Robust multiple change-point detection for covariance matrices using data depth

In this paper, two robust, nonparametric methods for multiple change-point detection in the covariance matrix of a multivariate sequence of observations are introduced. We demonstrate that changes in ranks generated from data depth functions can be used to detect certain types of changes in the covariance matrix of a sequence of observations. In order to detect more than one change, the first algorithm uses methods similar to that of wild-binary segmentation. The second algorithm estimates change-points by maximizing a penalized version of the classical Kruskal Wallis ANOVA test statistic. We show that this objective function can be maximized via the well-known PELT algorithm. Under mild, nonparametric assumptions both of these algorithms are shown to be consistent for the correct number of change-points and the correct location(s) of the change-point(s). We demonstrate the efficacy of these methods with a simulation study, where we compare our new methods to an competing method. We are able to estimate changes accurately when the data is heavy tailed or skewed. We are also able to detect second order change-points in a time series of multivariate financial returns, without first imposing a time series model on the data.