Average Run Lengths of an Optimal Method of Detecting a Change in Distribution

Abstract

Suppose one is able to observe sequentially a series of independent observations $X_1, X_2, \ldots$, such that $X_1, X_2, \ldots, X_{\nu - 1}$ are i.i.d. with known density $f_0$ and $X_\nu, X_{\nu + 1}, \ldots$ are i.i.d. with density $f_\theta$ where $\nu$ is unknown. Define $R(n, \theta) = \sum^n_{k = 1} \prod^n_{i = k} \frac{f_\theta(X_i)}{f_0(X_i)}.$ It is known that rules, which call for stopping and raising an alarm the first time $n$ that $R(n, \theta)$ or a mixture thereof exceeds a prespecified level $A$, are optimal methods of detecting that the density of the observations is not $f_0$ any more. Practical applications of such stopping rules require knowledge of their operating characteristics, whose exact evaluation is difficult. Here are presented asymptotic $(A \rightarrow \infty)$ expressions for the expected stopping times of such stopping rules (a) when $\nu = \infty$ and (b) when $\nu = 1$. We assume that the densities $f_\theta$ form an exponential family and that the distribution of $\log(f_\nu(X_1)/f_0(X_1))$ is (strongly) nonlattice. Monte Carlo studies indicate that the asymptotic expressions are very good approximations, even when the expected sample sizes are small.