On the restrictiveness of the hazard rate order

Abstract

Every element $θ=(θ_1,\ldots,θ_n)$ of the probability $n$-simplex induces a probability distribution $P_θ$ of a random variable $X$ that can assume only a finite number of real values $x_1 < \cdots < x_n$ by defining $P_θ(X=x_i) = θ_i, 1\leq i \leq n$. We show that if $Θ$ and $Θ'$ are two random vectors uniformly distributed on $Δ^n$, then $P(P_Θ\leq_{\rm hr} P_{Θ'})=\frac{1}{2^{n-1}}$ where $\leq_{\rm hr}$ denotes the hazard rate order.