Concavity and Convexity of Order Statistics in Sample Size

TL;DR

This paper demonstrates the convexity and concavity properties of the expectations of order statistics from specific distribution classes and applies these findings to auction theory, particularly in analyzing auctioneer objectives with convex costs.

Contribution

It establishes convexity and concavity of order statistic expectations for MRHR and MHR distributions and applies these results to auction models with private values and reserves.

Findings

Expectation of the $k$-th order statistic from MRHR is convex in sample size.

Expectation of the $(n-k+1)$-th order statistic from MHR is concave in sample size.

Concavity of auctioneer's objective function under MHR valuation distributions.

Abstract

We show that the expectation of the $k^{\mathrm{th}}$-order statistic of an i.i.d. sample of size $n$ from a monotone reverse hazard rate (MRHR) distribution is convex in $n$ and that the expectation of the $(n-k+1)^{\mathrm{th}}$-order statistic from a monotone hazard rate (MHR) distribution is concave in $n$ for $n\ge k$. We apply this result to the analysis of independent private value auctions in which the auctioneer faces a convex cost of attracting bidders. In this setting, MHR valuation distributions lead to concavity of the auctioneer's objective. We extend this analysis to auctions with reserve values, in which concavity is assured for sufficiently small reserves or for a sufficiently large number of bidders.