半参EV模型和缺失数据下估计方程的经验似然推断
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摘要
作为一种非参数统计方法,经验似然自Owen (1988)年提出以来已经得到越来越多的关注。它已经广泛用于构造兴趣参数和光滑函数的置信区域。许多文献表明相比于正态逼近方法,经验似然有许多的优势。比如,由经验似然构造的置信区域其形状完全由数据决定,而且还具有域保持性和变换不变性。如今,作为一种重要的非参数统计方法,经验似然已经成为非常有用的统计推断工具。许多学者已经把它应用到线性模型,非参模型及半参模型中。
然而,在许多的应用领域,比如工农业生产、社会调查、经济学、生物医学和流行病学等领域,由于各种各样的原因我们很难获得一些变量的精确测量或全部测量,因此会经常遇到诸如测量误差数据、缺失数据和删失数据等复杂的数据。如何处理这些复杂数据进而进行有效的统计推断已经成为当今统计界的研究热点之一。本学位论文将研究测量误差数据和缺失数据下的一些推断,那就是我们将运用经验似然这个工具来研究两类有测量误差的半参模型和有缺失数据的估计方程,我们所做的工作进一步拓宽了经验似然的应用领域。
由于不能直接对感兴趣的变量进行观测,而只能获得它的替代值。一个简单、经典的测量误差模型或者变量误差模型(EV)模型假设W=X+U,其中X是兴趣变量,W是变量X在与之独立的可加测量误差U下的一个替代值,满足E(U)=0。对于简单的线性EV模型及非线性EV模型人们已经研究的很多了。随着应用技术的发展,半参模型由于其灵活性和可解释性也得到了很好的研究及广泛的应用。在半参模型中,变系数部分线性模型(VCPLM)和可加部分线性模型(APLM)是两类很常用的模型,这两类模型都既有效地避免了非参模型的“维数祸根”问题("curse of dimensionality")又都具有线性模型的可解释能力。因此在本学位论文中,我们仅在经典测量误差模型下研究变系数部分线性模型(VCPLM)和可加部分线性模型(APLM)的推断。具体地讲,在第二章我们将运用经验似然方法对变系数部分线性EV模型的参数和非参部分进行推断,对可加部分线性EV模型的经验似然推断放到第三章。
半参数变系数部分线性EV模型有如下的形式：其中Y是响应变量,T,X和Z是回归变量,β=(β1,…,βp)'是p维的未知参数,α(T)=(α1(T),...,αq(T))'是q维的未知函数向量,ε是在给定X,Z,T的条件下零均值的随机误差。U是测量误差,其均值为零,且与(X,Z,T)独立。You和Chen(2006)研究该模型,对参数分量部分提出了修正的截面最小二乘估计,对非参部分提出了局部多项式估计,并证明了参数估计具有相合性和渐近正态性,非参估计达到了最优的收敛速度。但是他们没有考虑参数和非参部分置信区域的构造。如果我们采用正态逼近方法构造置信区域,You和Chen(2006)的结果告诉我们由于参数估计的方差很复杂,因而这样做很不方便。于是本学位论文将采用经验似然的方法对参数部分和非参部分分别构造他们的置信区域。我们首先给出参数的一个估计函数,基于该估计函数,我们定义出未知参数的经验对数似然比统计量log(R(β)),并在一定条件下证明统计量2log(R(β))渐近趋于标准卡方分布,因此可以用它来构造置信区域。我们同时也证明了未知参数β的极大经验似然估计β是渐近正态的。基于极大经验似然估计β,我们又提出了对未知函数α(t)基于残差调整的下辅助随机变量,并定义α(t)相应的残差调整经验对数似然比函数l(α(t)),证明了在一定条件之下—2l(α(t))的极限分布是标准卡方。
类似于第二章的思想,我们在第三章研究可加部分线性EV模型的经验似然推断。可加部分线性EV模型具有下面的形式：其中Y是响应变量,X和Z=(Z1,…,ZD)'分别是Rp和RD上的协变量,f1,…,fD是未知函数,β=(β1…,βp)'是p维的未知参数,ε是随机误差,满足在给定X和Z条件下均值为零。U是均值为零的测量误差,并且与(X,Z,Y)独立。为简单起见,我们研究D=2的情况。为保证非参函数的可识别性,假设E{f1(Z1)}=E{f2(Z2)}=0,同时假设X和Y已中心化。通过对衰减的修正(correction-for-attenuation),我们得到了未知参数基于纠衰(corrected-attenuation)下的辅助随机变量作为其估计函数,然后定义相应的基于纠衰的经验似然比函数。在没有对非参函数要求欠光滑(undcrsmoothing)的条件下,我们证明了相应统计量极限分布是标准卡方分布,因此基于该统计量很容易得到未知参数的置信区域。模拟结果表明：通过比较置信区域的覆盖概率和平均长度,我们提出的方法要优于Liang,Thurston,Duppert,Apanasovich和Hauser(2008)提出的截面最小二乘方法。基于参数β的经验似然比统计量,很容易得到它的极大经验似然估计β,进而得到非参函数修正的后拟(backfitting)估计.因此,对非参函数我们又给出了其残差调整的经验对数似然比统计量,并证明它仍具有非参的Wilk's定理。值得一提的是在对非参函数f1(z1)进行推断时并不需要精确估计非参函数f2(z2)在任意点的值,只需知道f2(z2)修正的后拟估计在样本观测点处的值即可。
在第四章我们研究缺失数据下的估计方程。在Zhou, Wan和Wang (2008)的文章中,他们基于观测到的数据用估计函数的非参估计进行借补,定义了未知参数新的估计函数。由于非参估计的插入,导致估计函数是有偏的,基于该估计函数的经验似然比不再收敛于标准卡方,而是卡方变量的加权和,其中的权重是未知的(具体结果见Zhou, Wan和Wang (2008)文章中的定理3)。因此为了得到标准卡方分布,需要进行调整,从而需要对未知的调整因子进行有效的估计。另外,非参估计时窗宽选择需要欠光滑条件。这些都使得在构造未知参数置信区域的时候使用起来很不方便。受到Xuc (2009a)和Xuc (2009b)文章的启发,我们提出了用加权修正的方法来减非参估计的偏,定义新的估计函数,证明基于新的估计函数下未知参数的经验对数似然比函数渐近于标准卡方,这一结果不同于Zhou, Wan和Wang (2008)得到的结果。因此我们的方法避免了对未知调整因子的估计,而且可以用基于数据(data-driven)的常用窗宽选择的方法来选择最优窗宽。数值模拟进一步验证了我们的方法。
