New two-sample tests for skewed populations and their connection to theoretical power of Bootstrap-t test

Abstract

Various tests are available to compare the means of two populations. Tests for skewed data, however, are not well studied even though they are often needed in pharmaceutical study and agricultural economics. In particular, there is no available result to give power and sample size calculation for a two-sample Bootstrap-t test in skewed populations. In this paper, we propose easy-to-compute new tests and study their theoretical properties. The proposed work starts with derivation of a second-order Edgeworth expansion for the pooled two-sample t-statistic. Then new test rejection regions are formed based on Cornish–Fisher expansion of quantiles. The new tests account for first-order and second-order population skewnesses that were ignored in two-sample t test. We report the theoretical type I error accuracy and power of the newly proposed tests and the large sample t test. We also provide the detailed conditions under which the proposed tests give better power than the two-sample large sample test. Our new tests, $\hbox {TCF}_1$ and TCF, are asymptotically equivalent to Bootstrap-t test up to $O(N^{-1})$ and $O(N^{-3/2})$, respectively. Compared with commonly used two-sample parametric and nonparametric tests, the new tests are computationally efficient, give better power for skewed data with moderate sample size, and provide sample size calculation to achieve desired power at a given significance level. Empirical studies confirmed our theoretical results.