# On computing the distribution function for the Poisson binomial distribution

Listed:
• Hong, Yili

## Abstract

The Poisson binomial distribution is the distribution of the sum of independent and non-identically distributed random indicators. Each indicator follows a Bernoulli distribution and the individual probabilities of success vary. When all success probabilities are equal, the Poisson binomial distribution is a binomial distribution. The Poisson binomial distribution has many applications in different areas such as reliability, actuarial science, survey sampling, econometrics, etc. The computing of the cumulative distribution function (cdf) of the Poisson binomial distribution, however, is not straightforward. Approximation methods such as the Poisson approximation and normal approximations have been used in literature. Recursive formulae also have been used to compute the cdf in some areas. In this paper, we present a simple derivation for an exact formula with a closed-form expression for the cdf of the Poisson binomial distribution. The derivation uses the discrete Fourier transform of the characteristic function of the distribution. We develop an algorithm that efficiently implements the exact formula. Numerical studies were conducted to study the accuracy of the developed algorithm and approximation methods. We also studied the computational efficiency of different methods. The paper is concluded with a discussion on the use of different methods in practice and some suggestions for practitioners.

## Suggested Citation

• Hong, Yili, 2013. "On computing the distribution function for the Poisson binomial distribution," Computational Statistics & Data Analysis, Elsevier, vol. 59(C), pages 41-51.
• Handle: RePEc:eee:csdana:v:59:y:2013:i:c:p:41-51
DOI: 10.1016/j.csda.2012.10.006
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File URL: http://www.sciencedirect.com/science/article/pii/S0167947312003568

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## References listed on IDEAS

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1. Duffie, Darrell & Saita, Leandro & Wang, Ke, 2007. "Multi-period corporate default prediction with stochastic covariates," Journal of Financial Economics, Elsevier, vol. 83(3), pages 635-665, March.
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## Citations

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Cited by:

1. Peizhou Liao & Hao Wu & Tianwei Yu, 0. "ROC Curve Analysis in the Presence of Imperfect Reference Standards," Statistics in Biosciences, Springer;International Chinese Statistical Association, vol. 0, pages 1-14.
2. repec:spr:stabio:v:9:y:2017:i:1:d:10.1007_s12561-016-9159-7 is not listed on IDEAS
3. Jeff Alstott & Giorgio Triulzi & Bowen Yan & Jianxi Luo, 2017. "Mapping technology space by normalizing patent networks," Scientometrics, Springer;Akadémiai Kiadó, vol. 110(1), pages 443-479, January.
4. Musa Çağlar & Sinan Gürel, 0. "Public R&D project portfolio selection problem with cancellations," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 0, pages 1-29.
5. Binchao Chen & Timothy Matis & James Benneyan, 2016. "Computing exact bundle compliance control charts via probability generating functions," Health Care Management Science, Springer, vol. 19(2), pages 103-110, June.
6. repec:spr:orspec:v:39:y:2017:i:3:d:10.1007_s00291-016-0468-5 is not listed on IDEAS

### Keywords

Characteristic function; k-out-of-n system; Longevity risk; Normal approximation; Sum of independent random indicators; Warranty returns;

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