# Statistical Inference for the Heteroscedastic Partially Linear Varying-Coefficient Errors-in-Variables Model with Missing Censoring Indicators

Authors: Yuye Zou; Chengxin Wu
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

In this paper, we focus on heteroscedastic partially linear varying-coefficient errors-in-variables models under right-censored data with censoring indicators missing at random. Based on regression calibration, imputation, and inverse probability weighted methods, we define a class of modified profile least square estimators of the parameter and local linear estimators of the coefficient function, which are applied to constructing estimators of the error variance function. In order to improve the estimation accuracy and take into account the heteroscedastic error, reweighted estimators of the parameter and coefficient function are developed. At the same time, we apply the empirical likelihood method to construct confidence regions and maximum empirical likelihood estimators of the parameter. Under appropriate assumptions, the asymptotic normality of the proposed estimators is studied. The strong uniform convergence rate for the estimators of the error variance function is considered. Also, the asymptotic chi-squared distribution of the empirical log-likelihood ratio statistics is proved. A simulation study is conducted to evaluate the finite sample performance of the proposed estimators. Meanwhile, one real data example is provided to illustrate our methods.


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## Body

## 1. Introduction

In regression analysis, for a long period of time, the flexible and refined statistical regression models are widely applied in theoretical study and practical application. The main results related to parameter regression models and nonparameter regression models are rather mature. Recently, semiparameter regression models can reduce the high risk of misspecification related to parameter regression models and avoid the “curse of dimensionality” for nonparametric regression models. Thanks to their advantage, semiparametric regression models enjoy consideration attention from statisticians. Semiparametric regression models have various forms. Specially, partially linear varying-coefficient errors-in-variables (PLVCEV) model, as a typical example, was introduced by You and Chen [1] and has the following form:(1)V=X⊤β+Z⊤αU+ε,ξ=X+e,where V is the response variable, X∈ℝp,Z∈ℝq are the covariates, β=β1,…,βp⊤ is a vector of p-dimensional unknown parameter, α⋅=α1⋅,…,αq⋅⊤ is an unknown q-dimensional vector of coefficient function, and ε is the random error. The measurement error e is independent of X,Z,U with mean zero and covariance matrix Σe. In order to identify the model, Σe is assumed to be known.As one general and flexible semiparametric model, model (1) includes a variety of models of interest. When X is observed exactly, model (1) boils down to be PLVC model [2, 3]. When Z=1, q=1 and X is observed exactly, model (1) reduces to partially linear regression model [4]. When X is observed exactly, and α⋅ is a constant vector, model (1) becomes a linear regression model. When q=1 and Z=1, model (1) reduces to partially linear EV model [5]. For model (1), You and Chen [1] proposed estimators of parametric and nonparametric components and showed their asymptotic properties. Liu and Liang [6] constructed the asymptotical normality of jackknife estimator for error variance and standard chi-square distribution of jackknife empirical log-likelihood statistic. Fan et al. [7] established penalized profile least squares estimation of parameter and nonparameter in the model.The literature mentioned above assumed that the random errors are homoscedastic, which means that the random errorε is independent of X,Z,U. However, in many practical application fields, the error variance function may change with the variables. Heteroscedastic error models have attracted much attention of many scholars. For example, You et al. [8] considered the estimation of parametric and nonparametric parts for partially linear regression models with heteroscedastic errors. Fan et al. [9] constructed confidence regions of parameter for heteroscedastic PLVCEV model based on empirical likelihood method. Shen et al. [10] discussed estimation and inference for PLVC model with heteroscedastic errors. Xu and Duan [11] extended the results of Shen et al. [10] to efficient estimation for PLVCEV model with heteroscedastic errors.The above related works assumed that the responses are observed completely. However, in many practical fields, especially in biomedical studies and survival analysis, the response cannot be completely observed due to censored variables. Huang and Huang [12, 13] discussed the constructed confidence regions of the parameters for varying-coefficient single-index model and partially linear single-index EV model by empirical likelihood method under censored data, respectively. The aforementioned results require that the censoring indicators be always observed. However, the censoring indicators may not be observed completely. For example, the death of individual is attributable to the cause of interest that may require information that is not gathered or lost due to various reasons [14]. In this paper, we assume that the censoring indicators are missing at random (MAR), which is common and reasonable in statistical analysis with missing data [15]. There are a lot of works related to missing censoring indicators. For example, Wang and Dinse [16] and Li and Wang [17] proposed weighted least square estimators of unknown parameter and proved their asymptotical normality for linear regression model. Shen and Liang [18] discussed the estimation and variable selection for PLVC quantile regression model. Wang et al. [19] considered composite quantile regression for linear regression model. However, there is no literature focusing on the estimation and confidence regions of heteroscedastic errors model with right-censored data when the censoring indicators are MAR.In this paper, we consider modified profile least square (PLS) estimators of the unknown parameter and local linear estimators of the coefficient function. Besides the point estimation, we are also interested in interval estimation in terms of empirical likelihood (EL) method, which, first introduced by Owen [20], is a very effective method for constructing confidence regions, which enjoys a lot of nice properties over the normal approximation-based methods and bootstrap approach. Thanks to its advantage, there are a lot of literature-related EL methods to refer to. For instance, Fan et al. [21] considered penalized EL for high-dimensional PLCVEV model. Wang and Drton [22] established estimation for linear structural equation models with dependent errors based on EL method. Fan et al. [23] discussed weighted EL for heteroscedastic varying-coefficient partially nonlinear model with missing data. Zou et al. [24] considered EL inference for partially linear single-index EV model with missing censoring indicators.It is worth pointing out that it is innovative and interesting in studying the PLVCEV model with heteroscedastic errors under censoring indicators MAR. Thus, we consider estimation and confidence regions based on modified profiled LS method and EL inference, respectively. The main aims of this paper include the following aspects: (1) define a class of modified PLS estimators of the parameter and local linear estimators of coefficient function based on regression calibration, imputation, and inverse probability weighted approaches, and prove the asymptotical normality of the proposed estimators; (2) construct reweighted estimators of the parameter and coefficient function based on estimators of the error variance function, and establish the asymptotic properties of the proposed estimators; (3) develop the asymptotic standard chi-squared distribution of the empirical log-likelihood ratio functions, construct the confidence regions for the parameter, and propose the asymptotic distribution of the corresponding maximum EL estimators. Finally, a simulation study and a real data analysis are conducted to demonstrate the finite sample performance of the proposed procedures.The rest of this paper is organized as follows. In Section2, we construct modified PLE estimators of the parameter and local linear estimators of the coefficient function. In Section 3, we proposed empirical log-likelihood ratio statistics and maximum EL estimators. The main results are shown in Section 4. Section 5 presents simulation and real data analysis. In Section 6, we show some conclusions. The proofs of the main results are shown in Appendix.


## 2. Methodology

Suppose thatVi,Xi,Zi,Ui,i=1,…,n is a sample from model (1), that is,(2)Vi=Xi⊤β+Zi⊤αUi+εi,ξi=Xi+ei,where the model error εi satisfies Eεi|Xi,Zi,Ui=0 and Varεi|Xi,Zi,Ui=σ2Ui, which is an unknown function of Ui representing heteroscedastic error. In the practical application, the response Vi may be right censored by various reasons. Let Ci be censoring time with distribution function (df) G⋅. One can only observe Yi=minVi,Ci with df H and censoring indicator δi=IVi≤Ci. Define the missing indicator to be ςi, which is 0 if δi is missing; otherwise, it is 1. Throughout this article, we assume that Vi is independent of Ci, and δi is MAR, which implies that ςi and δi are conditional independent given Ti=Yi,Xi,Zi,Ui, i.e.,(3)Pδi=1|Yi,Xi,Zi,Ui,ςi=Pδi=1|Yi,Xi,Zi,Ui≔mTi.


