A Brief Derivation of the Asymptotic Distribution of Pearson’s Statistic and an Accurate Approximation to Its Exact Distribution

  • Serge B. ProvostEmail author
Part of the Fields Institute Communications book series (FIC, volume 78)


A brief and accessible derivation of the asymptotic distribution of Pearson’s goodness-of-fit statistic is proposed. Additionally, a shifted gamma distribution is introduced as an accurate approximation to be utilized when the chi-squared distribution proves to be inadequate. It is also explained that the exact probability mass function of this test statistic can be readily determined from its moment-generating function via symbolic computations. Two illustrative numerical examples are included.


Pearson’s statistic Asymptotic distribution Goodness-of-fit tests Shifted gamma distribution 

AMS Mathematics Subject Classification (2010)

62E20 60E10 62E15 62E17 



The financial support of the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged. Thanks are also due to two referees for their valuable comments. This Festschrift, which was organized in recognition of Ian McLeod’s significant contributions to Time Series as well as several other areas of Statistics, is indeed a fitting tribute to his scholarly accomplishments. Ian has truly been a valued colleague over the years.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Statistical and Actuarial SciencesThe University of Western OntarioLondonCanada

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