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Empirical and Smoothed Bayes Factor Type Inferences Based on Empirical Likelihoods for Quantiles

Alan Hutson, State University of New York at Buffalo 
Nicole Lazar, University of Georgia 
*Ge Tao, State University of New York at Buffalo 
Albert Vexler, State University of New York at Buffalo 
Jihnhee Yu, State University of New York at Buffalo 

Keywords: Bayes factor, empirical likelihood, quantile, kernel function

The Bayes factor, a practical tool of applied statistics, has been dealt with extensively in the literature in the context of hypothesis testing. The Bayes factor based on parametric likelihoods can be considered both as a pure Bayesian approach as well as a standard technique for computing P-values for hypothesis testing when the functional forms of the data distributions are known. In this article, we employ empirical likelihood methodology in order to modify Bayes factor type procedures for the nonparametric setting. The proposed approach is applied towards developing testing methods involving quantiles, which are commonly used to characterize distributions. Comparing quantiles thus provides valuable information; however, very few tests for quantiles are available. We present and evaluate one- and two-sample distribution-free Bayes factor type methods for testing quantiles based on indicators and smooth kernel functions. Although the proposed procedures are nonparametric, their asymptotic behaviors are similar to those of the classical Bayes factor approach. An extensive Monte Carlo study and real data examples show that the developed tests have excellent operating characteristics for one-sample and two-sample data analysis.