### 2.1. Modified Profile Least Squares Estimation

The local linear regression technique is employed to estimate the coefficient functionα⋅. If α⋅ has twice continuous derivative at point u0, for u in a small neighborhood of u0, one can approximate α⋅ by the following expansion with Taylor expansion:(4)αju=αju0+αj′u0u−u0≔aj+bju−u0,j=1,…,q,where αj′u=∂αju/∂u. Then, aj,bj can be estimated by minimizing the following objective function:(5)∑i=1nδi1−GYiYi−Xi⊤β−∑j=1naj+bjUi−u0Zij2KUi−uh1n,where K⋅ is a kernel function, and 0<h1n⟶0 is a bandwidth sequence. Due to the missing indicators, some δi cannot be observed. Therefore, model (2) cannot be applied directly. One can replace δi with its conditional expectation mTi=Eδi|Ti. Thus, aj,bj can be defined as the minimizer of(6)∑i=1nmTi1−GYiYi−Xi⊤β−∑j=1naj+bjUi−u0Zij2KUi−uh1n.However, in practical fields, functionm⋅ is usually unknown. One can use parametric and nonparametric methods to estimate m⋅. However, when the covariates are high-dimensional, nonparametric estimation may cause “the curse of dimensionality.” Hence, throughout this paper, we assume that m⋅ follows a parametric model mT=mT,θ, where θ is an unknown parameter vector. Following Wang and Dinse [16], the estimator θ^n of θ can be obtained by maximizing the following likelihood function:(7)∏i=1nmTi,θςiδi1−mTi,θςi1−δi.Letδic=mTi,θ. Since Eδic=EmTi,θ=EEδi|Ti=Eδi, we replace δic with its estimator δ^ic=mTi,θ^n. Hence, aj,bj can be estimated by minimizing the following objective function:(8)∑i=1nδ^ic1−G^nYiYi−Xi⊤β−∑j=1naj+bjUi−u0Zij2KUi−uh1n,where G^nYi is the estimator of G⋅, which is defined by(9)G^ny=1−∏i:Yi≤yn−Rin−Ri+11−μ^ny,Ri=∑j=1nIYj≤Yi,μ^ny=∑i=1nδiςiLy−Yi/an∑i=1nςiLy−Yi/an,which is the Nadaraya–Watson estimator of μy=Eδi|Yi=y with the kernel function L⋅ and bandwidth sequence 0<an⟶0.For notational simplicity, letY=Y1,…,Yn⊤, X=X1,…,Xn⊤, Z=Z1,…,Zn⊤,(10)Du=Z1⊤U1−uh1nZ1⊤⋮⋮Zn⊤Un−uh1nZn⊤,M=Z1⊤αU1⋮Zn⊤αUn,Y˜i=Yi−SiY,X˜i=Xi⊤−SiX,ξ˜i=ξi−Siξ,ξ=ξ1,…,ξn,ωu=diagKU1−uh1n,…,KUn−uh1n,Si=Zi⊤0Du⊤ωuΔ^cDu−1Du⊤Δ^cωu,Δ^c=diagδ^1c1−G^nY1,…,δ^nc1−G^nYn.Ifβ is known, one can obtain the local linear estimator of coefficient function by(11)α¯cu=Iq0Du⊤ωuΔ^cDu−1Du⊤Δ^cωuY−Xβ.Substituting (11) into the original model (8) and eliminating bias produced by the measurement error, we get the following modified PLS estimator of the parameter based on regression calibration method,(12)β˜c=∑i=1nδ^ic1−G^nYiξ˜iξ˜i⊤−Σe−1∑i=1nδ^ic1−G^nYiξ˜iY˜i.Then, the local linear regression estimator ofαu is defined as follows: (13)α˜cu=Iq0Du⊤ωuΔ^cDu−1Du⊤Δ^cωuY−ξβ˜c.Letδim=ςiδi+1−ςimTi,θ. Since Eδim=Eςiδi+1−ςimTi,θi=Eδi under the missing mechanism, we can impute δi with δ^im=ςiδi+1−ςimTi,θ^n in expression (6). Hence, aj,bj can be estimated by minimizing the following objective function:(14)∑i=1nδ^im1−G^nYiYi−Xi⊤β−∑j=1naj+bjUi−u0Zij2KUi−uh1n.Ifβ is known, one can obtain the local linear estimator of coefficient function by(15)α¯mu=Iq0Du⊤ωuΔ^mDu−1Du⊤Δ^mωuY−Xβ,where Δ^m=diagδ^1m/1−G^nY1,…,δ^nm/1−G^nYn. Substituting (21) into the original model and eliminating bias produced by the measurement error, hence, we obtain the following modified PLS estimator of β based on imputation method:(16)β˜m=∑i=1nδ^im1−G^nYiξ˜iξ˜i⊤−Σe−1∑i=1nδ^im1−G^nYiξ˜iY˜i.Thus, the local linear regression estimator ofα⋅ is defined as follows:(17)α˜mu=Iq0Du⊤ωuΔ^mDu−1Du⊤Δ^mωuY−ξβ˜m.Letδiw=ςiδi/πYi+1−ςi/πYimTi,θ. Note that Eδiw=Eςiδi/πYi+1−ςi/πYimTi,θ=Eδ under MAR assumption. Hence, we substitute δi with δ^iw=ςδi/π^nYi+1−ςi/π^nYimTi,θ^n, where π^ny=∑i=1nςiΩy−Yi/bn/∑i=1nΩy−Yi/bn is a nonparametric estimator of πy=Eς|Y=y with kernel function Ω⋅ and bandwidth sequence 0<bn⟶0. Hence, aj,bj can be estimated by minimizing the following objective function:(18)∑i=1nδ^iw1−G^nYiYi−Xi⊤β−∑j=1naj+bjUi−u0Zij2KUi−uh1n.Ifβ is known, one can obtain the local linear estimator of coefficient function by(19)α¯wu=Iq0Du⊤ωuΔ^wDu−1Du⊤Δ^wωuY−Xβ,where Δ^w=diagδ^1w/1−G^nY1,…,δ^nw/1−G^nYn. Substituting (19) into the original model and eliminating bias produced by the measurement error. Hence, we can get the following modified PLS estimator of β based on inverse probability weighted method:(20)β˜w=∑i=1nδ^iw1−G^nYiξ˜iξ˜i⊤−Σe−1∑i=1nδ^iw1−G^nYiξ˜iY˜i.Hence, the local linear regression estimator ofα⋅ is defined as follows:(21)α˜wu=Iq0Du⊤ωuΔ^wDu−1Du⊤Δ^wωuY−ξβ˜w.


### 2.2. Estimation for Error Variance

In order to improve the estimation of parametric and nonparametric parts, we construct local linear estimators of the error variance functionσ2⋅ in this subsection. Note that σ2u=Eδic/1−GYiYi−ξi⊤β−ZiαUi2−β⊤Σeβ|Ui. By minimizing the following object function with respect to μ1,(22)∑i=1nδ^ic1−G^nYiYi−ξi⊤β˜c−Zi⊤α˜cUi2−β˜c⊤Σeβ˜c−μ1−v1Ui−u2KUi−uh2n,the local linear regression estimator of σ2u based on regression calibration method is defined by(23)σ^c2u=∑i=1nWnicYi−ξi⊤β˜c−Zi⊤α˜cUi2−β˜c⊤Σeβ˜c,where the weight function Wnic⋅ is defined by(24)Wnicu=nh2n−1δ^ic/1−G^nYiKUi−u/h2nAn2cu−Ui−uAn1uAn0cuAn2cu−An1cu2,with(25)Anjc=1nh2n∑i=1nδ^ic1−G^nYiKUi−uh2nUi−uj,forj=0,1,2.Note that(26)σ2u=Eδim1−GYiYi−ξi⊤β−ZiαUi2−β⊤Σeβ|Ui.By minimizing the following object function with respect toμ2,(27)∑i=1nδ^im1−G^nYiYi−ξi⊤β˜m−Zi⊤α˜mUi2−β˜m⊤Σeβ˜m−μ2−v2Ui−u2KUi−uh2n,the local linear regression estimator of σm2u based on imputation method is defined by(28)σ^m2u=∑i=1nWnimYi−ξi⊤β˜m−Zi⊤α˜mUi2−β˜m⊤Σeβ˜m,where the weight function Wnim⋅ is defined by(29)Wnimu=nh2n−1δ^im/1−G^nYiKUi−u/h2nAn2mu−Ui−uAn1muAn0muAn2mu−An1mu2,with(30)Anjm=1nh2n∑i=1nδ^im1−G^nYiKUi−uh2nUi−uj,forj=0,1,2.Note that(31)σ2u=Eδiw1−GYiYi−ξi⊤β−ZiαUi2−β⊤Σeβ|Ui.By minimizing the following object function with respect toμ3,(32)∑i=1nδ^iw1−G^nYiYi−ξi⊤β˜w−Zi⊤α˜wUi2−β˜w⊤Σeβ˜w−μ3−v3Ui−u2KUi−uh2n,the local linear regression estimator of σ2u based on inverse probability weighted method is defined by(33)σ^w2u=∑i=1nWniwYi−ξi⊤β˜w−Zi⊤α˜wUi2−β˜w⊤Σeβ˜w,where the weight function Wniw⋅ is defined by(34)Wniwu=nh2n−1δ^iw/1−G^nYiKUi−u/h2nAn2wu−Ui−uAn1wuAn0wuAn2wu−An1wu2,with(35)Anjw=1nh2n∑i=1nδ^iw1−G^nYiKUi−uh2nUi−uj,forj=0,1,2.


### 2.3. Reweighted Estimation

In this subsection, we construct the reweighted estimations of the parametric and nonparametric parts based on the error variance estimatorσ^c2u given in (23). By minimizing the following object function,(36)∑i=1nδ^ic1−G^nYiσ^c−2UiYi−ξi⊤β−∑j=1naj+bjUi−u0Zij2−β⊤ΣeβKUi−uh3n,then, we get the following reweighted estimator of β based on the regression calibration method:(37)β^c=∑i=1nδ^ic1−G^nYiσ^c−2Uiξ˜iξ˜i⊤−Σe−1∑i=1nδ^ic1−G^nYiσ^c−2Uiξ˜iY˜i.Furthermore, the reweighted estimator of the coefficient functionαu is defined by(38)α^cu=Iq0Du⊤ωuΔ^cDu−1Du⊤Δ^cωuY−ξβ^c.Similarly, based on the error variance estimatorσ^m2u given in (28) and minimizing the following object function(39)∑i=1nδ^im1−G^nYiσ^m−2UiYi−ξi⊤β−∑j=1naj+bjUi−u0Zij2−β⊤ΣeβKUi−uh3n,then, we get the reweighted estimator of β based on the imputation method:(40)β^m=∑i=1nδ^im1−G^nYiσ^m−2Uiξ˜iξ˜i⊤−Σe−1∑i=1nδ^im1−G^nYiσ^m−2Uiξ˜iY˜i.Hence, the reweighted estimator of the coefficient functionαu is defined by(41)α^mu=Iq0Du⊤ωuΔ^mDu−1Du⊤Δ^mωuY−ξβ^m.From the error variance estimatorσ^w2u given in (33) and minimizing the following object function(42)∑i=1nδ^iw1−G^nYiσ^w−2UiYi−ξi⊤β−∑j=1naj+bjUi−u0Zij2−β⊤ΣeβKUi−uh3n,then, we get the reweighted estimator of β based on the inverse probability weighted method:(43)β^w=∑i=1nδ^iw1−G^nYiσ^w−2Uiξ˜iξ˜i⊤−Σe−1∑i=1nδ^iw1−G^nYiσ^w−2Uiξ˜iY˜i.Thus, the reweighted estimator of the coefficient functionαu is defined by(44)α^wu=Iq0Du⊤ωuΔ^wDu−1Du⊤Δ^wωuY−ξβ^w.


