Monitoring for a change point in a sequence of distributions

Abstract

We propose a method for the detection of a change point in a sequence $\{{\mathit{F}}_{\mathit{i}}\}$ of distributions, which are available through a large number of observations at each $\mathit{i}\ge 1$. Under the null hypothesis, the distributions ${\mathit{F}}_{\mathit{i}}$ are equal. Under the alternative hypothesis, there is a change point ${\mathit{i}}^{\ast }>1$, such that ${\mathit{F}}_{\mathit{i}}=\mathit{G}$ for $\mathit{i}\ge {\mathit{i}}^{\ast }$ and some unknown distribution G, which is not equal to ${\mathit{F}}_1$. The change point, if it exists, is unknown, and the distributions before and after the potential change point are unknown. The decision about the existence of a change point is made sequentially, as new data arrive. At each time i, the count of observations, N, can increase to infinity. The detection procedure is based on a weighted version of the Wasserstein distance. Its asymptotic and finite sample validity is established. Its performance is illustrated by an application to returns on stocks in the S&P 500 index.