Some stochastic inequalities for weighted sums

Abstract

We compare weighted sums of i.i.d. positive random variables according to the usual stochastic order. The main inequalities are derived using majorization techniques under certain log-concavity assumptions. Specifically, let $Y_i$ be i.i.d. random variables on ${\mathbf{R}}_{+}$. Assuming that $\log Y_i$ has a log-concave density, we show that $\sum a_i{Y}_i$ is stochastically smaller than $\sum b_i{Y}_i$, if $(\log a_1,...,\log a_n)$ is majorized by $(\log b_1,...,\log b_n)$. On the other hand, assuming that $Y_i^p$ has a log-concave density for some $p>1$, we show that $\sum a_i{Y}_i$ is stochastically larger than $\sum b_i{Y}_i$, if $(a_1^q,...,a_n^q)$ is majorized by $(b_1^q,...,b_n^q)$, where $p^{-1}+q^{-1}=1$. These unify several stochastic ordering results for specific distributions. In particular, a conjecture of Hitczenko [Sankhyā A 60 (1998) 171--175] on Weibull variables is proved. Potential applications in reliability and wireless communications are mentioned.

Published in at http://dx.doi.org/10.3150/10-BEJ302 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (3)

• Stochastic Ordering of Exponential Family Distributions and Their Mixtures
• On an Inequality of Karlin and Rinott Concerning Weighted Sums of i.i.d. Random Variables
• Relative log-concavity and a pair of triangle inequalities

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• Some inequalities of linear combinations of independent random variables: II
• Schur properties of convolutions of gamma random variables