Optimal detection of a change in distribution (Q1062715)

This paper is concerned with an optimal detection rule of a change in distributions of a series of independent observations \(\{X_ n\}\). Suppose \(X_ 1,X_ 2,...\), are independent random variables such that \(X_ 1,...,X_{\nu -1}\) are each distributed according to a distribution \(F_ 0\) and \(X_{\nu},X_{\nu +1},...\), are each distributed according to a distribution \(F_ 1\), where \(1\leq \nu \leq \infty\) is unknown. The problem is to find a stopping rule N which minimizes E(N-\(\nu\) \(| N\geq \nu)\) subject to a restriction on false detections \(E(N| \nu =\infty)\geq B.\) It is shown that a stopping rule \(N_ A\) considered by \textit{S. W. Roberts} [A comparison of some control chart procedures. Technometrics 8, 411-430 (1966)] is a limit as \(p\to 0\) of the Bayes solution of the Bayesian problem [cf. \textit{A. N. Shiryaev}, Teor. Veroyatn. Primen. 8, 26-51 (1963; Zbl 0213.438); English translation in Theory Probab. Appl. 8, 22-46 (1963)] and asymptotically (p\(\to 0)\) Bayes risk efficient. The speed with which a stopping rule detects a true change of distribution is evaluated following the author and \textit{D. Siegmund} [Dept. Statist., Stanford Univ. Tech. Rept. (1984)]. An almost minimax rule is constructed based on ideas of \textit{T. S. Ferguson}, Mathematical statistics: A decision theoretic approach (1967; Zbl 0153.476)].