Coincidences, Goodness of Fit Test and Confidence Interval for Poisson Distribution Parameter via Coincidence

Victor Nijimbere^1,

¹School of Mathematics and Statistics, Carleton University, Ottawa, Canada

Cite this paper

Victor Nijimbere. Coincidences, Goodness of Fit Test and Confidence Interval for Poisson Distribution Parameter via Coincidence. American Journal of Applied Mathematics and Statistics. 2016; 4(6):185-193. doi: 10.12691/AJAMS-4-6-4

Abstract

The probability of the coincidence of some discrete random variables having a Poisson distribution with parameters λ₁, λ₂, …, λ_n, and moments are expressed in terms of the hypergeometric function ₁F_n or the modified Bessel function of the first kind if n=2. Considering the null hypothesis H₀: λ₁=λ₂=….= λ_n =θ, where θ is some positive constant number, asymptotic approximations of the probability and moments are derived for large θ using the asymptotic expansion of the hypergeometric function ₁F_n and that of the modified Bessel function of the first kind if n=2. Further, we show that if the sample mean is a minimum variance unbiased estimator (MVUE) for the parameter λ_i, then the probability that H₀ is true can be approximated by that of a coincidence. In that case, a chi-square χ² goodness of fit test can be established and a 100(1-α)% confidence interval (CI) for θ can be constructed using the variance of the coincidence (or via coincidence) and the Central Limit Theorem (CLT).

Keywords

probability, hypergeometric functions, modified bessel functions, asymptotic expansion, MVUE

Copyright

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

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