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 $\{F_i\}$ of distributions, which are available through a large number of observations at each $i \geq 1$. Under the null hypothesis, the distributions $F_i$ are equal. Under the alternative hypothesis, there is a change point $i^* > 1$, such that $F_i = G$ for $i \geq i^*$ and some unknown distribution $G$, which is not equal to $F_1$. 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.