Likelihood ratio ordering for convolution of random variables with respect to multidimensional parameter

Abstract

Suppose that Xθ_i,i=1,…,n are independent random variables with distribution functions F(x,θ_i), where θ_i=(θ_i1,…,θik) i = 1, …, n. In this paper, we study the likelihood ratio ordering of the convolution of random variables in terms of the multi-parameters majorization order. We prove that if the likelihood ratio ordering of convolution of independent random variables in terms of majorization order of one parameter is satisfied for t (t ≤ k) parameters, then the likelihood ratio ordering of the convolution of the random variables in terms of the majorization order of the vector parameters of dimension t holds. In the general case if the likelihood ratio ordering of convolution of independent random variables in terms of majorization order of one parameter holds with bigger in direction for some parameters, and less in direction for some, and with the remaining parameters fixed, we show that the likelihood ratio ordering of the convolution of the random variables in terms of the multi-parameters majoriza...