Cramer-von Mises tests for change points

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.

Automated summary

This study focuses on two nonparametric tests for testing the hypothesis of identical distribution of a sequence of independent observations, and analyzes the situation where the distribution changes at a single change point. By using the Cramer-von Mises two-sample test, one test utilizes the maximum test statistic over all possible change points, while the other test averages over all possible change points, showing that the average statistic provides useful p-values more quickly, particularly in long sequences. Power analysis for contiguous alternatives reveals that the average statistic has larger limiting power than its level, while the maximal statistic does not. Asymptotic methods and bootstrapping are employed to construct the test distribution, and the tests' performance is verified through a Monte Carlo power study with various alternative distributions.