Detection and localization of changes in a panel of densities

Abstract

We propose a new methodology for identifying and localizing changes in the Fréchet mean of a multivariate time series of probability densities. The functional data objects we study are random densities $f_{s,t}$ indexed by discrete time $t$ and a vector component $s$, which can be treated as a broadly understood spatial location. Our main objective is to identify the set of components $s$, where a change occurs with statistical certainty. A challenge of this analysis is that the densities $f_{s,t}$ are not directly observable and must be estimated from sparse and potentially imbalanced data. Such setups are motivated by the analysis of two data sets that we investigate in this work. First, a hitherto unpublished large data set of Brazilian Covid infections and a second, a financial data set derived from intraday prices of U.S. Exchange Traded Funds. Chief statistical advances are the development of change point tests and estimators of components of change for multivariate time series of densities. We prove the theoretical validity of our methodology and investigate its finite sample performance in a simulation study.