A Nonparametric Test for Equality of Distributions

Abstract

A computationally inexpensive equality test for multivariate distributions is useful inmany applications. The goal of this work is to investigate a proposed such test, the projective KS test. It involves projecting the multivariate data onto random lines through the origin and performing one-dimensional Kolmogorov-Smirnov tests on the resulting projections. The projective KS statistic is the maximum of the metrics obtained from those 1D tests. Two methods of implementation of the projective KS test were developed, one involving sub-sampling in the form of partitioning of the data, and the other using only the whole sample(s). The distribution of the projective KS statistics for both methods was investi- gated. For each method, numerical experiments were carried out to calculate the power of the tests for different sample and sub-sample sizes and different numbers of projections, for a range of normal distributions. Power characteristics for the two methods were compared to each other and to the performance of Z-tests. The proposed test (both methods) is less powerful than the Z-tests, but it has the advantage of being nonparametric. In addition, the projective KS test is potentially useful as a test of random variable independence.