Optimal rules for the sequential selection of uniform spacings

Abstract

ABSTRACT The unit interval is broken at random into n spacings with the breaking points given by a random sample of size n−1 from the uniform distribution on . A decision-maker observes the lengths of these spacings sequentially and must decide either to select the present spacing or to reject it and continue to observe the next one. No recall of preceding observations is permitted. In this paper, we first find an optimal stopping rule to maximize the probability of selecting the largest spacing. Furthermore, we establish a connection to the classical secretary problem, from which we derive a lower bound for the maximum probability. Moreover, we conjecture that the limiting optimal probability is the same as in the full-information best-choice problem. Next, we find an optimal stopping rule to maximize the expected length of the selected spacing. It is shown that the maximum obtainable expected length lies between the expected value of the second-largest and the largest spacing.