On the restrictiveness of the usual stochastic order and the likelihood ratio order

Abstract

Every element θ=(θ1,…,θn) of the probability n-simplex induces a probability distribution Pθ of a random variable X that can assume only a finite number of real values x1<⋯<xn by defining Pθ(X=xi)=θi,1≤i≤n. We show that if Θ and Θ′ are two random vectors uniformly distributed on Δn, then P(PΘ≤stPΘ′)=1n and P(PΘ≤lrPΘ′)=1n!, where ≤st and ≤lr denote the usual stochastic order and the likelihood ratio order, respectively.