On robust stopping times for detecting changes in distribution

TL;DR

This paper develops robust sequential change detection methods that do not rely on prior knowledge of the change point, achieving near-optimal detection delays comparable to Bayesian approaches.

Contribution

It introduces stopping times that are nearly Bayesian in performance without requiring prior information about the change point.

Findings

Constructed stopping times with false alarm probability at most α.

Achieved detection delays close to \\log(\\theta/α) as \\theta/α \\to \\infty.

Proved robustness of the proposed methods against lack of prior information.

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 $\theta$.