An Extension of the Sum-the-Odds Theorem to the Stopping Problems on the Bernoulli Trials of Random Length
Mitsushi Tamaki, Qi Wang, Aiko Kurushima · 2000
The optimal stopping problem of choosing the best item from the population of unknown size has been studied by Presman and Sonin [4], Irle [2] and Petruccelli [3]. As an extension of this problem, we consider the problem of maximizing the probability of stopping on the last success on the independent Bernoulli trials of random length and give a sufficient condition for the optimal rule to be of threshold type. This result can be considered as a generalization of the Sum-the-Odds Theorem by Bruss [1]. Bruss theorem is also extended to the problem of stopping on one of the last m successes on the independent Bernoulli trials and the problem of stopping on the last success when the Bernoulli random variables are Markov dependent. Moreover we consider the stopping problem with refusal. References [1] F.Thomas Bruss. Sum the odds to one and stop. Ann. Probab., 28(3):1384–