Bernoulli

  • Bernoulli
  • Volume 17, Number 3 (2011), 1044-1053.

Some stochastic inequalities for weighted sums

Yaming Yu

Full-text: Open access

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 Yi be i.i.d. random variables on R+. Assuming that logYi has a log-concave density, we show that ∑ aiYi is stochastically smaller than ∑ biYi, if (log a1, …, log an) is majorized by (log b1, …, log bn). On the other hand, assuming that Yip has a log-concave density for some p > 1, we show that ∑aiYi is stochastically larger than ∑ biYi, if (a1q, …, anq) is majorized by (b1q, …, bnq), 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.

Article information

Source
Bernoulli Volume 17, Number 3 (2011), 1044-1053.

Dates
First available in Project Euclid: 7 July 2011

Permanent link to this document
https://projecteuclid.org/euclid.bj/1310042855

Digital Object Identifier
doi:10.3150/10-BEJ302

Mathematical Reviews number (MathSciNet)
MR2817616

Zentralblatt MATH identifier
1225.60035

Keywords
gamma distribution log-concavity majorization Prékopa–Leindler inequality Rayleigh distribution tail probability usual stochastic order Weibull distribution weighted sum

Citation

Yu, Yaming. Some stochastic inequalities for weighted sums. Bernoulli 17 (2011), no. 3, 1044--1053. doi:10.3150/10-BEJ302. https://projecteuclid.org/euclid.bj/1310042855


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