## 2.1. Modified Profile Least Squares Estimation

The local linear regression technique is employed to estimate the coefficient functionα⋅. If α⋅ has twice continuous derivative at point u0, for u in a small neighborhood of u0, one can approximate α⋅ by the following expansion with Taylor expansion:(4)αju=αju0+αj′u0u−u0≔aj+bju−u0,j=1,…,q,where αj′u=∂αju/∂u. Then, aj,bj can be estimated by minimizing the following objective function:(5)∑i=1nδi1−GYiYi−Xi⊤β−∑j=1naj+bjUi−u0Zij2KUi−uh1n,where K⋅ is a kernel function, and 0<h1n⟶0 is a bandwidth sequence. Due to the missing indicators, some δi cannot be observed. Therefore, model (2) cannot be applied directly. One can replace δi with its conditional expectation mTi=Eδi|Ti. Thus, aj,bj can be defined as the minimizer of(6)∑i=1nmTi1−GYiYi−Xi⊤β−∑j=1naj+bjUi−u0Zij2KUi−uh1n.However, in practical fields, functionm⋅ is usually unknown. One can use parametric and nonparametric methods to estimate m⋅. However, when the covariates are high-dimensional, nonparametric estimation may cause “the curse of dimensionality.” Hence, throughout this paper, we assume that m⋅ follows a parametric model mT=mT,θ, where θ is an unknown parameter vector. Following Wang and Dinse [16], the estimator θ^n of θ can be obtained by maximizing the following likelihood function:(7)∏i=1nmTi,θςiδi1−mTi,θςi1−δi.Letδic=mTi,θ. Since Eδic=EmTi,θ=EEδi|Ti=Eδi, we replace δic with its estimator δ^ic=mTi,θ^n. Hence, aj,bj can be estimated by minimizing the following objective function:(8)∑i=1nδ^ic1−G^nYiYi−Xi⊤β−∑j=1naj+bjUi−u0Zij2KUi−uh1n,where G^nYi is the estimator of G⋅, which is defined by(9)G^ny=1−∏i:Yi≤yn−Rin−Ri+11−μ^ny,Ri=∑j=1nIYj≤Yi,μ^ny=∑i=1nδiςiLy−Yi/an∑i=1nςiLy−Yi/an,which is the Nadaraya–Watson estimator of μy=Eδi|Yi=y with the kernel function L⋅ and bandwidth sequence 0<an⟶0.For notational simplicity, letY=Y1,…,Yn⊤, X=X1,…,Xn⊤, Z=Z1,…,Zn⊤,(10)Du=Z1⊤U1−uh1nZ1⊤⋮⋮Zn⊤Un−uh1nZn⊤,M=Z1⊤αU1⋮Zn⊤αUn,Y˜i=Yi−SiY,X˜i=Xi⊤−SiX,ξ˜i=ξi−Siξ,ξ=ξ1,…,ξn,ωu=diagKU1−uh1n,…,KUn−uh1n,Si=Zi⊤0Du⊤ωuΔ^cDu−1Du⊤Δ^cωu,Δ^c=diagδ^1c1−G^nY1,…,δ^nc1−G^nYn.Ifβ is known, one can obtain the local linear estimator of coefficient function by(11)α¯cu=Iq0Du⊤ωuΔ^cDu−1Du⊤Δ^cωuY−Xβ.Substituting (11) into the original model (8) and eliminating bias produced by the measurement error, we get the following modified PLS estimator of the parameter based on regression calibration method,(12)β˜c=∑i=1nδ^ic1−G^nYiξ˜iξ˜i⊤−Σe−1∑i=1nδ^ic1−G^nYiξ˜iY˜i.Then, the local linear regression estimator ofαu is defined as follows: (13)α˜cu=Iq0Du⊤ωuΔ^cDu−1Du⊤Δ^cωuY−ξβ˜c.Letδim=ςiδi+1−ςimTi,θ. Since Eδim=Eςiδi+1−ςimTi,θi=Eδi under the missing mechanism, we can impute δi with δ^im=ςiδi+1−ςimTi,θ^n in expression (6). Hence, aj,bj can be estimated by minimizing the following objective function:(14)∑i=1nδ^im1−G^nYiYi−Xi⊤β−∑j=1naj+bjUi−u0Zij2KUi−uh1n.Ifβ is known, one can obtain the local linear estimator of coefficient function by(15)α¯mu=Iq0Du⊤ωuΔ^mDu−1Du⊤Δ^mωuY−Xβ,where Δ^m=diagδ^1m/1−G^nY1,…,δ^nm/1−G^nYn. Substituting (21) into the original model and eliminating bias produced by the measurement error, hence, we obtain the following modified PLS estimator of β based on imputation method:(16)β˜m=∑i=1nδ^im1−G^nYiξ˜iξ˜i⊤−Σe−1∑i=1nδ^im1−G^nYiξ˜iY˜i.Thus, the local linear regression estimator ofα⋅ is defined as follows:(17)α˜mu=Iq0Du⊤ωuΔ^mDu−1Du⊤Δ^mωuY−ξβ˜m.Letδiw=ςiδi/πYi+1−ςi/πYimTi,θ. Note that Eδiw=Eςiδi/πYi+1−ςi/πYimTi,θ=Eδ under MAR assumption. Hence, we substitute δi with δ^iw=ςδi/π^nYi+1−ςi/π^nYimTi,θ^n, where π^ny=∑i=1nςiΩy−Yi/bn/∑i=1nΩy−Yi/bn is a nonparametric estimator of πy=Eς|Y=y with kernel function Ω⋅ and bandwidth sequence 0<bn⟶0. Hence, aj,bj can be estimated by minimizing the following objective function:(18)∑i=1nδ^iw1−G^nYiYi−Xi⊤β−∑j=1naj+bjUi−u0Zij2KUi−uh1n.Ifβ is known, one can obtain the local linear estimator of coefficient function by(19)α¯wu=Iq0Du⊤ωuΔ^wDu−1Du⊤Δ^wωuY−Xβ,where Δ^w=diagδ^1w/1−G^nY1,…,δ^nw/1−G^nYn. Substituting (19) into the original model and eliminating bias produced by the measurement error. Hence, we can get the following modified PLS estimator of β based on inverse probability weighted method:(20)β˜w=∑i=1nδ^iw1−G^nYiξ˜iξ˜i⊤−Σe−1∑i=1nδ^iw1−G^nYiξ˜iY˜i.Hence, the local linear regression estimator ofα⋅ is defined as follows:(21)α˜wu=Iq0Du⊤ωuΔ^wDu−1Du⊤Δ^wωuY−ξβ˜w.


## 2.2. Estimation for Error Variance

In order to improve the estimation of parametric and nonparametric parts, we construct local linear estimators of the error variance functionσ2⋅ in this subsection. Note that σ2u=Eδic/1−GYiYi−ξi⊤β−ZiαUi2−β⊤Σeβ|Ui. By minimizing the following object function with respect to μ1,(22)∑i=1nδ^ic1−G^nYiYi−ξi⊤β˜c−Zi⊤α˜cUi2−β˜c⊤Σeβ˜c−μ1−v1Ui−u2KUi−uh2n,the local linear regression estimator of σ2u based on regression calibration method is defined by(23)σ^c2u=∑i=1nWnicYi−ξi⊤β˜c−Zi⊤α˜cUi2−β˜c⊤Σeβ˜c,where the weight function Wnic⋅ is defined by(24)Wnicu=nh2n−1δ^ic/1−G^nYiKUi−u/h2nAn2cu−Ui−uAn1uAn0cuAn2cu−An1cu2,with(25)Anjc=1nh2n∑i=1nδ^ic1−G^nYiKUi−uh2nUi−uj,forj=0,1,2.Note that(26)σ2u=Eδim1−GYiYi−ξi⊤β−ZiαUi2−β⊤Σeβ|Ui.By minimizing the following object function with respect toμ2,(27)∑i=1nδ^im1−G^nYiYi−ξi⊤β˜m−Zi⊤α˜mUi2−β˜m⊤Σeβ˜m−μ2−v2Ui−u2KUi−uh2n,the local linear regression estimator of σm2u based on imputation method is defined by(28)σ^m2u=∑i=1nWnimYi−ξi⊤β˜m−Zi⊤α˜mUi2−β˜m⊤Σeβ˜m,where the weight function Wnim⋅ is defined by(29)Wnimu=nh2n−1δ^im/1−G^nYiKUi−u/h2nAn2mu−Ui−uAn1muAn0muAn2mu−An1mu2,with(30)Anjm=1nh2n∑i=1nδ^im1−G^nYiKUi−uh2nUi−uj,forj=0,1,2.Note that(31)σ2u=Eδiw1−GYiYi−ξi⊤β−ZiαUi2−β⊤Σeβ|Ui.By minimizing the following object function with respect toμ3,(32)∑i=1nδ^iw1−G^nYiYi−ξi⊤β˜w−Zi⊤α˜wUi2−β˜w⊤Σeβ˜w−μ3−v3Ui−u2KUi−uh2n,the local linear regression estimator of σ2u based on inverse probability weighted method is defined by(33)σ^w2u=∑i=1nWniwYi−ξi⊤β˜w−Zi⊤α˜wUi2−β˜w⊤Σeβ˜w,where the weight function Wniw⋅ is defined by(34)Wniwu=nh2n−1δ^iw/1−G^nYiKUi−u/h2nAn2wu−Ui−uAn1wuAn0wuAn2wu−An1wu2,with(35)Anjw=1nh2n∑i=1nδ^iw1−G^nYiKUi−uh2nUi−uj,forj=0,1,2.


## 2.3. Reweighted Estimation

In this subsection, we construct the reweighted estimations of the parametric and nonparametric parts based on the error variance estimatorσ^c2u given in (23). By minimizing the following object function,(36)∑i=1nδ^ic1−G^nYiσ^c−2UiYi−ξi⊤β−∑j=1naj+bjUi−u0Zij2−β⊤ΣeβKUi−uh3n,then, we get the following reweighted estimator of β based on the regression calibration method:(37)β^c=∑i=1nδ^ic1−G^nYiσ^c−2Uiξ˜iξ˜i⊤−Σe−1∑i=1nδ^ic1−G^nYiσ^c−2Uiξ˜iY˜i.Furthermore, the reweighted estimator of the coefficient functionαu is defined by(38)α^cu=Iq0Du⊤ωuΔ^cDu−1Du⊤Δ^cωuY−ξβ^c.Similarly, based on the error variance estimatorσ^m2u given in (28) and minimizing the following object function(39)∑i=1nδ^im1−G^nYiσ^m−2UiYi−ξi⊤β−∑j=1naj+bjUi−u0Zij2−β⊤ΣeβKUi−uh3n,then, we get the reweighted estimator of β based on the imputation method:(40)β^m=∑i=1nδ^im1−G^nYiσ^m−2Uiξ˜iξ˜i⊤−Σe−1∑i=1nδ^im1−G^nYiσ^m−2Uiξ˜iY˜i.Hence, the reweighted estimator of the coefficient functionαu is defined by(41)α^mu=Iq0Du⊤ωuΔ^mDu−1Du⊤Δ^mωuY−ξβ^m.From the error variance estimatorσ^w2u given in (33) and minimizing the following object function(42)∑i=1nδ^iw1−G^nYiσ^w−2UiYi−ξi⊤β−∑j=1naj+bjUi−u0Zij2−β⊤ΣeβKUi−uh3n,then, we get the reweighted estimator of β based on the inverse probability weighted method:(43)β^w=∑i=1nδ^iw1−G^nYiσ^w−2Uiξ˜iξ˜i⊤−Σe−1∑i=1nδ^iw1−G^nYiσ^w−2Uiξ˜iY˜i.Thus, the reweighted estimator of the coefficient functionαu is defined by(44)α^wu=Iq0Du⊤ωuΔ^wDu−1Du⊤Δ^wωuY−ξβ^w.


