Empirical likelihood ratios applied to goodness-of-fit tests based on sample entropy

The likelihood approach based on the empirical distribution functions is a well-accepted statistical tool for testing. However, the proof schemes of the Neyman–Pearson type lemmas induce consideration of density-based likelihood ratios to obtain powerful test statistics. In this article, we introduce the distribution-free density-based likelihood technique, applied to test for goodness-of-fit. We focus on tests for normality and uniformity, which are common tasks in applied studies. The well-known goodness-of-fit tests based on sample entropy are shown to be a product of the proposed empirical likelihood (EL) methodology. Although the efficiency of test statistics based on classes of entropy estimators has been widely addressed in the statistical literature, estimation of the sample entropy has been not invariantly defined, and hence this estimation produces tests that are difficult to be applied to real data studies. The proposed EL approach defines clear forms of the entropy-based tests. Monte Carlo simulation results confirm the preference of the proposed method from a power perspective. Real data examples study the proposed approach in practice.