## Article citationsMore >>

Nemenman, E., Coincidences and estimetion of entropies of random variables with large cardinalities, Entropy, 13(12), 2013-2023. Dec. 2011.

has been cited by the following article:

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

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

American Journal of Applied Mathematics and Statistics. 2016, Vol. 4 No. 6, 185-193
DOI: 10.12691/ajams-4-6-4

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.

Correspondence to: Victor  Nijimbere, School of Mathematics and Statistics, Carleton University, Ottawa, Canada. Email: victornijimbere@gmail.com

## Abstract

The probability of the coincidence of some discrete random variables having a Poisson distribution with parameters λ1, λ2, …, λn, and moments are expressed in terms of the hypergeometric function 1Fn or the modified Bessel function of the first kind if n=2. Considering the null hypothesis H0: λ12=….= λn, 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 1Fn 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 λi, 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).