COMBINED NONPARAMETRIC TESTS FOR EQUALITY OF LOCATION AND SCALE UNDER ORDERED ALTERNATIVES

Abstract

This study proposes four combined nonparametric test statistics for detecting ordered alternatives in both location and scale across multiple populations. T₁ combines the Jonckheere–Terpstra (JT) statistic with the Moses-type scale statistic using equal weighting, while T₂ combines the Modified Jonckheere–Terpstra (MJT) statistic with the Modified Moses-type scale statistic using equal weighting. T₃ and T₄ apply the same respective combinations but with a weighted scheme that assigns greater emphasis to the scale component, with weight equal to the subgroup size m. Both mean-based and median-based centering approaches were implemented within a subgroup sampling framework. A Monte Carlo simulation study with 10,000 replications per configuration was conducted for k = 3 and k = 4 populations under different sample sizes and subgroup sizes. Six distributions were tested: Normal, Logistic, Laplace, Cauchy, Exponential, and Chi-square with multiple degrees of freedom. Three alternative structures were considered: pure location shifts, pure scale differences, and simultaneous changes in both parameters, along with null configurations for Type I error assessment. Simulation results showed consistent patterns across symmetric distributions. T₁ and T₂ showed higher power under location differences, while T₃ and T₄ showed higher power under scale differences. Under combined alternatives for symmetric distributions, T₃ and T₄ showed better performance, with T₄ having a slight advantage in many cases. Type I error control was adequate for symmetric and approximately symmetric distributions, including the heavy-tailed Cauchy. However, under strongly skewed distributions, Type I error rates were substantially inflated due to induced dependence between the location and scale components, rendering the asymptotic tests unreliable in such settings. Test selection should therefore account for both the anticipated alternative structure and the degree of distributional skewness.