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A Three-Parameter Binomial Approximation

Part of: Distribution theory - Probability Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Erol A. Peköz ,
Adrian Röllin ,
Vydas Čekanavičius  and
Michael Shwartz
Erol A. Peköz*
Affiliation:
Boston University
Adrian Röllin*
Affiliation:
National University of Singapore
Vydas Čekanavičius*
Affiliation:
Vilnius University
Michael Shwartz*
Affiliation:
Veterans' Health Administration, Boston, and Boston University
*
∗Postal address: Boston University School of Management, 595 Commonwealth Avenue, Boston, MA 02215, USA.
∗∗∗Postal address: National University of Singapore, 2 Science Drive 2, 117543, Singapore.
∗∗∗∗Postal address: Vilnius University, Faculty of Mathematics and Informatics, Naugarduko 24, Vilnius LT-03223, Lithuania.
∗Postal address: Boston University School of Management, 595 Commonwealth Avenue, Boston, MA 02215, USA.
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Abstract

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We approximate the distribution of the sum of independent but not necessarily identically distributed Bernoulli random variables using a shifted binomial distribution, where the three parameters (the number of trials, the probability of success, and the shift amount) are chosen to match the first three moments of the two distributions. We give a bound on the approximation error in terms of the total variation metric using Stein's method. A numerical study is discussed that shows shifted binomial approximations are typically more accurate than Poisson or standard binomial approximations. The application of the approximation to solving a problem arising in Bayesian hierarchical modeling is also discussed.

Keywords

Stein's methodPoisson binomial distributionbinomial approximation

MSC classification

Secondary: 60E15: Inequalities; stochastic orderings 60G50: Sums of independent random variables; random walks
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 4 , December 2009 , pp. 1073 - 1085
Copyright
Copyright © Applied Probability Trust 2009 

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