Empirical likelihood inference for the accelerated failure time model

Accelerated failure time (AFT) models are useful regression tools for studying the association between a survival time and covariates. Semiparametric inference procedures have been proposed in an extensive literature. Among these, use of an estimating equation which is monotone in the regression parameter and has some excellent properties was proposed by Fygenson and Ritov (1994). However, there is a serious under-coverage problem for small sample sizes. In this paper, we derive the limiting distribution of the empirical log-likelihood ratio for the regression parameter on the basis of the monotone estimating equations. Furthermore, the empirical likelihood (EL) confidence intervals/regions for the regression parameter are obtained. We conduct a simulation study in order to compare the proposed EL method with the normal approximation method. The simulation results suggest that the empirical likelihood based method outperforms the normal approximation based method in terms of coverage probability. Thus, the proposed EL method overcomes the under-coverage problem of the normal approximation method.     

Efficient empirical-likelihood-based inferences for the single-index model

This article proposes the efficient empirical-likelihood-based inferences for the single component of the parameter and the link function in the single-index model. Unlike the existing empirical likelihood procedures for the single-index model, the proposed profile empirical likelihood for the parameter is constructed by using some components of the maximum empirical likelihood estimator (MELE) based on a semiparametric efficient score. The empirical-likelihood-based inference for the link function is also considered. The resulting statistics are proved to follow a standard chi-squared limiting distribution. Simulation studies are undertaken to assess the finite sample performance of the proposed confidence intervals. An application to real data set is illustrated.     

Empirical likelihood inference for censored median regression with weighted empirical hazard functions

In recent years, median regression models have been shown to be useful for analyzing a variety of censored survival data in clinical trials. For inference on the regression parameter, there have been a variety of semiparametric procedures. However, the accuracy of such procedures in terms of coverage probability can be quite low when the censoring rate is heavy. In this paper, based on weighted empirical hazard functions, we apply an empirical likelihood (EL) ratio method to the median regression model with censoring data and derive the limiting distribution of EL ratio. Confidence region for the regression parameter can then be obtained accordingly. Furthermore, we compared the proposed method with the standard method through extensive simulation studies. The proposed method almost always outperformed the existing method.     

Empirical likelihood inference for median regression models for censored survival data

Recent advances in median regression model have made it possible to use this model for analyzing a variety of censored survival data. For inference on the model parameter vector, there are now semiparametric procedures based on normal approximation that are valid without strong conditions on the error distribution. However, the accuracy of such procedures can be quite low when the censoring proportion is high. In this paper, we propose an alternative semiparametric procedure based on the empirical likelihood. We define the empirical likelihood ratio for the parameter vector and show that its limiting distribution is a weighted sum of chi-square distributions. Numerical results from a simulation study suggest that the empirical likelihood method is more accurate than the normal approximation based method of Ying et al. (J. Amer. Statist. Assoc. 90 (1995) 178).     

Empirical likelihood for mixed-effects error-in-variables model

This paper mainly introduces the method of empirical likelihood and its applications on two different models. We discuss the empirical likelihood inference on fixed-effect parameter in mixed-effects model with error-in-variables. We first consider a linear mixed-effects model with measurement errors in both fixed and random effects. We construct the empirical likelihood confidence regions for the fixed-effects parameters and the mean parameters of random-effects. The limiting distribution of the empirical log likelihood ratio at the true parameter is X2p＋q, where p, q are dimension of fixed and random effects respectively. Then we discuss empirical likelihood inference in a semi-linear error-in-variable mixed-effects model. Under certain conditions, it is shown that the empirical log likelihood ratio at the true parameter also converges to X2p＋q. Simulations illustrate that the proposed confidence region has a coverage probability more closer to the nominal level than normal approximation based confidence region.     

Semiparametric inference for transformation models via empirical likelihood

Recent advances in the transformation model have made it possible to use this model for analyzing a variety of censored survival data. For inference on the regression parameters, there are semiparametric procedures based on the normal approximation. However, the accuracy of such procedures can be quite low when the censoring rate is heavy. In this paper, we apply an empirical likelihood ratio method and derive its limiting distribution via U-statistics. We obtain confidence regions for the regression parameters and compare the proposed method with the normal approximation based method in terms of coverage probability. The simulation results demonstrate that the proposed empirical likelihood method overcomes the under-coverage problem substantially and outperforms the normal approximation based method. The proposed method is illustrated with a real data example. Finally, our method can be applied to general U-statistic type estimating equations.     

