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

Let Xλ1,…,Xλn be independent random variables such that Xλi, i=1,…,n has probability density function fν,σ,λi(x)=2λiνΓ(ν2)(2σ2)ν2xν−1exp(−(λix)22σ2),ν>0,σ>0,λi>0, known as a generalized Rayleigh random variable. We show that for ν≥1, if (λ1∗2,…,λn∗2) majorizes (λ12,…,λn2), then ∑i=1nXλi∗ is larger than ∑i=1nXλi according to likelihood ratio ordering.