Empirical likelihood method, as a nonparametric method, has received more and more attention since it was first proposed by Owen (1988). It has been popularly used for constructing confidence regions for some interesting parameters and smooth function. Many advantages of empirical likelihood over normal approximation method have been shown in the literatures. For example, it is known that the shape and orientation of empirical likelihood based confidence regions arc determined entirely by the data, and also these regions are range preserving and transformation respecting. Today, as an important nonparamctric method, empirical likelihood has become a very useful tool for statistical inference. Many authors have used the method for linear, nonparametric and semiparametric regression models.
However, in many application fields, such as industry and agriculture production, society investigation, economics, biomedical sciences and epidemiology and so on, it is difficult for us to obtain the exact or complete measurement for some variables due to many different kinds of reasons, so complicated data such as measurement error data, missing data, censored data are often encountered. How to deal with these compli-cated data to derive efficient inferences has become one of the hot issues in modern statistical analysis. In this thesis, we shall study some inferences under measurement error data and missing data, that is we shall employ the empirical likelihood tool to investigate two classes of semiparametric models with measurement error data and es-timating equation with missing data, these work further broadens the application areas of empirical likelihood.
Instead of observing the interesting variable directly, we only observe its surrogate. A simple and classical measurement error model or crrors-in-variables (EV) model as-sume that W=X+U, where X is the variable of interest,W is the surrogate of X with additive measurement error U which is independent of X and E(U)=0. The simple linear EV model and nonlinear EV model have been well studied. With the development of applied sciences, semiparametric regression models have been well researched and popularly used for their flexibility and interpretability. Among semi-parametric models, varying-coefficient partially linear model (VCPLM) and additive partially linear model (APLM) arc two classes of commonly-used models because they effectively avoid the "curse of dimensionality" of nonparametric model and have the explanatory power of the linear regression model. So in this thesis, we study the in-ferences for VCPLM and APLM only under the classical measurement error model. More specifically, we employ the empirical likelihood method to infer the paramet-ric and nonparametric components for varying-coefficient partially linear EV model in Chapter 2 and empirical likelihood inferences for additive partially linear EV model in Chapter 3.
