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 $X_1,\ldots,X_{θ-1}$ have a common probability density $p_0$, while $X_θ,X_{θ+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 $θ\in \mathbb{Z}^+$ is unknown and the goal is to construct a stopping time $τ$ that detects the change-point $θ$ as soon as possible. The existing approaches to this problem rely essentially on some a priori information about $θ$. For instance, in Bayes approaches, it is assumed that $θ$ 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 $θ$, but have nearly Bayesian detection delays. More precisely, we propose stopping times solving approximately the following problem: \begin{equation*} \begin{split} &\quad Δ(θ;τ^α)\rightarrow\min_{τ^α}\quad \textbf{subject to}\quad α(θ;τ^α)\le α\ \textbf{ for any}\ θ\ge1, \end{split} \end{equation*} where $α(θ;τ)=\mathbf{P}_θ\bigl\{τ<θ\bigr\}$ is \textit{the false alarm probability} and $Δ(θ;τ)=\mathbf{E}_θ(τ-θ)_+$ is \textit{the average detection delay}, %In this paper, we construct $\widetildeτ^α$ such that %\[ % \max_{θ\ge 1}α(θ;\widetildeτ^α)\le α \text{and}\ %Δ(θ;\widetildeτ^α)\le (1+o(1))\log(θ/α), \ \text{as} \ θ/α%\rightarrow\infty, %\] and explain why such stopping times are robust w.r.t. a priori information about $θ$.