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Empirical likelihood is a popular nonparametric statistical tool that does not require any distributional assumptions. In this paper, we explore the possibility of conducting variable selection via Bayesian empirical likelihood. We show theoretically that when the prior distribution satisfies certain mild conditions, the corresponding Bayesian empirical likelihood estimators are posteriorly consistent and variable selection consistent. As special cases, we show the prior of Bayesian empirical likelihood LASSO and SCAD satisfies such conditions and thus can identify the non-zero elements of the parameters with probability tending to 1. In addition, it is easy to verify that those conditions are met for other widely used priors such as ridge, elastic net and adaptive LASSO. Empirical likelihood depends on a parameter that needs to be obtained by numerically solving a non-linear equation. Thus, there exists no conjugate prior for the posterior distribution, which causes the slow convergence of the MCMC sampling algorithm in some cases. To solve this problem, we propose a novel approach, which uses an approximation distribution as the proposal. The computational results demonstrate quick convergence for the examples used in the paper. We use both simulation and real data analyses to illustrate the advantages of the proposed methods.

The categorical Gini correlation is an alternative measure of dependence between a categorical and numerical variables, which characterizes the independence of the variables. A nonparametric test for the equality of K distributions has been developed based on the categorical Gini correlation. By applying the jackknife empirical likelihood approach, the standard limiting chi-square distribution with degree freedom of $K-1$ is established and is used to determine critical value and $p$-value of the test. Simulation studies show that the proposed method is competitive to existing methods in terms of power of the tests in most cases. The proposed method is illustrated in an application on a real data set.

In many applications, parameters of interest are estimated by solving some non-smooth estimating equations with $U$-statistic structure. Jackknife empirical likelihood (JEL) approach can solve this problem efficiently by reducing the computation complexity of the empirical likelihood (EL) method. However, as EL, JEL suffers the sensitivity problem to outliers. In this paper, we propose a weighted jackknife empirical likelihood (WJEL) to tackle the above limitation of JEL. The proposed WJEL tilts the JEL function by assigning smaller weights to outliers. The asymptotic of the WJEL ratio statistic is derived. It converges in distribution to a multiple of a chi-square random variable. The multiplying constant depends on the weighting scheme. The self-normalized version of WJEL ratio does not require to know the constant and hence yields the standard chi-square distribution in the limit. Robustness of the proposed method is illustrated by simulation studies and one real data application., Comment: 25 pages, 4 tables and one figure

The Gini correlation plays an important role in measuring dependence of random variables with heavy tailed distributions, whose properties are a mixture of Pearson's and Spearman's correlations. Due to the structure of this dependence measure, there are two Gini correlations between each pair of random variables, which are not equal in general. Both the Gini correlation and the equality of the two Gini correlations play important roles in Economics. In the literature, there are limited papers focusing on the inference of the Gini correlations and their equality testing. In this paper, we develop the jackknife empirical likelihood (JEL) approach for the single Gini correlation, for testing the equality of the two Gini correlations, and for the Gini correlations' differences of two independent samples. The standard limiting chi-square distributions of those jackknife empirical likelihood ratio statistics are established and used to construct confidence intervals, rejection regions, and to calculate $p$-values of the tests. Simulation studies show that our methods are competitive to existing methods in terms of coverage accuracy and shortness of confidence intervals, as well as in terms of power of the tests. The proposed methods are illustrated in an application on a real data set from UCI Machine Learning Repository., Comment: 20 pages, 6 tables, two figures