Decision Theoretic Optimality of the Cusum Procedure

Abstract

Suppose $X_1, X_2, \ldots$ are independent random variables such that for some unknown $\nu$, each of $X_1, \ldots, X_{\nu - 1}$ is distributed according to $F_0$, while $X_\nu, X_{\nu + 1}, \ldots$ are all distributed according to $F_1$. We prove a result of Moustakides that claims that the CUSUM procedures are optimal in the sense of Lorden. We do that by proving that the procedures are Bayes for some stochastic mechanism of generating $\nu$.