On the Expectations of Maxima of Sets of Independent Random Variables

TL;DR

This paper investigates the maximum expected value of the maximum of independent random variables, with applications to optimizing the lifetime of systems and populations, revealing concavity properties and conditions for optimal component choices.

Contribution

It establishes the concavity of the expected maximum function and provides conditions for optimal component selection in lifetime maximization problems.

Findings

The lattice of expected maxima is concave.

Conditions for when a single component type maximizes expected lifetime.

Bound on sign changes in the expected maximum difference sequence.

Abstract

Let $X^1, ..., X^k$ and $Y^1, ..., Y^m$ be jointly independent copies of random variables $X$ and $Y$, respectively. For a fixed total number $n$ of random variables, we aim at maximising $M(k,m):= E \max \{X^1, ..., X^k, Y^1, >..., Y^{m} \}$ in $k = n-m\ge 0$, which corresponds to maximising the expected lifetime of an $n$-component parallel system whose components can be chosen from two different types. We show that the lattice $\{M(k,m): k, m\ge 0\}$ is concave, give sufficient conditions on $X$ and $Y$ for M(n,0) to be always or ultimately maximal and derive a bound on the number of sign changes in the sequence $M(n,0)-M(0,n)$, $n\ge 1$. The results are applied to a mixed population of Bienayme-Galton-Watson processes, with the objective to derive the optimal initial composition to maximise the expected time to extinction.