Empirical likelihood inference for censored median regression model via nonparametric kernel estimation

An alternative to the accelerated failure time model is to regress the median of the failure time on the covariates. In the recent years, censored median regression models have been shown to be useful for analyzing a variety of censored survival data with the robustness property. Based on missing information principle, a semiparametric inference procedure for regression parameter has been developed when censoring variable depends on continuous covariate. In order to improve the low coverage accuracy of such procedure, we apply an empirical likelihood ratio method (EL) to the model and derive the limiting distributions of the estimated and adjusted empirical likelihood ratios for the vector of regression parameter. Two kinds of EL confidence regions for the unknown vector of regression parameters are obtained accordingly. We conduct an extensive simulation study to compare the performance of the proposed methods with that normal approximation based method. The simulation results suggest that the EL methods outperform the normal approximation based method in terms of coverage probability. Finally, we make some discussions about our methods.     

右删失数据下非线性回归模型的经验似然推断

考虑随机右删失数据下非线性回归模型,提出了模型中未知参数的调整的经验对数似然比统计量.在一定的条件下,证明了.所提出的的统计量具有渐近χ~2分布,由此结果构造了兴趣参数的置信域.通过模拟研究,对经典的经验似然、调整的经验似然和非线性最小二乘方法在有限样本下进行了比较,并对氯离子浓度试验数据进行了分析.     

Penalized Empirical Likelihood Estimation for Partially Linear Single Index Panel Model with Fixed Effects

This paper constructs a penalized empirical likelihood estimation method via quadratic inference function method, filter method and empirical likelihood estimation method. Under some regular conditions, we derived the large sample properties of estimators and show that the proposed empirical likelihood ratio is asymptotically to chi-square distribution. Furthermore, the infinite sample performance of the proposed method is evaluated by Monte Carlo simulation and real data analysis.     

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??This paper constructs a penalized empirical likelihood estimation method via quadratic inference function method, filter method and empirical likelihood estimation method. Under some regular conditions, we derived the large sample properties of estimators and show that the proposed empirical likelihood ratio is asymptotically to chi-square distribution. Furthermore, the infinite sample performance of the proposed method is evaluated by Monte Carlo simulation and real data analysis.     

Spline-based sieve maximum likelihood estimation in the partly linear model under monotonicity constraints

We study a spline-based likelihood method for the partly linear model with monotonicity constraints. We use monotone B-splines to approximate the monotone nonparametric function and apply the generalized Rosen algorithm to compute the estimators jointly. We show that the spline estimator of the nonparametric component achieves the possible optimal rate of convergence under the smooth assumption and that the estimator of the regression parameter is asymptotically normal and efficient. Moreover, a spline-based semiparametric likelihood ratio test is established to make inference of the regression parameter. Also an observed profile information method to consistently estimate the standard error of the spline estimator of the regression parameter is proposed. A simulation study is conducted to evaluate the finite sample performance of the proposed method. The method is illustrated by an air pollution study.     

具有随机右删失随机变量分位数的经验似然推断

本文研究了具有随机右删失随机变量分位数的置信域的构造.利用经验似然和截尾值估算相结合的方法,给出了分位数的对数经验似然比统计量,在较少的条件下证明了该统计量的极限分布为自由度为1的x~2分布.使得完全数据下的分位数的经验似然推断方法应用到非完全数据中.     

Risk analysis with categorical explanatory variables

To better forecast the Value-at-Risk of the aggregate insurance losses, Heras et al. (2018) propose a two-step inference of using logistic regression and quantile regression without providing detailed model assumptions, deriving the related asymptotic properties, and quantifying the inference uncertainty. This paper argues that the application of quantile regression at the second step is not necessary when explanatory variables are categorical. After describing the explicit model assumptions, we propose another two-step inference of using logistic regression and the sample quantile. Also, we provide an efficient empirical likelihood method to quantify the uncertainty. A simulation study confirms the good finite sample performance of the proposed method.     

