Robust Sequential Testing

Abstract

This paper considers the asymptotic minimax property of the sequential probability ratio test (SPRT) when the given distributions $P_{\pm \varepsilon}$ contain a small amount of contamination. Let $\mathscr{P}_{\pm \varepsilon}$ be the neighborhoods of $P_{\pm \varepsilon}.$ Suppose that $P_\varepsilon$ and $P_{-\varepsilon}$ approach each other as $\varepsilon \downarrow 0$ and that $\mathscr{P}_{\pm \varepsilon}$ shrink at an appropriate rate. We prove (under regularity assumptions) that the SPRT based on the least favorable pair of distributions $(Q^\ast_{-\varepsilon}, Q^\ast_\varepsilon)$ given by Huber (1965) is asymptotically least favorable for expected sample size and is asymptotically minimax, provided that the limiting maximum error probabilities do not exceed $1/2.$