Bias-corrected estimations in varying-coefficient partially nonlinear models with measurement error in the nonparametric part

In this paper, we consider the statistical inference for the varying-coefficient partially nonlinear model with additive measurement errors in the nonparametric part. The local bias-corrected profile nonlinear least-squares estimation procedure for parameter in nonlinear function and nonparametric function is proposed. Then, the asymptotic normality properties of the resulting estimators are established. With the empirical likelihood method, a local bias-corrected empirical log-likelihood ratio statistic for the unknown parameter, and a corrected and residual adjusted empirical log-likelihood ratio for the nonparametric component are constructed. It is shown that the resulting statistics are asymptotically chi-square distribution under some suitable conditions. Some simulations are conducted to evaluate the performance of the proposed methods. The results indicate that the empirical likelihood method is superior to the profile nonlinear least-squares method in terms of the confidence regions of parameter and point-wise confidence intervals of nonparametric function.     

Empirical likelihood inferences for varying coefficient partially nonlinear models

In this article, empirical likelihood inferences for the varying coefficient partially nonlinear models are investigated. An empirical log-likelihood ratio function for the unknown parameter vector in the nonlinear function part and a residual-adjusted empirical log-likelihood ratio function for the nonparametric component are proposed. The corresponding Wilks phenomena are proved and the confidence regions for parametric component and nonparametric component are constructed. Simulation studies indicate that, in terms of coverage probabilities and average areas of the confidence regions, the empirical likelihood method performs better than the normal approximation-based method. Furthermore, a real data set application is also provided to illustrate the proposed empirical likelihood estimation technique.     

Empirical Likelihood for Nonparametric Components in Additive Partially Linear Models

Empirical likelihood-based inference for the nonparametric components in additive partially linear models is investigated. An empirical likelihood approach to construct the confidence intervals of the nonparametric components is proposed when the linear covariate is measured with and without errors. We show that the proposed empirical log-likelihood ratio is asymptotically standard chi-squared without requiring the undersmoothing of the nonparametric components. Then, it can be directly used to construct the confidence intervals for the nonparametric functions. A simulation study indicates that, compared with a normal approximation-based approach, the proposed method works better in terms of coverage probabilities and widths of the pointwise confidence intervals.     

Empirical Likelihood Inference for the Parameter in Additive Partially Linear EV Models

In this article, we consider empirical likelihood inference for the parameter in the additive partially linear models when the linear covariate is measured with error. By correcting for attenuation, a corrected-attenuation empirical log-likelihood ratio statistic for the unknown parameter β, which is of primary interest, is suggested. We show that the proposed statistic is asymptotically standard chi-square distribution without requiring the undersmoothing of the nonparametric components, and hence it can be directly used to construct the confidence region for the parameter β. Some simulations indicate that, in terms of comparison between coverage probabilities and average lengths of the confidence intervals, the proposed method performs better than the profile-based least-squares method. We also give the maximum empirical likelihood estimator (MELE) for the unknown parameter β, and prove the MELE is asymptotically normal under some mild conditions.     

Two Adjusted Empirical-Likelihood-Based Methods in Generalized Varying-Coefficient Partially Linear Model

An empirical likelihood method was proposed by Owen and has been extended to many semiparametric and nonparametric models with a continuous response variable. However, there has been less attention focused on the generalized regression model. This article systematically studies two adjusted empirical-likelihood-based methods in the generalized varying-coefficient partially linear models. Based on the popular profile likelihood estimation procedure, the new adjusted empirical likelihood technology for the parameter is established and the resulting statistics are shown to be asymptotically standard chi-square distributed. Further, the adjusted empirical-likelihood-based confidence regions are established, and an efficient adjusted profile empirical-likelihood-based confidence intervals/regions for any components of the parameter, which are of primary interest, is also constructed. Their asymptotic properties are also derived. Some numerical studies are carried out to illustrate the performance of the proposed inference procedures.     

Generalized empirical likelihood inference in partial linear regression model for longitudinal data

In this paper, empirical likelihood inference for longitudinal data within the framework of partial linear regression models are investigated. The proposed procedures take into consideration the correlation within groups without involving direct estimation of nuisance parameters in the correlation matrix. The empirical likelihood method is used to estimate the regression coefficients and the baseline function, and to construct confidence intervals. A nonparametric version of Wilk's theorem for the limiting distribution of the empirical likelihood ratio is derived. Compared with methods based on normal approximations, the empirical likelihood does not require consistent estimators for the asymptotic variance and bias. The finite sample behaviour of the proposed method is evaluated with simulation and illustrated with an AIDS clinical trial data set.     

