A sharp lower bound for choosing the maximum of an independent sequence

Abstract

In this paper we consider a variation of the full-information secretary problem where the random variables to be observed are independent but not necessary identically distributed. The main result is a sharp lower bound for the optimal win probability. Precisely, if X₁,...,X_n are independent random variables with known continuous distributions and V_n(X₁,...,X_n):=sup_τℙ(X_τ=M_n), where M_n≔max{X₁,...,X_n} and the supremum is over all stopping times adapted to X₁,...,X_n then V_n(X₁,...,X_n)≥(1-1/n)^n-1, and this bound is attained. The method of proof consists in reducing the problem to that of a sequence of random variables taking at most two possible values, and then applying Bruss' sum-the-odds theorem, Bruss (2000). In order to obtain a sharp bound for each n, we improve Bruss' lower bound, Bruss (2003), for the sum-the-odds problem.