## 3. Empirical Likelihood

The confidence regions of the parameter can be constructed by the asymptotic distribution of Theorems1 and 4. However, the estimation of asymptotic covariance is quite complicated. In this section, we shall employ the EL method to construct confidence regions for β, which avoids to estimate the complicated covariance.


### 3.1. Regression Calibration Empirical Likelihood

We introduce the following auxiliary random vector based on regression calibration method:(45)η^i,cβ=mTi,θ^n1−G^nYiσ^c−2Uiξ˜iY˜i−ξ˜i⊤β+Σeβ.Thus, we define the empirical log-likelihood ratio function as follows:(46)L^cβ=sup∏i=1nnpi:pi≥0,∑i=1npi=1,∑i=1npiη^i,cβ=0.The optimal value ofpi satisfying (46) is given by pi=1/n⋅1/1+λ⊤η^i,cβ, where λ=λcβ is the solution to the equation ∑i=1nη^i,cβ/1+λ⊤η^i,cβ=0. By the Lagrange multiplier method, the corresponding empirical log-likelihood ratio function is represented as(47)l^cβ=2∑i=1nlog1+λ⊤η^i,cβ.By maximizing−l^cβ, we can obtain a maximum EL estimator β^cel of β with regression calibration method.


### 3.2. Imputation Empirical Likelihood

We introduce the following auxiliary random vector based on imputation method:(48)η^i,mβ=ςiδi+1−ςimTi,θ^n1−G^nYiσ^m−2Uiξ˜iY˜i−ξ˜i⊤β+Σeβ.Hence, we define the empirical log-likelihood ratio function as follows:(49)L^mβ=sup∏i=1nnpi:pi≥0,∑i=1npi=1,∑i=1npiη^i,mβ=0.The optimal value ofpi satisfying (49) is given by pi=1/n⋅1/1+λ⊤η^i,mβ, where λ=λmβ is the solution to the equation ∑i=1nη^i,mβ/1+λ⊤η^i,mβ=0. By the Lagrange multiplier method, the corresponding empirical log-likelihood ratio function is(50)l^mβ=2∑i=1nlog1+λ⊤η^i,mβ.By maximizing−l^mβ, we can obtain a maximum EL estimator β^mel of β with imputation method.


### 3.3. Inverse Probability Weighted Empirical Likelihood

We introduce the following auxiliary random vector based on inverse probability weighted method:(51)η^i,wβ=ςiδi/π^nYi+1−ςi/π^nYimTi,θ^n1−G^nYiσ^w−2Uiξ˜iY˜i−ς˜i⊤β+Σeβ.Then, we define the empirical log-likelihood ratio function as follows:(52)L^wβ=sup∏i=1nnpi:pi≥0,∑i=1npi=1,∑i=1npiη^i,wβ=0.The optimal value ofpi satisfying (52) is given by pi=1/n⋅1/1+λ⊤η^i,wβ, where λ=λwβ is the solution to the equation ∑i=1nη^i,wβ/1+λ⊤η^i,wβ=0. By the Lagrange multiplier method, the corresponding empirical log-likelihood ratio function is represented as(53)l^wβ=2∑i=1nlog1+λ⊤η^i,wβ.By maximizing−l^wβ, we can obtain a maximum EL estimator β^wel of β with inverse probability weighted method.


## 3.1. Regression Calibration Empirical Likelihood

We introduce the following auxiliary random vector based on regression calibration method:(45)η^i,cβ=mTi,θ^n1−G^nYiσ^c−2Uiξ˜iY˜i−ξ˜i⊤β+Σeβ.Thus, we define the empirical log-likelihood ratio function as follows:(46)L^cβ=sup∏i=1nnpi:pi≥0,∑i=1npi=1,∑i=1npiη^i,cβ=0.The optimal value ofpi satisfying (46) is given by pi=1/n⋅1/1+λ⊤η^i,cβ, where λ=λcβ is the solution to the equation ∑i=1nη^i,cβ/1+λ⊤η^i,cβ=0. By the Lagrange multiplier method, the corresponding empirical log-likelihood ratio function is represented as(47)l^cβ=2∑i=1nlog1+λ⊤η^i,cβ.By maximizing−l^cβ, we can obtain a maximum EL estimator β^cel of β with regression calibration method.


## 3.2. Imputation Empirical Likelihood

We introduce the following auxiliary random vector based on imputation method:(48)η^i,mβ=ςiδi+1−ςimTi,θ^n1−G^nYiσ^m−2Uiξ˜iY˜i−ξ˜i⊤β+Σeβ.Hence, we define the empirical log-likelihood ratio function as follows:(49)L^mβ=sup∏i=1nnpi:pi≥0,∑i=1npi=1,∑i=1npiη^i,mβ=0.The optimal value ofpi satisfying (49) is given by pi=1/n⋅1/1+λ⊤η^i,mβ, where λ=λmβ is the solution to the equation ∑i=1nη^i,mβ/1+λ⊤η^i,mβ=0. By the Lagrange multiplier method, the corresponding empirical log-likelihood ratio function is(50)l^mβ=2∑i=1nlog1+λ⊤η^i,mβ.By maximizing−l^mβ, we can obtain a maximum EL estimator β^mel of β with imputation method.


## 3.3. Inverse Probability Weighted Empirical Likelihood

We introduce the following auxiliary random vector based on inverse probability weighted method:(51)η^i,wβ=ςiδi/π^nYi+1−ςi/π^nYimTi,θ^n1−G^nYiσ^w−2Uiξ˜iY˜i−ς˜i⊤β+Σeβ.Then, we define the empirical log-likelihood ratio function as follows:(52)L^wβ=sup∏i=1nnpi:pi≥0,∑i=1npi=1,∑i=1npiη^i,wβ=0.The optimal value ofpi satisfying (52) is given by pi=1/n⋅1/1+λ⊤η^i,wβ, where λ=λwβ is the solution to the equation ∑i=1nη^i,wβ/1+λ⊤η^i,wβ=0. By the Lagrange multiplier method, the corresponding empirical log-likelihood ratio function is represented as(53)l^wβ=2∑i=1nlog1+λ⊤η^i,wβ.By maximizing−l^wβ, we can obtain a maximum EL estimator β^wel of β with inverse probability weighted method.


