Year: 2011
Type: article
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 R+. Assuming that $\log Y_i$ has a log-concave density, we show that $∑ a_iY_i$ is stochastically smaller than ∑ biYi, 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 $∑a_iY_i$ is stochastically larger than $∑ b_iY_i$, 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.
Source: Bernoulli
DOI: 10.3150/10-bej302
Author: Yaming Yu
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2106396612