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Bounds for the Distance Between the Distributions of Sums of Absolutely Continuous i.i.d. Convex-Ordered Random Variables with Applications

Part of: Distribution theory Stochastic processes Distribution theory - Probability Special processes

Published online by Cambridge University Press:  14 July 2016

Tasos C. Christofides  and
Eutichia Vaggelatou
Tasos C. Christofides*
Affiliation:
University of Cyprus
Eutichia Vaggelatou*
Affiliation:
University of Athens
*
∗Postal address: Department of Mathematics and Statistics, University of Cyprus, P.O. Box 20537, Nicosia, CY 1678, Cyprus. Email address: [email protected]
∗∗Postal address: Section of Statistics and Operations Research, Department of Mathematics, University of Athens, Panepistemiopolis, Athens 15784, Greece. Email address: [email protected]
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Abstract

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Let X1, X2,… and Y1, Y2,… be two sequences of absolutely continuous, independent and identically distributed (i.i.d.) random variables with equal means E(Xi)=E(Yi), i=1,2,… In this work we provide upper bounds for the total variation and Kolmogorov distances between the distributions of the partial sums ∑i=1nXi and ∑i=1nYi. In the case where the distributions of the Xis and the Yis are compared with respect to the convex order, the proposed upper bounds are further refined. Finally, in order to illustrate the applicability of the results presented, we consider specific examples concerning gamma and normal approximations.

Keywords

Zolotarev's ideal metric of order 2Kolmogorov distancetotal variation distanceconvex orderNBUE/NWUEWeibull distributionStudent distributiongamma approximationnormal approximation

MSC classification

Secondary: 60E15: Inequalities; stochastic orderings 62E17: Approximations to distributions (nonasymptotic) 62E20: Asymptotic distribution theory 60G50: Sums of independent random variables; random walks 60K10: Applications (reliability, demand theory, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 1 , March 2009 , pp. 255 - 271
Copyright
Copyright © Applied Probability Trust 2009 

Footnotes

Partially supported by the University of Athens research grant 70/4/8810.

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