Two-Sample Smooth Tests for the Equality of Distributions

TL;DR

This paper introduces a new two-sample test for equality of distributions that improves power over classical tests, extends to multivariate data, and employs bootstrap methods for critical value computation.

Contribution

It proposes a novel two-sample test based on Neyman's smooth test, extended to multivariate data with bootstrap calibration, enhancing detection of local features.

Findings

The new test outperforms classical methods in power.

Bootstrap method provides valid critical values in high dimensions.

Numerical studies confirm effectiveness for small samples.

Abstract

This paper considers the problem of testing the equality of two unspecified distributions. The classical omnibus tests such as the Kolmogorov-Smirnov and Cram\`er-von Mises are known to suffer from low power against essentially all but location-scale alternatives. We propose a new two-sample test that modifies the Neyman's smooth test and extend it to the multivariate case based on the idea of projection pursue. The asymptotic null property of the test and its power against local alternatives are studied. The multiplier bootstrap method is employed to compute the critical value of the multivariate test. We establish validity of the bootstrap approximation in the case where the dimension is allowed to grow with the sample size. Numerical studies show that the new testing procedures perform well even for small sample sizes and are powerful in detecting local features or high-frequency components.