@article{ajams2016464,
author={Nijimbere, Victor},
title={Coincidences, Goodness of Fit Test and Confidence Interval for Poisson Distribution Parameter via Coincidence},
journal={American Journal of Applied Mathematics and Statistics},
volume={4},
number={6},
pages={185--193},
year={2016},
url={http://pubs.sciepub.com/ajams/4/6/4},
issn={2328-7292},
abstract={The probability of the coincidence of some discrete random variables having a Poisson distribution with parameters <i>¦Ë</i><SUB><i>1</i></SUB><i>,</i><i> </i><i>¦Ë</i><SUB><i>2</i></SUB><i>,</i><i> </i><i>¡­, ¦Ë</i><SUB><i>n</i></SUB>, and moments are expressed in terms of the hypergeometric function <SUB><i>1</i></SUB><i>F</i><SUB><i>n</i></SUB> or the modified Bessel function of the first kind if <i>n=2</i>. Considering the null hypothesis <i>H</i><SUB><i>0</i></SUB><i>: ¦Ë</i><SUB><i>1</i></SUB><i>=¦Ë</i><SUB><i>2</i></SUB><i>=¡­.= ¦Ë</i><SUB><i>n</i></SUB><i> =¦È</i>, where <i>¦È</i> is some positive constant number, asymptotic approximations of the probability and moments are derived for large <i>¦È</i> using the asymptotic expansion of the hypergeometric function <SUB><i>1</i></SUB><i>F</i><SUB><i>n</i></SUB> and that of the modified Bessel function of the first kind if <i>n=2</i>. Further, we show that if the sample mean is a minimum variance unbiased estimator (MVUE) for the parameter <i>¦Ë</i><SUB><i>i</i></SUB>, then the probability that <i>H</i><SUB><i>0</i></SUB> is true can be approximated by that of a coincidence. In that case, a chi-square <i>¦Ö</i><SUP><i>2 </i></SUP>goodness of fit test can be established and a 100(1-<i>¦Á</i>)% confidence interval (CI) for <i>¦È</i> can be constructed using the variance of the coincidence (or via coincidence) and the Central Limit Theorem (CLT).},
doi={10.12691/ajams-4-6-4}
publisher={Science and Education Publishing}
}