A two-sample test for the equality of univariate marginal distributions for high-dimensional data

Abstract

A recurring theme in modern statistics is dealing with high-dimensional data whose main feature is a large number, p, of variables but a small sample size. In this context our aim is to address the problem of testing the null hypothesis that the marginal distributions of p variables are the same for two groups. We propose a test statistic motivated by the simple idea of comparing, for each of the p variables, the empirical characteristic functions computed from the two samples. The asymptotic normality of the test statistic is derived under mixing conditions. In our asymptotic analysis the number of variables tends to infinity, while the size of individual samples remains fixed. In order to obtain a practical test several estimators of the variance are proposed, leading to three somewhat different versions of the test. An alternative global test based on the P-values derived from permutation tests is also proposed. A simulation study to investigate the finite sample properties of the proposed tests is carried out, and a practical illustration involving microarray data is provided.