Empirical Likelihood for Partially Linear Single-Index Errors-in-Variables Model

In this article, we consider the application of the empirical likelihood method to a partially linear single-index model. We focus on the case where some covariates are measured with additive errors. It is shown that the empirical log-likelihood ratio at the true parameter converges to the standard chi-square distribution. Simulations show that the proposed confidence region has coverage probability which is closer to the nominal level, as well as narrower than those of normal approximation method. A real data example is given.     

Exact Size and Power Properties of Five Tests for Multinomial Proportions

Estimators of the expectations of order statistics, suggested by Harrell &; Davis, are used in place of the order statistics in a quantile estimator, proposed by Kappenman , to produce a modification of Kappenman's procedure. Simulation studies indicate that the modification generally results in a reduction of mean squared error     

Empirical likelihood-based inference for parameter and nonparametric function in partially nonlinear models

This paper is concerned with statistical inference for partially nonlinear models. Empirical likelihood method for parameter in nonlinear function and nonparametric function is investigated. The empirical log-likelihood ratios are shown to be asymptotically chi-square and then the corresponding confidence intervals are constructed. By the empirical likelihood ratio functions, we also obtain the maximum empirical likelihood estimators of the parameter in nonlinear function and nonparametric function, and prove the asymptotic normality. A simulation study indicates that, compared with normal approximation-based method and the bootstrap method, the empirical likelihood method performs better in terms of coverage probabilities and average length/widths of confidence intervals/bands. An application to a real dataset is illustrated.     

Adjusted empirical likelihood inference for additive hazards regression

ABSTRACT

This article develops an adjusted empirical likelihood (EL) method for the additive hazards model. The adjusted EL ratio is shown to have a central chi-squared limiting distribution under the null hypothesis. We also evaluate its asymptotic distribution as a non central chi-squared distribution under the local alternatives of order n^{? 1/2}, deriving the expression for the asymptotic power function. Simulation studies and a real example are conducted to evaluate the finite sample performance of the proposed method. Compared with the normal approximation-based method, the proposed method tends to have more larger empirical power and smaller confidence regions with comparable coverage probabilities.     

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.     

Exact average coverage probabilities and confidence coefficients of confidence intervals for discrete distributions

For a confidence interval (L(X),U(X)) of a parameter θ in one-parameter discrete distributions, the coverage probability is a variable function of θ. The confidence coefficient is the infimum of the coverage probabilities, inf  _θ P _θ (θ∈(L(X),U(X))). Since we do not know which point in the parameter space the infimum coverage probability occurs at, the exact confidence coefficients are unknown. Beside confidence coefficients, evaluation of a confidence intervals can be based on the average coverage probability. Usually, the exact average probability is also unknown and it was approximated by taking the mean of the coverage probabilities at some randomly chosen points in the parameter space. In this article, methodologies for computing the exact average coverage probabilities as well as the exact confidence coefficients of confidence intervals for one-parameter discrete distributions are proposed. With these methodologies, both exact values can be derived.     

Statistical inference for varying-coefficient partially linear errors-in-variables models with missing data

Abstract

The purpose of this paper is twofold. First, we investigate estimations in varying-coefficient partially linear errors-in-variables models with covariates missing at random. However, the estimators are often biased due to the existence of measurement errors, the bias-corrected profile least-squares estimator and local liner estimators for unknown parametric and coefficient functions are obtained based on inverse probability weighted method. The asymptotic properties of the proposed estimators both for the parameter and nonparametric parts are established. Second, we study asymptotic distributions of an empirical log-likelihood ratio statistic and maximum empirical likelihood estimator for the unknown parameter. Based on this, more accurate confidence regions of the unknown parameter can be constructed. The methods are examined through simulation studies and illustrated by a real data analysis.     

Empirical likelihood for high-dimensional partially linear model with martingale difference errors

In this paper, we focus on the empirical likelihood (EL) inference for high-dimensional partially linear model with martingale difference errors. An empirical log-likelihood ratio statistic of unknown parameter is constructed and is shown to have asymptotically normality distribution under some suitable conditions. This result is different from those derived before. Furthermore, an empirical log-likelihood ratio for a linear combination of unknown parameter is also proposed and its asymptotic distribution is chi-squared. Based on these results, the confidence regions both for unknown parameter and a linear combination of parameter can be obtained. A simulation study is carried out to show that our proposed approach performs better than normal approximation-based method.