Empirical Likelihood for Nonparametric Models Under Linear Process Errors

In this paper, we study the construction of confidence intervals for a nonparametric regression function under linear process errors by using the blockwise technique. It is shown that the blockwise empirical likelihood (EL) ratio statistic is asymptotically distributed. The result is used to obtain EL based confidence intervals for the nonparametric regression function. The finite‐sample performance of the method is evaluated through a simulation study.     

Empirical Likelihood for a Heteroscedastic Partial Linear Errors-in-Variables Model

The purpose of this article is to use the empirical likelihood method to study construction of the confidence region for the parameter of interest in heteroscedastic partially linear errors-in-variables model with martingale difference errors. When the variance functions of the errors are known or unknown, we propose the empirical log-likelihood ratio statistics for the parameter of interest. For each case, a nonparametric version of Wilks’ theorem is derived. The results are then used to construct confidence regions of the parameter. A simulation study is carried out to assess the performance of the empirical likelihood method.     

Empirical Likelihood for Semiparametric Varying-Coefficient Heteroscedastic Partially Linear Errors-in-Variables Models

The purpose of this article is to use the empirical likelihood method to study construction of the confidence region for the parameter of interest in semiparametric varying-coefficient heteroscedastic partially linear errors-in-variables models. When the variance functions of the errors are known or unknown, we propose the empirical log-likelihood ratio statistics for the parameter of interest. For each case, a nonparametric version of Wilks’ theorem is derived. The results are then used to construct confidence regions of the parameter. A simulation study is carried out to assess the performance of the empirical likelihood method.     

Empirical likelihood for truncation parameter in random truncation model

In this paper we use the empirical likelihood method to construct confidence interval for truncation parameter in random truncation model. The empirical log-likelihood ratio is derived and its asymptotic distribution is shown to be a weighted chi-square. Simulation studies are used to compare the confidence intervals based on empirical likelihood and those based on normal approximation. It is found that the empirical likelihood method provides improved confidence interval.     

Empirical likelihood‐based inference for genetic mixture models

The authors show how the genetic effect of a quantitative trait locus can be estimated by a nonparametric empirical likelihood method when the phenotype distributions are completely unspecified. They use an empirical likelihood ratio statistic for testing the genetic effect and obtaining confidence intervals. In addition to studying the asymptotic properties of these procedures, the authors present simulation results and illustrate their approach with a study on breast cancer resistance genes.     

Penalized Empirical Likelihood via Bridge Estimator in Cox's Proportional Hazard Model

We propose the penalized empirical likelihood method via bridge estimator in Cox's proportional hazard model for parameter estimation and variable selection. Under reasonable conditions, we show that penalized empirical likelihood in Cox's proportional hazard model has oracle property. A penalized empirical likelihood ratio for the vector of regression coefficients is defined and its limiting distribution is a chi-square distributions. The advantage of penalized empirical likelihood as a nonparametric likelihood approach is illustrated in testing hypothesis and constructing confidence sets. The method is illustrated by extensive simulation studies and a real example.     

Empirical likelihood dimension reduction inference for partially non-linear error-in-responses models with validation data

ABSTRACT

In this article, partially non linear models when the response variable is measured with error and explanatory variables are measured exactly are considered. Without specifying any error structure equation, a semiparametric dimension reduction technique is employed. Two estimators of unknown parameter in non linear function are obtained and asymptotic normality is proved. In addition, empirical likelihood method for parameter vector is provided. It is shown that the estimated empirical log-likelihood ratio has asymptotic Chi-square distribution. A simulation study indicates that, compared with normal approximation method, empirical likelihood method performs better in terms of coverage probabilities and average length of the confidence intervals.     

CONFIDENCE INTERVALS FOR THE DIFFERENCE BETWEEN TWO PARTIAL AUCS

As new diagnostic tests are developed and marketed, it is very important to be able to compare the accuracy of a given two continuous‐scale diagnostic tests. An effective method to evaluate the difference between the diagnostic accuracy of two tests is to compare partial areas under the receiver operating characteristic curves (AUCs). In this paper, we review existing parametric methods. Then, we propose a new semiparametric method and a new nonparametric method to investigate the difference between two partial AUCs. For the difference between two partial AUCs under each method, we derive a normal approximation, define an empirical log‐likelihood ratio, and show that the empirical log‐likelihood ratio follows a scaled chi‐square distribution. We construct five confidence intervals for the difference based on normal approximation, bootstrap, and empirical likelihood methods. Finally, extensive simulation studies are conducted to compare the finite‐sample performances of these intervals, and a real example is used as an application of our recommended intervals. The simulation results indicate that the proposed hybrid bootstrap and empirical likelihood intervals outperform other existing intervals in most cases.     

