Robust sequential testing (Q1073518)

The author considers the asymptotic minimax property of the sequential probability ratio test (SPRT) when the given distributions \(P_{+\epsilon}\) and \(P_{-\epsilon}\) contain a small amount of contamination. Let \(N_{+\epsilon}\) and \(N_{-\epsilon}\) be the neighborhoods of \(P_{+\epsilon}\) and \(P_{-\epsilon}\), respectively. Suppose that \(P_{\epsilon}\) and \(P_{-\epsilon}\) approach each other as \(\epsilon\) tends to zero and that \(N_{+\epsilon}\) and \(N_{- \epsilon}\) shrink at an appropriate rate. Under suitable regularity conditions it is proved that the SPRT based on the least favorable pair of distributions \((Q_{+\epsilon},Q_{- \epsilon})\) as defined by \textit{P. J. Huber} [Ann. Math. Stat. 36, 1753- 1758 (1965; Zbl 0137.127)] is asymptotically least favorable for expected sample size and is asymptotically minimax, provided that the limiting maximum error probabilities are no greater than 1/2.