Asymptotic distribution of the nonparametric distribution estimator based on a martingale approach in doubly censored data

For analysis of time-to-event data with incomplete information beyond right-censoring, many generalizations of the inference of the distribution and regression model have been proposed. However, the development of martingale approaches in this area has not progressed greatly, while for right-censored data such an approach has spread widely to study the asymptotic properties of estimators and to derive regression diagnosis methods. In this paper, focusing on doubly censored data, we discuss a martingale approach for inference of the nonparametric maximum likelihood estimator (NPMLE). We formulate a martingale structure of the NPMLE using a score function of the semiparametric profile likelihood. Finally, an expression of the asymptotic distribution of the NPMLE is derived more conveniently without depending on an infinite matrix expression as in previous research. A further useful point is that a variance-covariance formula of the NPMLE computable in a larger sample is obtained as an empirical version of the limit form presented here.     

Estimation and test of restricted linear EV model with nonignorable missing covariates

This paper deals with estimation and test procedures for restricted linear errors-invariables (EV) models with nonignorable missing covariates. We develop a restricted weighted corrected least squares (WCLS) estimator based on the propensity score, which is fitted by an exponentially tilted likelihood method. The limiting distributions of the proposed estimators are discussed when tilted parameter is known or unknown. To test the validity of the constraints, we construct two test procedures based on corrected residual sum of squares and empirical likelihood method and derive their asymptotic properties. Numerical studies are conducted to examine the finite sample performance of our proposed methods.     

Empirical likelihood semiparametric nonlinear regression analysis for longitudinal data with responses missing at random

This paper develops the empirical likelihood (EL) inference on parameters and baseline function in a semiparametric nonlinear regression model for longitudinal data in the presence of missing response variables. We propose two EL-based ratio statistics for regression coefficients by introducing the working covariance matrix and a residual-adjusted EL ratio statistic for baseline function. We establish asymptotic properties of the EL estimators for regression coefficients and baseline function. Simulation studies are used to investigate the finite sample performance of our proposed EL methodologies. An AIDS clinical trial data set is used to illustrate our proposed methodologies.     

泊松回归模型中协变量含误差的经验似然推断

在模型的协变量含有测量误差的情况下,考虑一类泊松回归模型的统计推断问题.通过巧妙地构造辅助随机向量,提出一个工具变量类型的经验似然统计推断方法.证明构造的经验对数似然比函数渐近服从标准卡方分布,进而给出了回归系数的置信区间.所提出的估计方法可以有效地消除测量误差对估计精度的影响,并且具有较好的有限样本性质.     

核实数据下响应变量缺失的线性EV模型经验似然推断

考虑响应变量随机缺失而协变量带有误差的线性模型,借助于核实数据和借补方法,构造了回归系数的两种经验似然比,证明了所提出的估计的经验对数似然比渐近于一个自由度为1的独立χ2变量的加权和;而经调整后所得的调整经验对数似然比渐近于自由度为p的χ2分布,该结果可以用来构造未知参数的置信域.此外,我们也构造了响应均值的调整经验对数似然比统计量,并证明了所提出的统计量渐近于x2分布,可用此结果构造响应均值的置信域.通过模拟研究比较了置信域的精度及其平均区间长度.     

Corrected empirical likelihood inference for right-censored partially linear single-index model

This article deals with the inference on a right-censored partially linear single-index model (RCPLSIM). The main focus is the local empirical likelihood-based inference on the nonparametric part in RCPLSIM. With a synthetic data approach, an empirical log-likelihood ratio statistic for the nonparametric part is defined and it is shown that its limiting distribution is not a central chi-squared distribution. To increase the accuracy of the confidence interval, we also propose a corrected empirical log-likelihood ratio statistic for the nonparametric function. The resulting statistic is proved to follow a standard chi-squared limiting distribution. Simulation studies are undertaken to assess the finite sample performance of the proposed confidence intervals. A real example is also considered.     

Empirical likelihood inferences for semiparametric instrumental variable models

This paper studies the empirical likelihood inferences for a class of semiparametric instrumental variable models. We focus on the case that some covariates are endogenous variables, and some auxiliary instrumental variables are available. An instrumental variable based empirical likelihood method is proposed, and it is shown that the proposed empirical log-likelihood ratio is asymptotically chi-squared. Then, the confidence intervals for the regression coefficients are constructed. Some simulation studies are undertaken to assess the finite sample performance of the proposed empirical likelihood procedure.