On the restrictiveness of the hazard rate order

TL;DR

This paper investigates the probability that two randomly chosen distributions, uniformly distributed over the probability simplex, are ordered by the hazard rate, revealing a specific probability related to the hazard rate order.

Contribution

It establishes the exact probability that two independent uniform distributions on the simplex are ordered by the hazard rate, providing insight into the restrictiveness of the hazard rate order.

Findings

Probability that P_Theta ≤_hr P_Theta' is 1/2^{n-1}

Hazard rate order's restrictiveness quantified for uniform distributions

Analytical result for hazard rate order probability in finite support distributions

Abstract

Every element $\theta=(\theta_1,\ldots,\theta_n)$ of the probability $n$-simplex induces a probability distribution $P_\theta$ of a random variable $X$ that can assume only a finite number of real values $x_1 < \cdots < x_n$ by defining $P_\theta(X=x_i) = \theta_i, 1\leq i \leq n$. We show that if $\Theta$ and $\Theta'$ are two random vectors uniformly distributed on $\Delta^n$, then $P(P_\Theta\leq_{\rm hr} P_{\Theta'})=\frac{1}{2^{n-1}}$ where $\leq_{\rm hr}$ denotes the hazard rate order.