A Multiple Hypothesis Testing Approach to Detection Changes in Distribution

• Published in: Mathematical methods of statistics 2019-04, Vol.28 (2), p.155-167
• Main Authors: Golubev, G., Safarian, M.
• Format: Article
• Language: English
• Subjects: Algorithms; Change detection; False alarms; Hypotheses; Hypothesis testing; Independent variables; Mathematics and Statistics; Random variables; Statistical Theory and Methods; Statistics
• ISSN: 1066-5307; 1934-8045
• DOI: 10.3103/S1066530719020054

Let X 1 , X 2 ,... be independent random variables observed sequentially and such that X 1 ,..., X θ −1 have a common probability density p 0 , while X θ , X θ +1 ,... are all distributed according to p 1 ≠ p 0 . It is assumed that p 0 and p 1 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: Δ ( θ ; τ α ) → min τ α subject to α ( θ ; τ α ) ≤ α for any θ ≥ 1 , 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.