Selecting a sequence of last successes in independent trials

Abstract

Let I 1,I 2,…,I n be a sequence of independent indicator functions defined on a probability space (Ω, A, P). We say that index k is a success time if I k = 1. The sequence I 1,I 2,…,I n is observed sequentially. The objective of this article is to predict the lth last success, if any, with maximum probability at the time of its occurrence. We find the optimal rule and discuss briefly an algorithm to compute it in an efficient way. This generalizes the result of Bruss (1998) for l = 1, and is equivalent to the problem of (multiple) stopping with l stops on the last l successes. We then extend the model to a larger class allowing for an unknown number N of indicator functions, and present, in particular, a convenient method for an approximate solution if the success probabilities are small. We also discuss some applications of the results.