Stochastic Comparison For Linear Cominations Of Random Variables

Let∑i=1n aiXi be linear combinations of independent nonnegative random variables Xi,..., Xn,where a=(a1,..., an)∈R+n. Linear combinations of in-dependent nonnegative random variables arise naturally in statistics, operation research, reliability theory, computer science, economic theory, actuarial science and other fields. There are a large number of extensive studies on this topic in the literature. It should be remarked that most of the work in the literature is under some specific distribution assumptions such as exponential, Weibull, Pareto and gamma, etc. Few of them consider this problem under the general frame-work yet with independent and identically distributed assumptions. Motivated by Karlin and Rinott (1983) and Yu (2011), we unify the study of linear combi-nations of independent random variables under the general setup by using some monotone transforms. Motivated by Xu and Hu (2011), we further generalize the results to the case of independent but not necessarily identically distributed ran-dom variables. What’s more, the results are further generalized to the cases of permutation invariant random variables, of independent but not necessarily iden-tically distributed random variables which are ordered in the likelihood ratio or the hazard ratio order, and of comonotonic random variables.The main results complement and generalize the results in the literature in-cluding Karlin and Rinott (1983), Yu (2011), Tong (1980) and Xu and Hu (2012).