Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Andrés, A. M., & Tejedor, I. H. (2009). Comments on ‘Tests for the homogeneity of two binomial proportions in extremely unbalanced 2\(\times \)2 contingency tables’. Statistics in Medicine, 28, 528–531.
Cochran, W. G. (1952). The \(\chi ^2\) test of goodness of fit. The Annals of Mathematical Statistics, 25, 315–345.
Cox, M. K., & Key, C. H. (1993). Post hoc pair-wise comparisons for the chi-square test of homogeneity of proportions. Key Educational and Psychological Measurement, 53, 951–962.
DasGupta, A. (2008). Asymptotic theory of statistics and probability, Springer Texts in Statistics. New York: Springer.
Ferguson, T. S. (1996). A course in large sample theory. Boca Raton: Chapman & Hall.
Graybill, F. A. (1983). Matrices with applications in statistics. Belmont: Wadsworth.
Koehler, K. J., & Larntz, K. (1980). An empirical investigation of goodness-of-fit statistics for sparse multinomials. Journal of the American Statistical Association, 75, 336–344.
Mathai, A. M., & Pederzoli, G. (1996). A simple derivation of the chi-square approximation to Pearson’s statistic. Statistica, 56, 407–413.
Moore, D. S., & Spruill, M. C. (1975). Unified large-sample theory of general chi-squared statistics for tests of fit. Annals of Statistics, 3, 599–616.
Nathan, G. (1972). On the asymptotic power of tests for independence in contingency tables from stratified samples. Journal of the American Statistical Association, 67, 917–920.
Pearson, K. (1900). On the criterion that a given system of deviations from the probable in the case of a correlated system of variables is such that it can be reasonably supposed to have arisen from random sampling. Philosophical Magazine, 50, 157–175.
Roscoe, J. T., & Byars, J. A. (1971). An investigation of the restraints with respect to sample size commonly imposed on the use of the chi-square statistic. Journal of the American Statistical Association, 66, 755–759.
Rosner, B. (2006). Fundamentals of biostatistics. Chapter 10. Chi-square goodness-of-fit (6th ed.). Duxbury, MA: Thomson Learning Academic Resource Center.
Tobin, J. (1958). Estimation of relationship for limited dependent variables. Econometrica, 26, 24–36.
Watson, G. S. (1959). Some recent results in chi-square goodness-of-fit tests. Biometrics, 15, 440–468.
Yates, F. (1934). Contingency table involving small numbers and the \(\chi ^2\) test. Journal of the Royal Statistical Society, 1, 217–235.
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer Science+Business Media New York
About this chapter
Cite this chapter
Provost, S.B. (2016). A Brief Derivation of the Asymptotic Distribution of Pearson’s Statistic and an Accurate Approximation to Its Exact Distribution. In: Li, W., Stanford, D., Yu, H. (eds) Advances in Time Series Methods and Applications . Fields Institute Communications, vol 78. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6568-7_11
Download citation
DOI: https://doi.org/10.1007/978-1-4939-6568-7_11
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4939-6567-0
Online ISBN: 978-1-4939-6568-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)