β˜ is suitable for the link function h·,· which is not differentiable. Table 3 demonstrates that β˜ is reasonable for model (10) with right censoring data and nondifferential link function.


## 4.2. Application to Oscar Data

Redelmeier and Singh [14] compiled the Oscar Awards from 1929 to 2000 which list 766 nominees. We treat the data set as right-censored data. Because 327 died before the end of 2001, hence the censoring ratio is about 57.3%. Han et al. [15] elaborated the description of the Oscar data. Whether winning an Oscar Award increases life expectancy is an interesting topic.The number of total films in career, number of times the person was nominated for an Oscar (Num.), and whether the person won an Oscar (Id.) are chosen as covariates. We have intuitive thinking that there is a positive correlation between life expectancy and the number of total films in a career. To identify the parameter, we assume the coefficient of the number of total films in a career is 1. Random variableU follows a standard exponential distribution with 500 replications, and we choose σn=on−1/2. Because the survival time of nominees belongs to censoring data, we exploit the parameter estimator β˜ for mean residual life transformation model (10) and βrc for proportional mean residual life model (3) with censoring data.From Table4, the coefficients of whether the person won an Oscar for β˜ and βrc are positive, and the 95% confidence intervals of whether the person won an Oscar contain zero. So, we can deduce that a performer winning Oscar may not have a longer life span than those without winning.


## 5. Discussion

The mean residual lifemt is often of interest and obviously important in many fields of application, such as industrial reliability study, actuarial research, and survival analysis. This paper presents a general class of mean residual life regression models, which contain the proportional mean residual life models and partially extends a class of mean residual life regression models studied by Sun and Zhao (2010). The link function is assumed to be unknown and increasing in its seconding argument, but it is permitted to be not differentiable. Traditional methods like the counting process approach cannot be used to estimate the unknown regression parameter β0. Under maximum rank correlation estimation, we propose the estimation procedures for model (10) with complete data and right censoring data.In addition to right censoring data, length-biased data often appear in survival analysis. For example, in unemployment data surveys, short-term unemployment does not register with the unemployment center. In the modeling process, if the length-biased data is not corrected, the duration of unemployment will be overestimated. Therefore, one interesting aspect is the parameter estimation of model (10) with length-biased data. Second, it is possible to construct an estimator for the link function. We leave these possible extensions for future research.



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*Source: 4864128-2020-12-21.xml*