Semiparametric varying-coefficient partially linear EV model has the form as fol-lows where Y is the response, T. X and Z arc regressors,β= (β1,...,βp)' is a p-dimensional vector of unknown parameters,α(T)= (α1(T),....,αq(T))' is a q-dimensional vector of unknown functions andεis the random error with conditional mean zero given X, Z and T. U is the measurement error with mean zero and independent of (X, Z, T). You and Chen (2006) studied this model and proposed a modified profile least squares estimator for the parametric component and local polynomial estimator for the non-parametric component. They showed that the former is consistent and asymptotically normal distributed and the latter achieves the optimal strong convergence rate of the nonparametric regression. But they did not consider the construction of confidence region for the parametric and nonparametric component. If we use the popularly used normal approximation method to derive the confidence region, the result in You and Chen (2006) tells us that the limiting variance of the parameter estimator is very com-plicated, thus it is inconvenient to be used for confidence region construction. So in this thesis we use empirical likelihood to construct the confidence regions for the para-metric and nonparametric components. We first derive an estimator function for the parameter, based on this we define an empirical log-likelihood ratio statistic log(R(β)) for the unknown parameterβ. We show that the statistic -2log(R(β)) is asymptoti-cally standard chi-square distribution under some suitable conditions and can be used to construct the confidence region directly. We also prove the maximum empirical likelihood estimator (MELE)βof the unknown parameter vectorβis asymptotically normal. Then based on theβ, we propose a residual-adjusted auxiliary random vector for the unknown functions a(t) and define the corresponding residual-adjusted empir-ical log-likelihood ratio function l(a(t)) forα(t). Under some suitable conditions the limiting distribution of the -2l(α(t)) is asymptotically a standard chi-square.
Similar to the ideas of chapter 2, in chapter 3, we study the empirical likelihood inferences for additive partially linear EV model, which can be written as where Y is the response, X and Z=(Z1,..., ZD)' are covariates on Rp and RD respec-tively,f1,...,fD are unknown smooth functions,β=(β1,...,βp)' is a p-dimensional vector of unknown parameters andεis the random error with conditional mean zero given X and Z. U is the measurement error with mean zero and independent of (X,Z,Y). For simplicity, we study the case of D=2 and assume E{f1(Z1)}= E{f2(Z2)}=0 to ensure identifiability of the nonparametric functions, and X and Y are centered. By correction-for-attenuation, we get a corrected-attenuation auxiliary vector as an estimating function for the unknown parameter and then define the corre-sponding corrected-attenuation empirical likelihood ratio function. Without requiring the undersmoothing of the nonparametric components, we prove that the proposed statistic for the unknown parameter has a standard chi-square limiting distribution asymptotically, and so it can be conveniently used to derive the confidence regions. Sim-ulation studies indicate that, by comparing coverage probabilities and average lengths of the confidence intervals, the proposed method outperforms the profile-based least-squares method which has been studied by Liang, Thurston, Duppert, Apanasovich and Hauser (2008). Based on the proposed empirical likelihood ratio for the parameterβ, we can easily obtain the maximum empirical likelihood estimator (MELE)βofβ, and further the corrected backfitting estimators of the nonparametric functions. So the residual-adjusted empirical log-likelihood ratio statistics for nonparametric func-tions are given and the nonparametric Wilk's theorems are also obtained. It is worth to point out that our inference for f1(z1) does not need to accurately estimate the nonparametric function f2(z2) at any point, we only need to know some values of the corrected backfitting estimator for f2(z2) at the sample observations.
In chapter 4, we investigate estimating equation with missing data. In Zhou, Wan and Wang (2008), they imputed the estimating function by nonparametric estimator using the observed data, and then defined a new estimating function for unknown parameter. Since the nonparametric estimator is plugged in, the resulting estimating function becomes biased, the empirical likelihood ratio based on the biased estimating function cannot converge in distribution to a standard chi-square distribution, but a weighted sum of chi-square variables, where the weights are unknown (see Theorem 3 of Zhou, Wan and Wang (2008)). In order to obtain a standard chi-square distribution, adjustment is needed and an unknown adjustment factor needs to be efficiently esti-mated. Besides, under-smoothing involved in nonparametric estimation is needed in selecting the bandwidth, and so they arc inconvenient to use to construct a confidence region for the parameter of interest. Inspired by Xue (2009a) and Xue (2009b), we propose to use the weighted-corrected method to reduce the nonparametric bias and define the augmented inverse probability-weighted estimating function, and under the mild conditions, the resulting empirical log-likelihood ratio for unknown parameter is proved to be a standard chi-square distribution asymptotically, which is different from the result derived in Zhou, Wan and Wang (2008). So our approach avoids estimating an unknown adjustment factor and the commonly used data-driven algorithm can be applied to select an optimal bandwidth. Some simulations further verify our method.
