On the likelihood ratio order for convolutions of independent generalized Rayleigh random variables

Let X-lambda 1, ... , X-lambda n be independent random variables such that X-lambda i, i = 1, ... , n has probability density function f(nu,sigma,lambda i)(x) = 2 lambda(nu)(i)/Gamma(nu/2)(2 sigma(2))nu/2 x(nu-1) exp (-(lambda(i)x)(2)/2 sigma(2)), nu > 0, sigma > 0, lambda(i) > 0, known as a generalized Rayleigh random variable. We show that for nu >= 1, if (lambda(*2)(1), ... , lambda(*2)(n)) majorizes (lambda(2)(1), ... , lambda(2)(n)), then Sigma(n)(i=1) X-lambda i(*) is larger than Sigma(n)(i=1) X-lambda i according to likelihood ratio ordering. (C) 2008 Elsevier B.V. All rights reserved.