Robust multivariate change point analysis based on data depth

Modern methods for detecting changes in the scale or covariance of multivariate distributions rely primarily on testing for the constancy of the covariance matrix. These depend on higher-order moment conditions, and also do not work well when the dimension of the data is large or even moderate relative to the sample size. In this paper, we propose a nonparametric change point test for multivariate data using rankings obtained from data depth measures. As the data depth of an observation measures its centrality relative to the sample, changes in data depth may signify a change of scale of the underlying distribution, and the proposed test is particularly responsive to detecting such changes. We provide a full asymptotic theory for the proposed test statistic under the null hypothesis that the observations are stable, and natural conditions under which the test is consistent. The finite sample properties are investigated by means of a Monte Carlo simulation, and these along with the theoretical results confirm that the test is robust to heavy tails, skewness and high dimensionality. The proposed methods are demonstrated with an application to structural break detection in the rate of change of pollutants linked to acid rain measured in Turkey lake, a lake in central Ontario, Canada. Our test suggests a change in the rate of acid rain in the late 1980s/early 1990s, which coincides with clean air legislation in Canada and the US. The Canadian Journal of Statistics 48: 417–446; 2020 © 2020 Statistical Society of Canada