然而,在许多的应用领域,比如工农业生产、社会调查、经济学、生物医学和流行病学等领域,由于各种各样的原因我们很难获得一些变量的精确测量或全部测量,因此会经常遇到诸如测量误差数据、缺失数据和删失数据等复杂的数据。如何处理这些复杂数据进而进行有效的统计推断已经成为当今统计界的研究热点之一。本学位论文将研究测量误差数据和缺失数据下的一些推断,那就是我们将运用经验似然这个工具来研究两类有测量误差的半参模型和有缺失数据的估计方程,我们所做的工作进一步拓宽了经验似然的应用领域。
由于不能直接对感兴趣的变量进行观测,而只能获得它的替代值。一个简单、经典的测量误差模型或者变量误差模型(EV)模型假设W=X+U,其中X是兴趣变量,W是变量X在与之独立的可加测量误差U下的一个替代值,满足E(U)=0。对于简单的线性EV模型及非线性EV模型人们已经研究的很多了。随着应用技术的发展,半参模型由于其灵活性和可解释性也得到了很好的研究及广泛的应用。在半参模型中,变系数部分线性模型(VCPLM)和可加部分线性模型(APLM)是两类很常用的模型,这两类模型都既有效地避免了非参模型的“维数祸根”问题("curse of dimensionality")又都具有线性模型的可解释能力。因此在本学位论文中,我们仅在经典测量误差模型下研究变系数部分线性模型(VCPLM)和可加部分线性模型(APLM)的推断。具体地讲,在第二章我们将运用经验似然方法对变系数部分线性EV模型的参数和非参部分进行推断,对可加部分线性EV模型的经验似然推断放到第三章。
半参数变系数部分线性EV模型有如下的形式：其中Y是响应变量,T,X和Z是回归变量,β=(β1,…,βp)'是p维的未知参数,α(T)=(α1(T),...,αq(T))'是q维的未知函数向量,ε是在给定X,Z,T的条件下零均值的随机误差。U是测量误差,其均值为零,且与(X,Z,T)独立。You和Chen(2006)研究该模型,对参数分量部分提出了修正的截面最小二乘估计,对非参部分提出了局部多项式估计,并证明了参数估计具有相合性和渐近正态性,非参估计达到了最优的收敛速度。但是他们没有考虑参数和非参部分置信区域的构造。如果我们采用正态逼近方法构造置信区域,You和Chen(2006)的结果告诉我们由于参数估计的方差很复杂,因而这样做很不方便。于是本学位论文将采用经验似然的方法对参数部分和非参部分分别构造他们的置信区域。我们首先给出参数的一个估计函数,基于该估计函数,我们定义出未知参数的经验对数似然比统计量log(R(β)),并在一定条件下证明统计量2log(R(β))渐近趋于标准卡方分布,因此可以用它来构造置信区域。我们同时也证明了未知参数β的极大经验似然估计β是渐近正态的。基于极大经验似然估计β,我们又提出了对未知函数α(t)基于残差调整的下辅助随机变量,并定义α(t)相应的残差调整经验对数似然比函数l(α(t)),证明了在一定条件之下—2l(α(t))的极限分布是标准卡方。
类似于第二章的思想,我们在第三章研究可加部分线性EV模型的经验似然推断。可加部分线性EV模型具有下面的形式：其中Y是响应变量,X和Z=(Z1,…,ZD)'分别是Rp和RD上的协变量,f1,…,fD是未知函数,β=(β1…,βp)'是p维的未知参数,ε是随机误差,满足在给定X和Z条件下均值为零。U是均值为零的测量误差,并且与(X,Z,Y)独立。为简单起见,我们研究D=2的情况。为保证非参函数的可识别性,假设E{f1(Z1)}=E{f2(Z2)}=0,同时假设X和Y已中心化。通过对衰减的修正(correction-for-attenuation),我们得到了未知参数基于纠衰(corrected-attenuation)下的辅助随机变量作为其估计函数,然后定义相应的基于纠衰的经验似然比函数。在没有对非参函数要求欠光滑(undcrsmoothing)的条件下,我们证明了相应统计量极限分布是标准卡方分布,因此基于该统计量很容易得到未知参数的置信区域。模拟结果表明：通过比较置信区域的覆盖概率和平均长度,我们提出的方法要优于Liang,Thurston,Duppert,Apanasovich和Hauser(2008)提出的截面最小二乘方法。基于参数β的经验似然比统计量,很容易得到它的极大经验似然估计β,进而得到非参函数修正的后拟(backfitting)估计.因此,对非参函数我们又给出了其残差调整的经验对数似然比统计量,并证明它仍具有非参的Wilk's定理。值得一提的是在对非参函数f1(z1)进行推断时并不需要精确估计非参函数f2(z2)在任意点的值,只需知道f2(z2)修正的后拟估计在样本观测点处的值即可。
在第四章我们研究缺失数据下的估计方程。在Zhou, Wan和Wang (2008)的文章中,他们基于观测到的数据用估计函数的非参估计进行借补,定义了未知参数新的估计函数。由于非参估计的插入,导致估计函数是有偏的,基于该估计函数的经验似然比不再收敛于标准卡方,而是卡方变量的加权和,其中的权重是未知的(具体结果见Zhou, Wan和Wang (2008)文章中的定理3)。因此为了得到标准卡方分布,需要进行调整,从而需要对未知的调整因子进行有效的估计。另外,非参估计时窗宽选择需要欠光滑条件。这些都使得在构造未知参数置信区域的时候使用起来很不方便。受到Xuc (2009a)和Xuc (2009b)文章的启发,我们提出了用加权修正的方法来减非参估计的偏,定义新的估计函数,证明基于新的估计函数下未知参数的经验对数似然比函数渐近于标准卡方,这一结果不同于Zhou, Wan和Wang (2008)得到的结果。因此我们的方法避免了对未知调整因子的估计,而且可以用基于数据(data-driven)的常用窗宽选择的方法来选择最优窗宽。数值模拟进一步验证了我们的方法。