## 4. Main Results

For convenience and simplicity, we useC0,C1,… and c0,c1,… generically to represent any positive constants, which may take different values for each appearance. Let Γu=EZZ⊤|U=u, Φu=EZX⊤|U=u, Ψ=X−Φ−1UΓ−1UZ, Θ=X+e−ΦUΓ−1UΦ⊤Uε−eβ and M⊗2=MM⊤. Denote(54)Σ11=EmT,θ1−GYΨΨ⊤,Σ12=EmTi,θ1−GYiΘ+Σeβ⊗2,Σ21=Eςδ+1−ςmT,θ1−GYΨΨ⊤,Σ22=Eςδ+1−ξmT,θ1−GYΘ+Σeβ⊗2,Σ31=Eςδ+1−ξmT,θ1−GYΨΨ⊤,Σ32=Eς/πYδ+1−ς/πYmT,θ1−GYΘ+Σeβ⊗2,Σr,11=EmT,θ1−GYσ−2UΨΨ⊤,Σr,12=EmT,θ1−GYσ−2UΘ+Σeβ⊗2,Σr,21=Eςδ+1−ςmT,θ1−GYσ−2UΨΨ⊤,Σr,22=Eςδ+1−ςmT,θ1−GYσ−2UΘ+Σeβ⊗2,Σr,31=Eςδ+1−ςmT,θ1−GYσ−2UΨΨ⊤,Σr,32=Eς/πYδ+1−ς/πYmT,θ1−GYσ−2UΘ+Σeβ⊗2,Λ1=ν0EmT,θ1−GYg−1UΓ−1Uε−eβ2,Λ2=ν0Eςδ+1−ςmTi,θ1−GYig−1UΓ−1Uε−eβ2,Λ3=ν0Eς/πYδ+1−ς/πYmT,θ1−GYig−1UΓ−1Uε−eβ2.In order to prove the main results, we give a set of assumptions that are stated in the following theorems:(C1) The random variableU has bounded support U and its density function g⋅ is Lipschitz continuous and away from zero on its support.(C2) There iss>2 such that EX12s|U1<∞a.s., EZ12s|U1<∞a.s., Eε12s|X1,Z1<∞a.s. Γu and Φu are nonsingular matrixes.(C3)αj⋅,j=1,…,q has continuous second derivatives in U∈U.(C4) The variance functionσ2⋅ with uniform boundedness has continuous second-order derivation and is bounded away from zero.(C5) The kernelK⋅ as a symmetric density function has compact support −1,1, which is Lipschitz continuous, and satisfies ∫ℝtjKtdt=κj<∞,∫ℝtjK2tdt=νj<∞,j=0,1,2.(C6) DenoteaH=inft:Ht=1 and aG=inft:Gt=1. Let aH<aG and G⋅ is continuous. Ht is continuous for t>aH.(C7) The kernel functionsΩ⋅ and L⋅ are bounded with bounded compact supports, and nan⟶∞, nan2⟶0, nbn⟶∞ and nbn2⟶0.(C8)π⋅ and μ⋅ have bounded derivatives of order 1, and there exists c>0 such that infxπx≥c>0.(C9)Iθ0 is a positive definite. ∇mTi,θ0 is continuous at θ0.(C10) The bandwidthhin satisfies hin⟶0, nhin⟶∞, nhin8⟶0 and nhin2/logn2⟶∞ for i=1,2,3.Remark 1.
(a)
Assumptions (C1) and (C2) are used to establish the asymptotic normality and the oracle property of the estimators. Assumptions (C3) and (C4) are common conditions for varying-coefficients models with heteroscedastic error. Assumption (C5) requires that the kernel function is a proper density with finite second moment, which is required to derive the asymptotic variance of estimators. Assumption (C6) implies that1−GYi is bounded away from zero. Assumptions (C7)–(C9) are needed for the properties of G^ny and mTi,θ^n. Assumption (C10) underlines the relationship bandwidth with sample size n, which implies the optimal bandwidth in nonparametric estimation.(b)
From the Taylor expansion and conclusion in Li and Wang [17], one can get(55)nθ^n−θ=I−1θ01n∑i=1nξiδi−mTi,θ0∇mTi,θ0mTi,θ01−mTi,θ0+op1,which, together with assumption (C9), givesθ^n−θ=Opn−1/2.The asymptotic properties of the proposed estimators are shown in the following theorems.Theorem 1.
Suppose that assumptions (C1)–(C10) are satisfied; then, we have(56)nβ˜n−β⟶DN0,Σj1−1Σj2Σj1−1,where β˜n is taken to be β˜cu, β˜mu and β˜wu. j=1,2,3 correspond to β˜c, β˜m and β˜w, respectively.Theorem 2.
Suppose that assumptions (C1)–(C10) are satisfied; then, we have(57)nh1nα˜nu−αu−12hn2κ2α″u⟶DN0,Λj,where α˜nu is taken to be α˜cu, α˜mu and α˜wu. j=1,2,3 correspond to α˜cu, α˜mu and α˜wu, respectively.Theorem 3.
Suppose that assumptions (C1)–(C10) are satisfied; letρn=logn/nh2n1/2+h2n2; then, we have(58)supuσ^n2u−σ2u=Opρn,where σ˜n2u is taken to be one of σ˜c2u, σ˜m2u and σ˜w2u.Theorem 4.
Suppose that assumptions (C1)–(C10) are satisfied; then, we have(59)nβ^n−β⟶DN0,Σr,j1−1Σr,j2Σr,j1−1,where β^n is taken to be one of β^cu, β^mu and β^wu. j=1,2,3 correspond to β^cu, β^mu and β^wu, respectively.Theorem 5.
Suppose that assumptions (C1)–(C10) are satisfied; then, we have(60)nh1nα^nu−αu−12hn2κ2α″u⟶DN0,Λj,where α^nu is taken to be one of α^cu, α^mu and α^wu. j=1,2,3 correspond to α^cu, α^mu and α^wu, respectively.Theorem 6.
Suppose that assumptions (C1)–(C10) are satisfied; ifβ is the true value, then we have(61)l^nβ⟶Dχ12,where l^nβ denotes one of l^cβ, l^mβ, and l^wβ. χ12 is a standard chi-squared random variable with 1 degree of freedom.Theorem 7.
Suppose that assumptions (C1)–(C10) are satisfied; then, we have(62)nβ^el−β⟶DN0,Σr,j1−1Σr,j2Σr,j1−1,where β^el denotes one of β^cel, β^mel and β^wel. j=1,2,3 correspond to β^cel, β^mel and β^wel, respectively.Remark 2.
(a)
From Theorems1 and 4, the asymptotic variance of the reweighted estimator β^n is not greater than that of the modified profile LS estimator β˜n; that is, Σj1−1Σj2Σj1−1−Σr,j1−1Σr,j2Σr,j1−1 is a positive semidefinite matrix. The asymptotic variance of the reweighted estimator β^m is smaller than that of β^w, and is larger than that of β^c, which indicates that β^c performs the best, and β^w performs the worst. The modified PLS estimators β˜c, β˜m,β˜w enjoy the same conclusion.(b)
From Theorems2 and 5, the local polynomial estimator α˜n⋅ and reweighted estimator α^n⋅ have the same asymptotic distribution, which reflects the characteristic of the local regression in nonparametric models.(c)
From Theorem6, the 1001−τ% EL confidence region for β can be established as Iτ=β:l^nβ≤χ121−τ, where χ121−τ is the upper 1−τ-quantile of distribution of χ12.


## 5. Simulation

In this subsection, we carry out some numerical simulation to investigate the finite sample behavior of the proposed estimators. We compare the performance of the estimators based on the regression calibration method (CA), imputation method (IM) and inverse probability weighted method (IPW), and their corresponding reweighted estimators (R-CA, R-IM, R-IPW). Besides, we conduct a comparison of the EL method with the normal approximation (NA) approach in terms of coverage probabilities (CP) and average interval lengths (AL) under different settings. At the same time, we give a real data analysis. The kernel functions are taken asKu=3/41−u2Iu≤1, Lu15/161−u22Iu≤1 and Ωu=1/2Iu≤1. The bandwidths h1n, h2n and h3n have taken the same values by leave-one-sample-out cross-validation. The following simulation is based on 500 replications. The sample size n is chosen to be 100 and 400, repeatedly.


