Expected difference of order statistics in terms of hazard rate

TL;DR

This paper investigates how the expected difference between the largest and second-largest order statistics behaves when the hazard rate is increasing, showing it decreases with sample size and is completely monotone.

Contribution

It establishes a relationship between increasing hazard rates and the monotonicity of the expected gap between top order statistics.

Findings

Expected difference decreases as sample size increases.

Expected difference is completely monotone under increasing hazard rate.

Provides theoretical insight into order statistics behavior with hazard rate conditions.

Abstract

If the hazard rate $ \frac{ F'(x) }{ 1-F(x) } $ is increasing (in $x$), then $ \mathbb E\, ( X_{n:n} - X_{n-1:n} ) $ is decreasing (in $n$), and moreover, completely monotone.