Cramer-von Mises tests for Change Points

TL;DR

This paper introduces two nonparametric change point detection tests based on the Cramer-von Mises statistic, offering faster p-value computation and improved power analysis for long sequences and contiguous alternatives.

Contribution

It proposes new change point tests using Cramer-von Mises statistics, with theoretical and empirical analysis showing advantages over existing methods.

Findings

Average statistic provides quicker p-values than bootstrapping.

Average statistic has higher limiting power for contiguous alternatives.

Performance confirmed through Monte Carlo simulations.

Abstract

We study two nonparametric tests of the hypothesis that a sequence of independent observations is identically distributed against the alternative that at a single change point the distribution changes. The tests are based on the Cramer-von Mises two-sample test computed at every possible change point. One test uses the largest such test statistic over all possible change points; the other averages over all possible change points. Large sample theory for the average statistic is shown to provide useful p-values much more quickly than bootstrapping, particularly in long sequences. Power is analyzed for contiguous alternatives. The average statistic is shown to have limiting power larger than its level for such alternative sequences. Evidence is presented that this is not true for the maximal statistic. Asymptotic methods and bootstrapping are used for constructing the test distribution. Performance of the tests is checked with a Monte Carlo power study for various alternative distributions.