Empirical likelihood method, as a nonparametric method, has received more and more attention since it was first proposed by Owen (1988). It has been popularly used for constructing confidence regions for some interesting parameters and smooth function. Many advantages of empirical likelihood over normal approximation method have been shown in the literatures. For example, it is known that the shape and orientation of empirical likelihood based confidence regions arc determined entirely by the data, and also these regions are range preserving and transformation respecting. Today, as an important nonparamctric method, empirical likelihood has become a very useful tool for statistical inference. Many authors have used the method for linear, nonparametric and semiparametric regression models.
However, in many application fields, such as industry and agriculture production, society investigation, economics, biomedical sciences and epidemiology and so on, it is difficult for us to obtain the exact or complete measurement for some variables due to many different kinds of reasons, so complicated data such as measurement error data, missing data, censored data are often encountered. How to deal with these compli-cated data to derive efficient inferences has become one of the hot issues in modern statistical analysis. In this thesis, we shall study some inferences under measurement error data and missing data, that is we shall employ the empirical likelihood tool to investigate two classes of semiparametric models with measurement error data and es-timating equation with missing data, these work further broadens the application areas of empirical likelihood.
Instead of observing the interesting variable directly, we only observe its surrogate. A simple and classical measurement error model or crrors-in-variables (EV) model as-sume that W=X+U, where X is the variable of interest,W is the surrogate of X with additive measurement error U which is independent of X and E(U)=0. The simple linear EV model and nonlinear EV model have been well studied. With the development of applied sciences, semiparametric regression models have been well researched and popularly used for their flexibility and interpretability. Among semi-parametric models, varying-coefficient partially linear model (VCPLM) and additive partially linear model (APLM) arc two classes of commonly-used models because they effectively avoid the "curse of dimensionality" of nonparametric model and have the explanatory power of the linear regression model. So in this thesis, we study the in-ferences for VCPLM and APLM only under the classical measurement error model. More specifically, we employ the empirical likelihood method to infer the paramet-ric and nonparametric components for varying-coefficient partially linear EV model in Chapter 2 and empirical likelihood inferences for additive partially linear EV model in Chapter 3.
