Moments of random sums and Robbins' problem of optimal stopping

TL;DR

This paper addresses Robbins' optimal stopping problem by embedding it into a broader selection framework, resolving a conjecture about the value of the stopped variable under rank-based optimal rules.

Contribution

It introduces a generalized context for selection problems with nonanticipation constraints lifted, and proves a conjecture about the value of the stopped variable.

Findings

Resolved a conjecture on the value of the stopped variable.

Embedded Robbins' problem into a more general selection framework.

Provided new insights into rank-based optimal stopping rules.

Abstract

Robbins' problem of optimal stopping asks one to minimise the expected {\it rank} of observation chosen by some nonanticipating stopping rule. We settle a conjecture regarding the {\it value} of the stopped variable under the rule optimal in the sense of the rank, by embedding the problem in a much more general context of selection problems with the nonanticipation constraint lifted, and with the payoff growing like a power function of the rank.