### 5.1. Simulation Experiments

The data are generated from the following the PLVCEV model:(63)Ti=Xi1⊤β1+Xi2⊤β2+Zi⊤αUi+εi,ξi1=Xi1+ei1,ξi2=Xi2+ei2,i=1,…,n,where β1=1, β2=2, αu=sin2πu, the covariates Xi1 and Xi2 are from N0,1 and pairwise covariance covXij,Xik=0.5j−k. Zi is from U−1,1. Ui is from U0,1, the model error εi∼N0,1. The error variance function is taken as σ2u=0.25+γ1+sin2πu2 for γ=1 and 4. The measurement error ei∼N0,Σe. To represent different levels of measurement errors, we take Σe be Σe1=diag0.5,0.5 and Σe2=diag0.25,0.25 in the simulations, respectively. Let the censoring time Ci be from expυ, where υ is adapted to get different censoring ratios (CR). For the missing mechanism, we take Pςi|Xi,Zi,Ui,Yi=1/1+exp−a1−a2Xi1+Zi+Ui−a3Yi. We choose different υ and a=a1,a2,a3 to get different CR and missing ratios (MR). Suppose that mTi,θ follows a logistic model, that is, mTi,θ=1/1+exp−θ1−θ2Xi2−θ3Yi. Then, the parameter θ=θ1,θ2,θ3 is estimated by maximum likelihood method. See Table 1 for details.Table 1
Choosing forυ and a=a1,a2,a3 and corresponding CR and MR in model (63).
nCR (%)MR (%)υa=a1,a2,a3θ=θ1,θ2,θ310010109.25(−6.0,−1.1,−1.1)(−2.546,−0.595,−0.215)409.25(−1.3,−0.9,−0.8)(−2.551,−0.351,−0.311)40100.79(−6.1,−1.3,−1.0)(−0.152,−0.222,−1.650)400.79(−1.5,−0.9,−0.7)(−0.144,−0.213,−1.550)40010109.25(−6.0,−1.1,−1.1)(−2.567,−0.601,−0.221)409.25(−1.3,−0.9,−0.8)(−2.541,−0.355,−0.302)40100.79(−6.1,−1.3,−1.0)(−0.151,−0.213,−1.620)400.79(−1.5,−0.9,−0.7)(−0.145,−0.214,−1.550)In the first simulation, we study the finite sample performance of the proposed modified PLS estimators and reweighted estimators ofβ based on mean squared error (MSE) defined as(64)MSEβ^n=β^n−β⊤β^n−β,and the global mean square error (GMSE) of α^nu defined as(65)GMSE=1ngridM∑m=1M∑j=1ngridα^nuj−αuj2,where uj:j=1,2,…,ngrid is a sequence of grid points. In addition, we plot QQ-plots of the reweighted estimator α˜wu for αu under different settings in Figures 1 and 2. In the second simulations, we plot the curves of the proposed estimators σ^c2u, σ^m2u, and σ^w2u under different settings in Figures 3 and 4. In the third simulations, we consider CP and AL of the confidence regions for β based on the EL method (CPE, ALE) and NA method (CPN, ALN) with nominal level 0.95 under different settings in Table 2.Figure 1
The QQ-plots ofα˜w⋅ based on Σe1 (a) and Σe2 (b) with heteroscedasticity coefficient γ=1 (c) and γ=4 (d) under CR = 10% and MR = 10%.Figure 2
The QQ-plots ofα˜w⋅ based on Σe1 (a) and Σe2 (b) with heteroscedasticity coefficient γ=1 (c) and γ=4 (d) under CR = 40% and MR = 40%.Figure 3
The curves ofσ2⋅ (solid line), σ^c2⋅ (dotted line), σ^m2⋅ (dashed line), and σ^w2⋅ (dash-dotted line) based on Σe1 (a) and Σe2 (b) with heteroscedasticity coefficient γ=1 (c) and γ=4 (d) under CR = 10% and MR = 10%.Figure 4
The curves ofσ2⋅ (solid line), σ^c2⋅ (dotted line), σ^m2⋅ (dashed line), and σ^w2⋅ (dash-dotted line) based on Σe1 (a) and Σe2 (b) with heteroscedasticity coefficient γ=1 (c) and γ=4 (d) under CR = 40% and MR = 40%.Table 2
CPE and CPN ofβ with confidence 95% and their corresponding ALE and ALN with Σe1.
nCR (%)MR (%)Methodγ=1γ=4CPECPNALEALNCPECPNALEALN1001010IPW0.83100.59801.35770.53140.72700.58804.84361.9884IM0.89000.68901.19430.52880.86600.64502.47601.5248RC0.89300.69701.19230.51400.86900.66102.46091.5125R-IPW0.88100.64300.84650.49140.82500.61201.01700.9900R-IM0.92700.72400.79310.50900.87800.66000.95480.9740R-RC0.92700.73400.79220.50670.87800.66500.94940.92351040IPW0.84100.59701.36130.64030.75800.59304.67171.9853IM0.89000.67901.20790.51900.85800.64602.56341.5672RC0.89200.68801.20260.50800.86600.65402.53011.5410R-IPW0.87800.62800.87200.50630.81800.59901.03021.0481R-IM0.91200.70200.80150.50070.87200.66000.96380.9703R-RC0.92200.71500.79390.50060.87500.66800.96190.92564010IPW0.67000.48301.91490.65950.70600.50105.51291.7238IM0.88700.66201.33870.61350.83000.59602.43911.4932RC0.89600.68801.29970.56890.83600.60802.27431.4751R-IPW0.79900.48101.10880.54710.74000.54701.50441.3151R-IM0.91300.68300.86720.53530.84800.61001.13331.1330R-RC0.92400.70800.86520.50050.86800.63301.09500.95224040IPW0.69000.46001.93810.68550.70700.51605.43361.6817IM0.86200.61701.39930.63860.83300.58002.63161.5084RC0.88300.67301.33040.55540.83800.60302.37781.4733R-IPW0.77300.46901.14090.55220.73900.54801.47171.1889R-IM0.87500.64000.88830.53550.83500.61301.19071.0976R-RC0.89400.70000.87990.50500.85200.63201.14440.95094001010IPW0.73300.41700.74140.26310.67100.45804.96521.6687IM0.93100.65200.63700.25600.87200.54401.90040.9526RC0.93400.66400.63560.25410.87500.55901.85590.9386R-IPW0.83200.46700.52510.26100.81900.58000.54440.4033R-IM0.93900.66600.44850.24570.93300.60700.50920.3914R-RC0.94000.68600.44130.24560.93600.62800.50250.38451040IPW0.72200.40400.75020.26560.65600.44904.88171.6279IM0.92300.59200.64700.25880.83900.54302.12941.0194RC0.92900.65300.64550.24680.84700.58502.06010.9871R-IPW0.79700.46900.55180.36130.80400.58500.54930.3999R-IM0.94100.62200.44500.25900.92400.62800.51080.3921R-RC0.95600.66500.44150.25400.92700.63000.51020.38584010IPW0.36500.17301.07030.29530.59600.35306.60311.7275IM0.89700.54100.74400.27680.80500.46402.01050.9612RC0.92400.62100.72500.27590.83300.49301.83250.9396R-IPW0.52700.30200.76780.29460.61200.52700.77630.4621R-IM0.91800.58700.49910.27620.91700.56400.64000.4559R-RC0.94500.63600.48950.27430.92400.58200.62840.43704040IPW0.36400.16401.18500.31230.57500.36107.09901.7615IM0.84000.47300.84920.30050.76500.43102.49791.0689RC0.90300.60700.80580.29960.78100.48202.19001.0364R-IPW0.54100.28400.77990.29200.60100.53500.80700.4799R-IM0.84900.51200.57520.29180.88100.54100.66920.4606R-RC0.91400.61800.55990.28400.90600.54800.66140.4472From Tables2–4 and Figures 1–4, it can be seen that(1)
In Tables3 and 4, the MSE and GMSE of reweighted estimators are smaller than those of modified PLS estimators under the same setting. The results of IM estimators are smaller than those of IPW estimators, and bigger than those of RC estimators. The results increase as measurement error, heteroscedasticity error, and CR and/or MR increase. The results decrease as the sample size increases. The results above imply that the reweighted estimators perform better than the modified PLS estimators. The RC method performs best, and IPW method performs worst, which confirms the theoretical results.(2)
In Table2, the CP of reweighted estimators is larger than that of the modified PLS estimators. The CP of the RC method is the smallest, and that of the IPW method is the biggest under the same settings. The CP decreases as heteroscedasticity error and CR and/or MR increase. The results increase as sample size increases. The CP based on the EL method is smaller than that of the NA method. The AL performs in the opposite way.(3)
In Figures1 and 2, the fit is better as decreasing the heteroscedasticity error and measurement error. The fit is worse as increasing CR and/or MR.(4)
In Figures3 and 4, the proposed estimators of error variance perform better as decreasing measurement error, heteroscedasticity error, and CR and/or MR. The estimator σ^c2⋅ performs the best, and σ^w2⋅ performs the worst under the same settings.Table 3
The MSE of the estimators forβ1 and β2 and GMSE of the estimators for αu with Σe1.
nCR (%)MR (%)Methodγ=1γ=4γ=1γ=4β1β2β1β2GMSEGMSE1001010IPW0.06960.05930.24060.25800.17970.3782IM0.04880.04800.12370.11110.16700.2327RC0.04320.04440.11530.11090.16650.2314R-IPW0.03830.04070.36210.16190.17710.3097R-IM0.03810.03720.09150.08540.16650.2299R-RC0.03740.03670.08960.08380.16630.22931040IPW0.06170.06100.28640.25780.18020.3654IM0.04750.04550.18140.16080.16950.2738RC0.04240.04300.17420.15400.16460.2601R-IPW0.04830.03990.10300.09610.17710.3338R-IM0.03690.03650.08880.09000.16860.2654R-RC0.03620.03580.06730.07910.16440.25314010IPW0.22050.21950.31680.33540.28590.4654IM0.19180.17440.24670.26750.25530.4144RC0.12860.12990.24560.24660.25050.3727R-IPW0.17320.16700.29110.22140.28090.4626R-IM0.11640.11380.21970.22360.24900.4124R-RC0.11080.10980.20970.21070.23610.36514040IPW0.22370.21440.32450.31490.29260.4709IM0.17590.21300.28930.29100.27080.3585RC0.17400.17890.26110.28930.26590.3460R-IPW0.17970.16960.23120.24180.28380.4656R-IM0.15570.15740.22270.22810.26400.3473R-RC0.11100.12790.20800.20870.25020.34404001010IPW0.02460.02700.27490.27560.12880.3326IM0.01120.01250.04820.05260.11410.1503RC0.01000.00990.04620.05130.11230.1417R-IPW0.01390.01370.02930.03160.12650.2653R-IM0.00920.00950.02880.03150.11360.1475R-RC0.00900.00890.02820.02830.11210.13791040IPW0.03590.03590.33230.35800.13870.3695IM0.01950.01860.14350.14060.12190.1909RC0.01300.01850.11770.10300.11660.1777R-IPW0.01280.01310.04010.04170.13380.2959R-IM0.01100.01070.03650.04140.12060.1826R-RC0.01010.01040.03100.03000.11590.16904010IPW0.17150.16840.24300.25990.26040.3664IM0.07280.07080.14370.14320.16970.2188RC0.05050.04660.11680.11160.14870.2157R-IPW0.13360.13250.15960.15430.25620.3543R-IM0.07020.06570.12920.12790.16900.2174R-RC0.04680.04170.10650.10120.14820.21584040IPW0.17190.16390.26800.27060.26720.3895IM0.12590.11930.22200.20370.21800.2578RC0.06020.06920.13980.13400.15270.2230R-IPW0.13600.13390.18200.17290.26480.3752R-IM0.12080.11690.15360.15210.21610.2515R-RC0.04920.05060.11610.11310.15230.2219Table 4
The MSE of the estimators forβ1 and β2 and GMSE of the estimators for αu with Σe2.
nCR (%)MR (%)Methodγ=1γ=4γ=1γ=4β1β2β1β2GMSEGMSE1001010IPW0.06500.05360.34700.29340.17980.3396IM0.05340.04910.13320.13540.17250.2407RC0.05290.04800.13010.13300.17230.2383R-IPW0.05060.04930.10640.11160.17930.3138R-IM0.04930.04600.10380.10820.17240.2404R-RC0.04840.04520.08320.09260.17230.23791040IPW0.06330.07280.34040.35110.25190.3550IM0.05270.06230.18040.19410.17150.2726RC0.05110.06130.16420.16090.16990.2599R-IPW0.92620.75870.10460.10670.18110.3335R-IM0.04450.04590.09640.09860.17130.2670R-RC0.04180.04340.07760.08440.16960.25724010IPW0.22380.22350.34080.32840.29430.3486IM0.22320.22110.27610.27560.27520.3010RC0.20580.20080.26390.26130.25910.2633R-IPW0.17270.18760.30450.30140.28680.3441R-IM0.13840.13590.26830.26300.25760.2743R-RC0.13560.14110.25810.25250.24650.26134040IPW0.23710.23410.34040.35090.28850.3601IM0.22820.22660.30970.30840.27350.3565RC0.21320.21170.29590.29310.27340.3037R-IPW0.18600.18710.30820.32890.27090.3564R-IM0.15890.15920.27900.27700.26900.3548R-RC0.15730.15240.26420.26090.25650.29684001010IPW0.02250.02180.30160.25080.12860.3290IM0.01220.01330.05380.05970.11300.1439RC0.01190.01260.04140.05210.11230.1364R-IPW0.01340.01360.02930.04150.12610.2696R-IM0.01010.01010.03170.04070.11280.1409R-RC0.01010.00990.03040.03490.11190.13521040IPW0.03510.03020.33380.39390.13190.3946IM0.02690.02060.17800.13770.12100.2414RC0.02770.02150.13190.12420.11690.2034R-IPW0.01520.01400.04960.04650.12980.3259R-IM0.01490.01310.04800.04520.11930.2226R-RC0.01430.01260.03180.03220.11510.18614010IPW0.17420.17340.30180.29010.26270.3830IM0.07550.07290.16880.17260.17230.2229RC0.05650.05240.13750.13940.15080.2186R-IPW0.12930.13070.16070.16070.25810.3701R-IM0.07100.06850.13560.14050.17200.2207R- RC0.05210.04780.11140.11680.15040.21834040IPW0.17870.17430.28170.27860.26810.4099IM0.12850.12680.21700.23050.21270.2755RC0.08630.08410.15280.17150.15460.2333R-IPW0.12820.12610.17510.17500.26470.2353R-IM0.12030.11490.14940.14920.21260.2667R-RC0.05800.05660.11950.12700.15420.3959


