JOINT NONPARAMETRIC TESTS FOR LOCATION AND SCALE DIFFERENCES IN MULTIPLE POPULATIONS

Abstract

Nonparametric statistical methods are widely used when data do not satisfy the assumptions required for parametric procedures, such as normality and homogeneity of variances. While many existing nonparametric tests are designed to detect differences in either location or scale parameters separately, practical applications often require the simultaneous assessment of both. When location and scale vary together, procedures that focus on only one parameter may suffer from reduced power or lead to misleading conclusions. This dissertation develops new combined nonparametric tests for jointly assessing differences in location and scale across multiple populations. The proposed methods are constructed by combining the Kruskal–Wallis test for location with scale components derived from the Moses Kruskal–Wallis and Levene tests. Two groups of test statistics are introduced, each incorporating measures of central tendency based on the mean, median, and trimmed mean, to enhance robustness across symmetric, skewed, and heavy-tailed distributions. The performance of the proposed tests is evaluated through extensive simulation studies under a wide range of distributions and sample size configurations for three and four populations. The results demonstrate that the tests generally maintain nominal Type I error rates and achieve improved power when scale differences or joint location–scale differences are present. Among the proposed methods, Moses-based procedures show superior performance in heavy-tailed distributions, whereas Levene-based procedures exhibit more stable behavior across a broad range of distributional settings. Overall, the proposed tests provide reliable and effective tools for joint nonparametric inference on location and scale in multi-sample problems.