Published in The Annals of Applied Probability, Volume 2, Issue 2, May 1, 1992, pages 503-517. Copyright © 1992 Institute of Mathematical Statistics. The definitive version is available at http://dx.doi.org/10.1214/aoap/1177005713.
NOTE: At the time of publication, the author Theodore P. Hill was not yet affiliated with Cal Poly.
A universal bound for the maximal expected reward is obtained for stopping a sequence of independent random variables where the reward is a nonincreasing function of the rank of the variable selected. This bound is shown to be sharp in three classical cases: (i) when maximizing the probability of choosing one of the k best; (ii) when minimizing the expected rank; and (iii) for an exponential function of the rank.