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Some Inequalities of Linear Combinations of Independent Random Variables. I.

Part of: Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Maochao Xu  and
Taizhong Hu
Maochao Xu*
Affiliation:
Illinois State University
Taizhong Hu*
Affiliation:
University of Science and Technology of China
*
∗ Postal address: Department of Mathematics, Illinois State University, Normal, IL 61790-4520, USA. Email address: [email protected]
∗∗ Postal address: Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui 230026, P. R. China. Email address: [email protected]
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Abstract

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In this paper we provide some sufficient conditions to stochastically compare linear combinations of independent random variables. The main results extend those given in Proschan (1965), Ma (1998), Zhao et al. (2011), and Yu (2011). In particular, we propose a new sufficient condition to compare the peakedness of linear combinations of independent random variables which may have heavy-tailed properties.

Keywords

Likelihood ratio ordermajorizationpeakedness orderstochastic order

MSC classification

Secondary: 60E15: Inequalities; stochastic orderings
Type
Research Papers
Information
Journal of Applied Probability , Volume 48 , Issue 4 , December 2011 , pp. 1179 - 1188
Copyright
Copyright © Applied Probability Trust 2011 

Footnotes

Supported by the NNSF of China (grant numbers 11071232 and 70821001) and the National Basic Research Program of China (973 Program, grant number 2007CB814901).

References

An, M. Y. (1998). Logconcavity versus logconvexity: a complete characterization. J. Econom. Theory 80, 350–369.Google Scholar
Bagnoli, M. and Bergstrom, T. (2005). Log-concave probability and its applications. Econom. Theory 26, 445–469.Google Scholar
Birnbaum, Z. W. (1948). On random variables with comparable peakedness. Ann. Math. Statist. 19, 76–81.Google Scholar
Dharmadhikari, S. and Joag-Dev, K. (1988). Unimodality, Convexity, and Applications. Academic Press, Boston, MA.Google Scholar
Hu, T., Nanda, A. K., Xie, H. and Zhu, Z. (2004). Properties of some stochastic orders: A unified study. Naval Res. Logistics 51, 193–216.Google Scholar
Ibragimov, I. A. (1956). On the composition of unimodal distributions. Theory Prob. Appl. 1, 255–280.Google Scholar
Ibragimov, R. (2007). Efficiency of linear estimators under heavy-tailedness: convolutions of α-symmetric distributions. Econometric Theory 23, 501–517.Google Scholar
Jensen, D. R. (1997). Peakedness of linear forms in ensembles and mixtures. Statist. Prob. Lett. 35, 277–282.Google Scholar
Ma, C. (1998). On peakedness of distributions of convex combinations. J. Statist. Planning Infer. 70, 51–56.Google Scholar
Marshall, A. W., Olkin, I. and Arnold, B. C. (2011). Inequalities: Theory of Majorization and Its Applications, 2nd edn. Springer, New York.Google Scholar
Pan, X., Xu, M. and Hu, T. (2011). Some inequalities of linear combinations of independent random variables: II. Tech. Rep., University of Science and Technology of China.Google Scholar
Proschan, F. (1965). Peakedness of distributions of convex combinations. Ann. Math. Statist. 36, 1703–1706.Google Scholar
Purkayastha, S. (1998). Simple proofs of two results on convolutions of unimodal distributions. Statist. Prob. Lett. 39, 97–100.Google Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.Google Scholar
Tong, Y. L. (1994). Some recent developments on majorization inequalities in probability and statistics. Linear Algebra Appl. 199, 69–90.Google Scholar
Wintner, A. (1938). Asymptotic Distributions and Infinite Convolutions. Edwards Brothers, Ann Arbor, MI.Google Scholar
Yu, Y. (2011). Some stochastic inequalities for weighted sums. To appear in Bernoulli.Google Scholar
Zhao, P., Chan, P. S. and Ng, H. K. T. (2011). Peakedness for weighted sums of symmetric random variables. J. Statist. Planning Infer. 141, 1737–1743.Google Scholar