A Multiple Hypothesis Testing Approach to Detection Changes in Distribution

Abstract

Let X₁, X₂,... be independent random variables observed sequentially and such that X₁,..., X_θ−1 have a common probability density p₀, while X_θ, X_θ+1,... are all distributed according to p₁ ≠ p₀. It is assumed that p₀ and p₁ are known, but the time change θ ∈ ℤ⁺ is unknown and the goal is to construct a stopping time τ that detects the change-point θ as soon as possible. The standard approaches to this problem rely essentially on some prior information about θ. For instance, in the Bayes approach, it is assumed that θ is a random variable with a known probability distribution. In the 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 that are free from a priori information about θ. More formally, we propose an approach to solving approximately the following minimization problem:

$$\Delta(\theta;{\tau^\alpha})\rightarrow\min_{\tau^\alpha}\;\;\text{subject}\;\text{to}\;\;\alpha(\theta;{\tau^\alpha})\leq\alpha\;\text{for}\;\text{any}\;\theta\geq1,$$

where α(θ; τ) = P_θ{τ < θ} is the false alarm probability and Δ(θ; τ) = E_θ(τ − θ)₊ is the average detection delay computed for a given stopping time τ. In contrast to the standard CUSUM algorithm based on the sequential maximum likelihood test, our approach is related to a multiple hypothesis testing methods and permits, in particular, to construct universal stopping times with nearly Bayes detection delays.