### 5.2. A Real Data Analysis

In real data analysis, we illustrate the methodology via an application to a dataset from a breast cancer clinical trial [25]. This clinical trial was conducted by the Eastern Cooperative Oncology Group, whose target was evaluating tamoxifen as a treatment for stage II breast cancer among elderly women, who are older than 65. There are 169 elderly women participating in the trial, and we focus on 79 women who died by the end of the trial. But, unfortunately, the cause of death is incomplete. Among them, 44 women died from breast cancer, 17 died from other known causes, and 18 died from unknown causes. Let the censoring indicator δ show whether the death was caused by breast cancer, and let the missing indicator ς show whether the cause of death was known. The dataset contains four covariates: whether the patients accepted the treatment (1, tamoxifen; 0, placebo), denoted as X1; whether the estrogen receptor status was positive (1, yes; 0, unknown), denoted as X2; whether there were four or more axillary lymph positive nodes (1, yes; 0, no), denoted as Z; and whether the primary tumor is 3 cm or larger (1, yes; 0, no), denoted as U. Then, we employ the following model to fit the data,(66)Y=X1⊤β1+X2⊤β2+Z⊤αU+ε,where Y is the logarithm of the time to death due to breast cancer, which is censored, and the censoring indicator is MAR. The heteroscedastic error follows the form of Varε|X,Z,U=σ2Ui. For the purpose of comparison, we compute both mean squared prediction error (MSE) and mean absolute deviation (MAD) of the predictions, which are defined as follows: MSE=1/n∑i=1169Yi−Y^i2, and MAD=1/n∑i=1169Yi−Y^i, where Y^i is the fitted value of Yi. The values of MSE and MAD based on different methods are given in Table 5. In addition, the estimated curves of αu based on RC, IM, and IPW methods are reported in Figure 5.Table 5
The estimators, MSEs, and MADs for the breast cancer dataset.
MethodEstimatesMSEMADβ1β2RC0.10156.77871.05720.6311IM0.50635.31961.17871.9918IPW0.53295.03361.79342.2677R-RC0.10877.76051.33510.6693R-IM0.59946.13031.18692.5233R-IPW0.64655.80902.23852.8778Figure 5
The curves ofα^n⋅ for the breast cancer dataset based on RC, IM, and IPW methods.From Table5 and Figure 5, it can be seen that, (1) in Table 5, the estimators of β1 are positive, which indicates that the breast-cancer deaths may live longer if they received the treatment. Among these estimators, the MSE and MAD based on the R-RC method are smallest, which confirms the conclusions in Theorems 1 and 4. (2) In Figure 5, the primary tumor size of patients is mainly from 0 to 3 cm. The survival time decreases obviously as the tumor size increases.


## 5.1. Simulation Experiments

The data are generated from the following the PLVCEV model:(63)Ti=Xi1⊤β1+Xi2⊤β2+Zi⊤αUi+εi,ξi1=Xi1+ei1,ξi2=Xi2+ei2,i=1,…,n,where β1=1, β2=2, αu=sin2πu, the covariates Xi1 and Xi2 are from N0,1 and pairwise covariance covXij,Xik=0.5j−k. Zi is from U−1,1. Ui is from U0,1, the model error εi∼N0,1. The error variance function is taken as σ2u=0.25+γ1+sin2πu2 for γ=1 and 4. The measurement error ei∼N0,Σe. To represent different levels of measurement errors, we take Σe be Σe1=diag0.5,0.5 and Σe2=diag0.25,0.25 in the simulations, respectively. Let the censoring time Ci be from expυ, where υ is adapted to get different censoring ratios (CR). For the missing mechanism, we take Pςi|Xi,Zi,Ui,Yi=1/1+exp−a1−a2Xi1+Zi+Ui−a3Yi. We choose different υ and a=a1,a2,a3 to get different CR and missing ratios (MR). Suppose that mTi,θ follows a logistic model, that is, mTi,θ=1/1+exp−θ1−θ2Xi2−θ3Yi. Then, the parameter θ=θ1,θ2,θ3 is estimated by maximum likelihood method. See Table 1 for details.Table 1
Choosing forυ and a=a1,a2,a3 and corresponding CR and MR in model (63).
nCR (%)MR (%)υa=a1,a2,a3θ=θ1,θ2,θ310010109.25(−6.0,−1.1,−1.1)(−2.546,−0.595,−0.215)409.25(−1.3,−0.9,−0.8)(−2.551,−0.351,−0.311)40100.79(−6.1,−1.3,−1.0)(−0.152,−0.222,−1.650)400.79(−1.5,−0.9,−0.7)(−0.144,−0.213,−1.550)40010109.25(−6.0,−1.1,−1.1)(−2.567,−0.601,−0.221)409.25(−1.3,−0.9,−0.8)(−2.541,−0.355,−0.302)40100.79(−6.1,−1.3,−1.0)(−0.151,−0.213,−1.620)400.79(−1.5,−0.9,−0.7)(−0.145,−0.214,−1.550)In the first simulation, we study the finite sample performance of the proposed modified PLS estimators and reweighted estimators ofβ based on mean squared error (MSE) defined as(64)MSEβ^n=β^n−β⊤β^n−β,and the global mean square error (GMSE) of α^nu defined as(65)GMSE=1ngridM∑m=1M∑j=1ngridα^nuj−αuj2,where uj:j=1,2,…,ngrid is a sequence of grid points. In addition, we plot QQ-plots of the reweighted estimator α˜wu for αu under different settings in Figures 1 and 2. In the second simulations, we plot the curves of the proposed estimators σ^c2u, σ^m2u, and σ^w2u under different settings in Figures 3 and 4. In the third simulations, we consider CP and AL of the confidence regions for β based on the EL method (CPE, ALE) and NA method (CPN, ALN) with nominal level 0.95 under different settings in Table 2.Figure 1
The QQ-plots ofα˜w⋅ based on Σe1 (a) and Σe2 (b) with heteroscedasticity coefficient γ=1 (c) and γ=4 (d) under CR = 10% and MR = 10%.Figure 2
The QQ-plots ofα˜w⋅ based on Σe1 (a) and Σe2 (b) with heteroscedasticity coefficient γ=1 (c) and γ=4 (d) under CR = 40% and MR = 40%.Figure 3
The curves ofσ2⋅ (solid line), σ^c2⋅ (dotted line), σ^m2⋅ (dashed line), and σ^w2⋅ (dash-dotted line) based on Σe1 (a) and Σe2 (b) with heteroscedasticity coefficient γ=1 (c) and γ=4 (d) under CR = 10% and MR = 10%.Figure 4
The curves ofσ2⋅ (solid line), σ^c2⋅ (dotted line), σ^m2⋅ (dashed line), and σ^w2⋅ (dash-dotted line) based on Σe1 (a) and Σe2 (b) with heteroscedasticity coefficient γ=1 (c) and γ=4 (d) under CR = 40% and MR = 40%.Table 2
CPE and CPN ofβ with confidence 95% and their corresponding ALE and ALN with Σe1.
nCR (%)MR (%)Methodγ=1γ=4CPECPNALEALNCPECPNALEALN1001010IPW0.83100.59801.35770.53140.72700.58804.84361.9884IM0.89000.68901.19430.52880.86600.64502.47601.5248RC0.89300.69701.19230.51400.86900.66102.46091.5125R-IPW0.88100.64300.84650.49140.82500.61201.01700.9900R-IM0.92700.72400.79310.50900.87800.66000.95480.9740R-RC0.92700.73400.79220.50670.87800.66500.94940.92351040IPW0.84100.59701.36130.64030.75800.59304.67171.9853IM0.89000.67901.20790.51900.85800.64602.56341.5672RC0.89200.68801.20260.50800.86600.65402.53011.5410R-IPW0.87800.62800.87200.50630.81800.59901.03021.0481R-IM0.91200.70200.80150.50070.87200.66000.96380.9703R-RC0.92200.71500.79390.50060.87500.66800.96190.92564010IPW0.67000.48301.91490.65950.70600.50105.51291.7238IM0.88700.66201.33870.61350.83000.59602.43911.4932RC0.89600.68801.29970.56890.83600.60802.27431.4751R-IPW0.79900.48101.10880.54710.74000.54701.50441.3151R-IM0.91300.68300.86720.53530.84800.61001.13331.1330R-RC0.92400.70800.86520.50050.86800.63301.09500.95224040IPW0.69000.46001.93810.68550.70700.51605.43361.6817IM0.86200.61701.39930.63860.83300.58002.63161.5084RC0.88300.67301.33040.55540.83800.60302.37781.4733R-IPW0.77300.46901.14090.55220.73900.54801.47171.1889R-IM0.87500.64000.88830.53550.83500.61301.19071.0976R-RC0.89400.70000.87990.50500.85200.63201.14440.95094001010IPW0.73300.41700.74140.26310.67100.45804.96521.6687IM0.93100.65200.63700.25600.87200.54401.90040.9526RC0.93400.66400.63560.25410.87500.55901.85590.9386R-IPW0.83200.46700.52510.26100.81900.58000.54440.4033R-IM0.93900.66600.44850.24570.93300.60700.50920.3914R-RC0.94000.68600.44130.24560.93600.62800.50250.38451040IPW0.72200.40400.75020.26560.65600.44904.88171.6279IM0.92300.59200.64700.25880.83900.54302.12941.0194RC0.92900.65300.64550.24680.84700.58502.06010.9871R-IPW0.79700.46900.55180.36130.80400.58500.54930.3999R-IM0.94100.62200.44500.25900.92400.62800.51080.3921R-RC0.95600.66500.44150.25400.92700.63000.51020.38584010IPW0.36500.17301.07030.29530.59600.35306.60311.7275IM0.89700.54100.74400.27680.80500.46402.01050.9612RC0.92400.62100.72500.27590.83300.49301.83250.9396R-IPW0.52700.30200.76780.29460.61200.52700.77630.4621R-IM0.91800.58700.49910.27620.91700.56400.64000.4559R-RC0.94500.63600.48950.27430.92400.58200.62840.43704040IPW0.36400.16401.18500.31230.57500.36107.09901.7615IM0.84000.47300.84920.30050.76500.43102.49791.0689RC0.90300.60700.80580.29960.78100.48202.19001.0364R-IPW0.54100.28400.77990.29200.60100.53500.80700.4799R-IM0.84900.51200.57520.29180.88100.54100.66920.4606R-RC0.91400.61800.55990.28400.90600.54800.66140.4472From Tables2–4 and Figures 1–4, it can be seen that(1)
In Tables3 and 4, the MSE and GMSE of reweighted estimators are smaller than those of modified PLS estimators under the same setting. The results of IM estimators are smaller than those of IPW estimators, and bigger than those of RC estimators. The results increase as measurement error, heteroscedasticity error, and CR and/or MR increase. The results decrease as the sample size increases. The results above imply that the reweighted estimators perform better than the modified PLS estimators. The RC method performs best, and IPW method performs worst, which confirms the theoretical results.(2)
In Table2, the CP of reweighted estimators is larger than that of the modified PLS estimators. The CP of the RC method is the smallest, and that of the IPW method is the biggest under the same settings. The CP decreases as heteroscedasticity error and CR and/or MR increase. The results increase as sample size increases. The CP based on the EL method is smaller than that of the NA method. The AL performs in the opposite way.(3)
In Figures1 and 2, the fit is better as decreasing the heteroscedasticity error and measurement error. The fit is worse as increasing CR and/or MR.(4)
In Figures3 and 4, the proposed estimators of error variance perform better as decreasing measurement error, heteroscedasticity error, and CR and/or MR. The estimator σ^c2⋅ performs the best, and σ^w2⋅ performs the worst under the same settings.Table 3
The MSE of the estimators forβ1 and β2 and GMSE of the estimators for αu with Σe1.
nCR (%)MR (%)Methodγ=1γ=4γ=1γ=4β1β2β1β2GMSEGMSE1001010IPW0.06960.05930.24060.25800.17970.3782IM0.04880.04800.12370.11110.16700.2327RC0.04320.04440.11530.11090.16650.2314R-IPW0.03830.04070.36210.16190.17710.3097R-IM0.03810.03720.09150.08540.16650.2299R-RC0.03740.03670.08960.08380.16630.22931040IPW0.06170.06100.28640.25780.18020.3654IM0.04750.04550.18140.16080.16950.2738RC0.04240.04300.17420.15400.16460.2601R-IPW0.04830.03990.10300.09610.17710.3338R-IM0.03690.03650.08880.09000.16860.2654R-RC0.03620.03580.06730.07910.16440.25314010IPW0.22050.21950.31680.33540.28590.4654IM0.19180.17440.24670.26750.25530.4144RC0.12860.12990.24560.24660.25050.3727R-IPW0.17320.16700.29110.22140.28090.4626R-IM0.11640.11380.21970.22360.24900.4124R-RC0.11080.10980.20970.21070.23610.36514040IPW0.22370.21440.32450.31490.29260.4709IM0.17590.21300.28930.29100.27080.3585RC0.17400.17890.26110.28930.26590.3460R-IPW0.17970.16960.23120.24180.28380.4656R-IM0.15570.15740.22270.22810.26400.3473R-RC0.11100.12790.20800.20870.25020.34404001010IPW0.02460.02700.27490.27560.12880.3326IM0.01120.01250.04820.05260.11410.1503RC0.01000.00990.04620.05130.11230.1417R-IPW0.01390.01370.02930.03160.12650.2653R-IM0.00920.00950.02880.03150.11360.1475R-RC0.00900.00890.02820.02830.11210.13791040IPW0.03590.03590.33230.35800.13870.3695IM0.01950.01860.14350.14060.12190.1909RC0.01300.01850.11770.10300.11660.1777R-IPW0.01280.01310.04010.04170.13380.2959R-IM0.01100.01070.03650.04140.12060.1826R-RC0.01010.01040.03100.03000.11590.16904010IPW0.17150.16840.24300.25990.26040.3664IM0.07280.07080.14370.14320.16970.2188RC0.05050.04660.11680.11160.14870.2157R-IPW0.13360.13250.15960.15430.25620.3543R-IM0.07020.06570.12920.12790.16900.2174R-RC0.04680.04170.10650.10120.14820.21584040IPW0.17190.16390.26800.27060.26720.3895IM0.12590.11930.22200.20370.21800.2578RC0.06020.06920.13980.13400.15270.2230R-IPW0.13600.13390.18200.17290.26480.3752R-IM0.12080.11690.15360.15210.21610.2515R-RC0.04920.05060.11610.11310.15230.2219Table 4
The MSE of the estimators forβ1 and β2 and GMSE of the estimators for αu with Σe2.
nCR (%)MR (%)Methodγ=1γ=4γ=1γ=4β1β2β1β2GMSEGMSE1001010IPW0.06500.05360.34700.29340.17980.3396IM0.05340.04910.13320.13540.17250.2407RC0.05290.04800.13010.13300.17230.2383R-IPW0.05060.04930.10640.11160.17930.3138R-IM0.04930.04600.10380.10820.17240.2404R-RC0.04840.04520.08320.09260.17230.23791040IPW0.06330.07280.34040.35110.25190.3550IM0.05270.06230.18040.19410.17150.2726RC0.05110.06130.16420.16090.16990.2599R-IPW0.92620.75870.10460.10670.18110.3335R-IM0.04450.04590.09640.09860.17130.2670R-RC0.04180.04340.07760.08440.16960.25724010IPW0.22380.22350.34080.32840.29430.3486IM0.22320.22110.27610.27560.27520.3010RC0.20580.20080.26390.26130.25910.2633R-IPW0.17270.18760.30450.30140.28680.3441R-IM0.13840.13590.26830.26300.25760.2743R-RC0.13560.14110.25810.25250.24650.26134040IPW0.23710.23410.34040.35090.28850.3601IM0.22820.22660.30970.30840.27350.3565RC0.21320.21170.29590.29310.27340.3037R-IPW0.18600.18710.30820.32890.27090.3564R-IM0.15890.15920.27900.27700.26900.3548R-RC0.15730.15240.26420.26090.25650.29684001010IPW0.02250.02180.30160.25080.12860.3290IM0.01220.01330.05380.05970.11300.1439RC0.01190.01260.04140.05210.11230.1364R-IPW0.01340.01360.02930.04150.12610.2696R-IM0.01010.01010.03170.04070.11280.1409R-RC0.01010.00990.03040.03490.11190.13521040IPW0.03510.03020.33380.39390.13190.3946IM0.02690.02060.17800.13770.12100.2414RC0.02770.02150.13190.12420.11690.2034R-IPW0.01520.01400.04960.04650.12980.3259R-IM0.01490.01310.04800.04520.11930.2226R-RC0.01430.01260.03180.03220.11510.18614010IPW0.17420.17340.30180.29010.26270.3830IM0.07550.07290.16880.17260.17230.2229RC0.05650.05240.13750.13940.15080.2186R-IPW0.12930.13070.16070.16070.25810.3701R-IM0.07100.06850.13560.14050.17200.2207R- RC0.05210.04780.11140.11680.15040.21834040IPW0.17870.17430.28170.27860.26810.4099IM0.12850.12680.21700.23050.21270.2755RC0.08630.08410.15280.17150.15460.2333R-IPW0.12820.12610.17510.17500.26470.2353R-IM0.12030.11490.14940.14920.21260.2667R-RC0.05800.05660.11950.12700.15420.3959


