On robust stopping times for detecting changes in distribution

Abstract

Let $X_1,X_2,\ldots $ be independent random variables observed sequentially and such that $X_1,\ldots,X_{\theta-1}$ have a common probability density $p_0$, while $X_\theta,X_{\theta+1},\ldots $ are all distributed according to $p_1\neq p_0$. It is assumed that $p_0$ and $p_1$ are known, but the time change $\theta\in \mathbb{Z}^+$ is unknown and the goal is to construct a stopping time $\tau$ that detects the change-point $\theta$ as soon as possible. The existing approaches to this problem rely essentially on some a priori information about $\theta$. For instance, in Bayes approaches, it is assumed that $\theta$ is a random variable with a known probability distribution. In methods related to hypothesis testing, this a priori information is hidden in the so-called average run length. The main goal in this paper is to construct stopping times which do not make use of a priori information about $\theta$, but have nearly Bayesian detection delays. More precisely, we propose stopping times solving approximately the following problem: \begin{equation*} \begin{split} &\quad \Delta(\theta;\tau^\alpha)\rightarrow\min_{\tau^\alpha}\quad \textbf{subject to}\quad \alpha(\theta;\tau^\alpha)\le \alpha \ \textbf{ for any}\ \theta\ge1, \end{split} \end{equation*} where $\alpha(\theta;\tau)=\mathbf{P}_\theta\bigl\{\tau<\theta \bigr\}$ is \textit{the false alarm probability} and $\Delta(\theta;\tau)=\mathbf{E}_\theta(\tau-\theta)_+$ is \textit{the average detection delay}, %In this paper, we construct $\widetilde{\tau}^\alpha$ such that %\[ % \max_{\theta\ge 1}\alpha(\theta;\widetilde{\tau}^\alpha)\le \alpha\ \text{and}\ %\Delta(\theta;\widetilde{\tau}^\alpha)\le (1+o(1))\log(\theta/\alpha), \ \text{as} \ \theta/\alpha%\rightarrow\infty, %\] and explain why such stopping times are robust w.r.t. a priori information about $\theta$.