Jackknife Empirical Likelihood for the Accelerated Failure Time Model with Censored Data

Kendall and Gehan estimating functions are commonly used to estimate the regression parameter in accelerated failure time model with censored observations in survival analysis. In this paper, we apply the jackknife empirical likelihood method to overcome the computation difficulty about interval estimation. A Wilks’ theorem of jackknife empirical likelihood for U-statistic type estimating equations is established, which is used to construct the confidence intervals for the regression parameter. We carry out an extensive simulation study to compare the Wald-type procedure, the empirical likelihood method, and the jackknife empirical likelihood method. The proposed jackknife empirical likelihood method has a better performance than the existing methods. We also use a real data set to compare the proposed methods.     

Empirical Likelihood Inferences for Semiparametric Varying-Coefficient Partially Linear Models with Longitudinal Data

In this article, empirical likelihood inferences for semiparametric varying-coefficient partially linear models with longitudinal data are investigated. We propose a groupwise empirical likelihood procedure to handle the inter-series dependence of the longitudinal data. By using residual-adjustment, an empirical likelihood ratio function for the nonparametric component is constructed, and a nonparametric version Wilks' phenomenons is proved. Compared with methods based on normal approximations, the empirical likelihood does not require consistent estimators for the asymptotic variance and bias. A simulation study is undertaken to assess the finite sample performance of the proposed confidence regions.     

Empirical Likelihood Confidence Intervals for Response Mean with Data Missing at Random

Abstract.  A kernel regression imputation method for missing response data is developed. A class of bias-corrected empirical log-likelihood ratios for the response mean is defined. It is shown that any member of our class of ratios is asymptotically chi-squared, and the corresponding empirical likelihood confidence interval for the response mean is constructed. Our ratios share some of the desired features of the existing methods: they are self-scale invariant and no plug-in estimators for the adjustment factor and asymptotic variance are needed; when estimating the non-parametric function in the model, undersmoothing to ensure root- n consistency of the estimator for the parameter is avoided. Since the range of bandwidths contains the optimal bandwidth for estimating the regression function, the existing data-driven algorithm is valid for selecting an optimal bandwidth. We also study the normal approximation-based method. A simulation study is undertaken to compare the empirical likelihood with the normal approximation method in terms of coverage accuracies and average lengths of confidence intervals.     

Asymptotic normality of locally modelled regression estimator for functional data

We focus on the nonparametric regression of a scalar response on a functional explanatory variable. As an alternative to the well-known Nadaraya-Watson estimator for regression function in this framework, the locally modelled regression estimator performs very well [cf. [Barrientos-Marin, J., Ferraty, F., and Vieu, P. (2010), ‘Locally Modelled Regression and Functional Data’, Journal of Nonparametric Statistics, 22, 617–632]. In this paper, the asymptotic properties of locally modelled regression estimator for functional data are considered. The mean-squared convergence as well as asymptotic normality for the estimator are established. We also adapt the empirical likelihood method to construct the point-wise confidence intervals for the regression function and derive the Wilk's phenomenon for the empirical likelihood inference. Furthermore, a simulation study is presented to illustrate our theoretical results.     

Empirical likelihood for generalized partially linear varying-coefficient models

Generalized partially linear varying-coefficient models (GPLVCM) are frequently used in statistical modeling. However, the statistical inference of the GPLVCM, such as confidence region/interval construction, has not been very well developed. In this article, empirical likelihood-based inference for the parametric components in the GPLVCM is investigated. Based on the local linear estimators of the GPLVCM, an estimated empirical likelihood-based statistic is proposed. We show that the resulting statistic is asymptotically non-standard chi-squared. By the proposed empirical likelihood method, the confidence regions for the parametric components are constructed. In addition, when some components of the parameter are of particular interest, the construction of their confidence intervals is also considered. A simulation study is undertaken to compare the empirical likelihood and the other existing methods in terms of coverage accuracies and average lengths. The proposed method is applied to a real example.     

Smoothed empirical likelihood confidence intervals for the relative distribution with left‐truncated and right‐censored data

The study of differences among groups is an interesting statistical topic in many applied fields. It is very common in this context to have data that are subject to mechanisms of loss of information, such as censoring and truncation. In the setting of a two‐sample problem with data subject to left truncation and right censoring, we develop an empirical likelihood method to do inference for the relative distribution. We obtain a nonparametric generalization of Wilks' theorem and construct nonparametric pointwise confidence intervals for the relative distribution. Finally, we analyse the coverage probability and length of these confidence intervals through a simulation study and illustrate their use with a real data set on gastric cancer. The Canadian Journal of Statistics 38: 453–473; 2010 © 2010 Statistical Society of Canada