Nonparametric Comparisons of Distributions
Parametric methods for comparing distributions, whose justification in probability is based on specific distributional assumptions, may lead to loss of efficiency when the distributional assumptions are violated or may lead to biased tests. This chapter presents several commonly used nonparametric tests for comparison of distributions, including the Wilcoxon rank-sum test and the Kolmogorov–Smirnov test, as well as novel density-based empirical likelihood (EL) ratio tests based on two samples and paired data. Multiple-group comparison methodology is also introduced. As a nonparametric analogue to the two-sample t-test, the Wilcoxon rank-sum test (also called the Mann–Whitney U-test or the Mann–Whitney–Wilcoxon test) can be used primarily when investigators do not want to, or cannot, assume that data distributions are known. The chapter also introduces exact density-based empirical likelihood ratio (LR) tests for composite hypotheses related to treatment effects in order to provide efficient tools that compare study groups proposed by Vexler.