## 5.2. A Real Data Analysis

In real data analysis, we illustrate the methodology via an application to a dataset from a breast cancer clinical trial [25]. This clinical trial was conducted by the Eastern Cooperative Oncology Group, whose target was evaluating tamoxifen as a treatment for stage II breast cancer among elderly women, who are older than 65. There are 169 elderly women participating in the trial, and we focus on 79 women who died by the end of the trial. But, unfortunately, the cause of death is incomplete. Among them, 44 women died from breast cancer, 17 died from other known causes, and 18 died from unknown causes. Let the censoring indicator δ show whether the death was caused by breast cancer, and let the missing indicator ς show whether the cause of death was known. The dataset contains four covariates: whether the patients accepted the treatment (1, tamoxifen; 0, placebo), denoted as X1; whether the estrogen receptor status was positive (1, yes; 0, unknown), denoted as X2; whether there were four or more axillary lymph positive nodes (1, yes; 0, no), denoted as Z; and whether the primary tumor is 3 cm or larger (1, yes; 0, no), denoted as U. Then, we employ the following model to fit the data,(66)Y=X1⊤β1+X2⊤β2+Z⊤αU+ε,where Y is the logarithm of the time to death due to breast cancer, which is censored, and the censoring indicator is MAR. The heteroscedastic error follows the form of Varε|X,Z,U=σ2Ui. For the purpose of comparison, we compute both mean squared prediction error (MSE) and mean absolute deviation (MAD) of the predictions, which are defined as follows: MSE=1/n∑i=1169Yi−Y^i2, and MAD=1/n∑i=1169Yi−Y^i, where Y^i is the fitted value of Yi. The values of MSE and MAD based on different methods are given in Table 5. In addition, the estimated curves of αu based on RC, IM, and IPW methods are reported in Figure 5.Table 5
The estimators, MSEs, and MADs for the breast cancer dataset.
MethodEstimatesMSEMADβ1β2RC0.10156.77871.05720.6311IM0.50635.31961.17871.9918IPW0.53295.03361.79342.2677R-RC0.10877.76051.33510.6693R-IM0.59946.13031.18692.5233R-IPW0.64655.80902.23852.8778Figure 5
The curves ofα^n⋅ for the breast cancer dataset based on RC, IM, and IPW methods.From Table5 and Figure 5, it can be seen that, (1) in Table 5, the estimators of β1 are positive, which indicates that the breast-cancer deaths may live longer if they received the treatment. Among these estimators, the MSE and MAD based on the R-RC method are smallest, which confirms the conclusions in Theorems 1 and 4. (2) In Figure 5, the primary tumor size of patients is mainly from 0 to 3 cm. The survival time decreases obviously as the tumor size increases.


## 6. Conclusion

In this paper, we consider the estimation and confidence regions based on modified PLS method and EL inference for PLVCEV model with heteroscedastic errors under censoring indicators MAR, respectively. Asymptotic properties of the proposed estimators are established, and the confidence regions of parameter are constructed. In addition, a simulation study and real data analysis are conducted to illustrate our proposed method.Xu and Duan [11] established efficient estimation for varying-coefficient heteroscedastic partially linear model with additive errors, but their results are confined in responses observed completely. It is an innovative and challenging topic to study the statistical inference for heteroscedastic PLVCEV model under right-censored data with censoring indicators MAR.An interesting problem is whether we can extend the estimation method and incomplete data to functional regression models.



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*Source: 1141022-2021-06-07.xml*