An exact solution for choosing the largest measurement from a sample drawn from an uniform distribution

TL;DR

This paper derives an exact formula for selecting the largest measurement from a uniform sample, correcting previous assumptions, and compares the new solution with existing formulas through simulations and asymptotic analysis.

Contribution

It provides an exact solution for the optimal stopping problem in selecting the maximum from uniform samples, improving upon prior approximate methods.

Findings

Exact formula for selection probabilities derived

New solution outperforms previous approximations in simulations

Asymptotic analysis confirms the improved accuracy

Abstract

In "Recognizing the Maximum of a Sequence", Gilbert and Mosteller analyze a full information game where n measurements from an uniform distribution are drawn and a player (knowing n) must decide at each draw whether or not to choose that draw. The goal is to maximize the probability of choosing the draw that corresponds to the maximum of the sample. In their calculations of the optimal strategy, the optimal probability and the asymptotic probability, they assume that after a draw x the probability that the next i numbers are all smaller than x is $x^i$; but this fails to recognize that continuing the game (not choosing a draw because it is lower than a cutoff and waiting for the next draw) conditions the distribution of the following i numbers such that their expected maximum is higher then i/(i+1). The problem is now redefined with each draw leading to a win, a false positive loss, a false negative loss and a continuation. An exact formula for these probabilities is deduced, both for the general case of n-1 different indifference numbers (assuming 0 as the last cutoff) and the particular case of the same indifference number for all cutoffs but the last. An approximation is found that preserves the main characteristics of the optimal solution (slow decay of win probability, quick decay of false positives and linear decay of false negatives). This new solution and the original Gilbert and Mosteller formula are compared against simulations, and their asymptotic behavior is studied.