Semiparametric varying-coefficient partially linear EV model has the form as fol-lows where Y is the response, T. X and Z arc regressors,β= (β1,...,βp)' is a p-dimensional vector of unknown parameters,α(T)= (α1(T),....,αq(T))' is a q-dimensional vector of unknown functions andεis the random error with conditional mean zero given X, Z and T. U is the measurement error with mean zero and independent of (X, Z, T). You and Chen (2006) studied this model and proposed a modified profile least squares estimator for the parametric component and local polynomial estimator for the non-parametric component. They showed that the former is consistent and asymptotically normal distributed and the latter achieves the optimal strong convergence rate of the nonparametric regression. But they did not consider the construction of confidence region for the parametric and nonparametric component. If we use the popularly used normal approximation method to derive the confidence region, the result in You and Chen (2006) tells us that the limiting variance of the parameter estimator is very com-plicated, thus it is inconvenient to be used for confidence region construction. So in this thesis we use empirical likelihood to construct the confidence regions for the para-metric and nonparametric components. We first derive an estimator function for the parameter, based on this we define an empirical log-likelihood ratio statistic log(R(β)) for the unknown parameterβ. We show that the statistic -2log(R(β)) is asymptoti-cally standard chi-square distribution under some suitable conditions and can be used to construct the confidence region directly. We also prove the maximum empirical likelihood estimator (MELE)βof the unknown parameter vectorβis asymptotically normal. Then based on theβ, we propose a residual-adjusted auxiliary random vector for the unknown functions a(t) and define the corresponding residual-adjusted empir-ical log-likelihood ratio function l(a(t)) forα(t). Under some suitable conditions the limiting distribution of the -2l(α(t)) is asymptotically a standard chi-square.
Similar to the ideas of chapter 2, in chapter 3, we study the empirical likelihood inferences for additive partially linear EV model, which can be written as where Y is the response, X and Z=(Z1,..., ZD)' are covariates on Rp and RD respec-tively,f1,...,fD are unknown smooth functions,β=(β1,...,βp)' is a p-dimensional vector of unknown parameters andεis the random error with conditional mean zero given X and Z. U is the measurement error with mean zero and independent of (X,Z,Y). For simplicity, we study the case of D=2 and assume E{f1(Z1)}= E{f2(Z2)}=0 to ensure identifiability of the nonparametric functions, and X and Y are centered. By correction-for-attenuation, we get a corrected-attenuation auxiliary vector as an estimating function for the unknown parameter and then define the corre-sponding corrected-attenuation empirical likelihood ratio function. Without requiring the undersmoothing of the nonparametric components, we prove that the proposed statistic for the unknown parameter has a standard chi-square limiting distribution asymptotically, and so it can be conveniently used to derive the confidence regions. Sim-ulation studies indicate that, by comparing coverage probabilities and average lengths of the confidence intervals, the proposed method outperforms the profile-based least-squares method which has been studied by Liang, Thurston, Duppert, Apanasovich and Hauser (2008). Based on the proposed empirical likelihood ratio for the parameterβ, we can easily obtain the maximum empirical likelihood estimator (MELE)βofβ, and further the corrected backfitting estimators of the nonparametric functions. So the residual-adjusted empirical log-likelihood ratio statistics for nonparametric func-tions are given and the nonparametric Wilk's theorems are also obtained. It is worth to point out that our inference for f1(z1) does not need to accurately estimate the nonparametric function f2(z2) at any point, we only need to know some values of the corrected backfitting estimator for f2(z2) at the sample observations.
In chapter 4, we investigate estimating equation with missing data. In Zhou, Wan and Wang (2008), they imputed the estimating function by nonparametric estimator using the observed data, and then defined a new estimating function for unknown parameter. Since the nonparametric estimator is plugged in, the resulting estimating function becomes biased, the empirical likelihood ratio based on the biased estimating function cannot converge in distribution to a standard chi-square distribution, but a weighted sum of chi-square variables, where the weights are unknown (see Theorem 3 of Zhou, Wan and Wang (2008)). In order to obtain a standard chi-square distribution, adjustment is needed and an unknown adjustment factor needs to be efficiently esti-mated. Besides, under-smoothing involved in nonparametric estimation is needed in selecting the bandwidth, and so they arc inconvenient to use to construct a confidence region for the parameter of interest. Inspired by Xue (2009a) and Xue (2009b), we propose to use the weighted-corrected method to reduce the nonparametric bias and define the augmented inverse probability-weighted estimating function, and under the mild conditions, the resulting empirical log-likelihood ratio for unknown parameter is proved to be a standard chi-square distribution asymptotically, which is different from the result derived in Zhou, Wan and Wang (2008). So our approach avoids estimating an unknown adjustment factor and the commonly used data-driven algorithm can be applied to select an optimal bandwidth. Some simulations further verify our method.
引文
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