Density-based empirical likelihood procedures for testing symmetry of data distributions and K-sample comparisons

In practice, parametric likelihood-ratio techniques are powerful statistical tools. In this article, we propose and examine novel and simple distribution-free test statistics that efficiently approximate parametric likelihood ratios to analyze and compare distributions of K groups of observations. Using the density-based empirical likelihood methodology, we develop a Stata package that applies to a test for symmetry of data distributions and compares K-sample distributions. Recognizing that recent statistical software packages do not sufficiently address K-sample nonparametric comparisons of data distributions, we propose a new Stata command, vxdbel, to execute exact density-based empirical likelihood-ratio tests using K samples. To calculate p-values of the proposed tests, we use the following methods: 1) a classical technique based on Monte Carlo p-value evaluations; 2) an interpolation technique based on tabulated critical values; and 3) a new hybrid technique that combines methods 1 and 2. The third, cutting-edge method is shown to be very efficient in the context of exact-test p-value computations. This Bayesian-type method considers tabulated critical values as prior information and Monte Carlo generations of test statistic values as data used to depict the likelihood function. In this case, a nonparametric Bayesian method is proposed to compute critical values of exact tests.