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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: Stochastic processes Distribution theory 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: tasos@ucy.ac.cy
∗∗Postal address: Section of Statistics and Operations Research, Department of Mathematics, University of Athens, Panepistemiopolis, Athens 15784, Greece. Email address: evagel@math.uoa.gr
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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.

References

Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
Boutsikas, M. V. and Vaggelatou, E. (2008). A new method for obtaining sharp compound Poisson approximation error estimates for sums of locally dependent random variables. Preprint.Google Scholar
Cacoullos, T., Papadatos, N. and Papathanasiou, V. (1997). Variance inequalities for covariance kernels and applications to central limit theorems. Theory Prob. Appl. 42, 195–201.Google Scholar
Cacoullos, T., Papadatos, N. and Papathanasiou, V. (2001). An application of a density transform and the local limit theorem. Theory Prob. Appl. 46, 803–810.Google Scholar
Cheng, K. and He, Z. F. (1988). Exponential approximations in the classes of life distributions. Acta Math. Appl. Sinica (English Ser.) 4, 234–244.CrossRefGoogle Scholar
Daley, D. J. (1988). Tight bounds on the exponential approximation of some aging distributions. Ann. Prob. 16, 414–423.CrossRefGoogle Scholar
He, Z. and Cheng, K. (1987). Exponential approximations for some classes of life distributions. In Reliability Theory and Applications, eds Osaki, S. et al., World Scientific, Singapore, pp. 113–126.Google Scholar
Kaas, R. (1993). How to and how not to compute stop-loss premiums in practice. Insurance Math. Ecomom. 13, 241–254.CrossRefGoogle ScholarPubMed
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
Papadatos, N. and Papathanasiou, V. (1995). Distance in variation between two arbitrary distributions via the associated w-functions. Theory Prob. Appl. 40, 685–694.Google Scholar
Rachev, S. T. (1991). Probability Metrics and the Stability of Stochastic Models. John Wiley, Chichester.Google Scholar
Rachev, S. T. and Rüschendorf, L. (1990). Approximation of sums by compound Poisson distributions with respect to stop-loss distances. Adv. Appl. Prob. 22, 350–374.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.CrossRefGoogle Scholar
Szekli, R. (1995). Stochastic Ordering and Dependence in Applied Probability (Lecture Notes Statist. 97). Springer, New York.CrossRefGoogle Scholar
Vaggelatou, E. (2009). A new method for bounding the distance between sums of independent integer-valued random variables. To appear in Methodology Comput. Appl. Prob. Google Scholar
Zolotarev, V. M. (1983). Probability metrics. Theory Prob. Appl. 28, 278–302.CrossRefGoogle Scholar