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The exact distribution of the maximum, minimum and the range of Multinomial/Dirichlet and Multivariate Hypergeometric frequencies

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Abstract

The exact distribution of the maximum and minimum frequencies of Multinomial/Dirichlet and Multivariate Hypergeometric distributions of n balls in m urns is compactly represented as a product of stochastic matrices. This representation does not require equal urn probabilities, is invariant to urn order, and permits rapid calculation of exact probabilities. The exact distribution of the range is also obtained. These algorithms satisfy a long-standing need for routines to compute exact Multinomial/Dirichlet and Multivariate Hypergeometric maximum, minimum, and range probabilities in statistical computation libraries and software packages.

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References

  • Bennett, B.M., Holmes, E.N.: Small sample exact power of the range from a symmetrical distribution. Biom. J. 32, 277–280 (1990)

    Article  Google Scholar 

  • Bennett, B.M., Nakamura, E.: Percentage points of the range from a symmetric multinomial distribution. Biometrika 55, 377–379 (1968)

    Article  MathSciNet  Google Scholar 

  • Butler, R.W., Sutton, R.K.: Saddlepoint approximation for multivariate distribution functions and probability computations in sampling theory and outlier testing. J. Am. Stat. Assoc. 93, 596–604 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Cochran, W.G.: The χ 2 correction for continuity. J. Sci. 16, 421–436 (1942)

    MathSciNet  Google Scholar 

  • de Montmort, P.R.: Essay d’Analyse sur les Jeux de Hazard (1708). Reprinted by Chelsea, New York (1980)

    Google Scholar 

  • Feller, W.: An Introduction to Probability Theory and Its Applications. Wiley, New York (1968)

    MATH  Google Scholar 

  • Freeman, P.R.: Exact distribution of the largest multinomial frequency. Ann. Stat. 28, 333–336 (1979)

    MathSciNet  MATH  Google Scholar 

  • Fuchs, C., Kenett, R.: A test for detecting outlying cells in the multinomial distribution and two-way contingency tables. J. Am. Stat. Assoc. 75, 395–398 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  • Good, I.J.: Saddlepoint methods for the multinomial distributions. Ann. Math. Stat. 28, 861–881 (1957)

    Article  MathSciNet  Google Scholar 

  • Gupta, R.D., Richards, D.St.P.: A history of the Dirichlet and Liouville distributions. Int. Stat. Rev. 69, 433–446 (2001)

    Article  MATH  Google Scholar 

  • Hald, A.: A History of Probability and Statistics and Their Applications Before 1750. Wiley, New York (1990)

    Book  MATH  Google Scholar 

  • Johnson, N.L., Kotz, S.N.: Urn Models and Their Application: An Approach to Modern Discrete Probability Theory. Wiley, New York (1977)

    MATH  Google Scholar 

  • Johnson, N.L., Young, D.H.: Some applications of two approximations to the multinomial distribution. Biometrika 47, 463–469 (1960)

    MATH  Google Scholar 

  • Johnson, N.L., Kotz, S., Balakrishnan, N.: Discrete Multivariate Distributions. Wiley, New York (1997)

    MATH  Google Scholar 

  • Kemp, C.D., Kemp, A.W.: Rapid generation of frequency tables. Appl. Stat. 36, 277–282 (1987)

    Article  Google Scholar 

  • Lancaster, G.A., Green, M., Lane, S.: Reducing bias in ecological studies: an evaluation of different methodologies. J. R. Stat. Soc. A 169, 681–700 (2006)

    Article  MathSciNet  Google Scholar 

  • Levin, B.: A representation for multinomial cumulative distribution functions. Ann. Stat. 9, 1123–1126 (1981)

    Article  MATH  Google Scholar 

  • Levin, B.: On calculations involving the maximum cell frequency. Commun. Stat. Part A Theory Methods 12, 1299–1327 (1983)

    Article  MATH  Google Scholar 

  • Mallows, C.L.: An inequality involving multinomial probabilities. Biometrika 55, 422–424 (1968)

    Article  MATH  Google Scholar 

  • Mosimann, J.E.: On the compound multinomial distribution, the multivariate β-distribution, and correlations among proportions. Biometrika 49, 65–82 (1962)

    MathSciNet  MATH  Google Scholar 

  • Pearson, K.: Contributions to the mathematical theory of evolution I. Skew distribution in homogeneous material. Philos. Trans. R. Soc. Lond., Ser. A 186, 343–414 (1895)

    Article  Google Scholar 

  • Pearson, K.: On certain properties of the hypergeometrical series, and on the fitting of such series to observation polygons in the theory of chance. Philos. Mag., 5th Ser. 47, 236–246 (1899)

    Google Scholar 

  • Shonkwiler, J.S., Hanley, N.: A new approach to random utility modeling using the Dirichlet multinomial distribution. Environ. Resour. Econ. 26, 401–416 (2003)

    Article  Google Scholar 

  • Taylor, H.M., Karlin, S.: An Introduction to Stochastic Modeling, 3rd edn. Academic Press, San Diego (1998)

    MATH  Google Scholar 

  • Young, D.H.: Two alternatives to the standard χ 2-test of the hypothesis of equal cell frequencies. Biometrika 49, 107–116 (1962)

    MathSciNet  MATH  Google Scholar 

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  1. Faculty of Business & Law, Deakin University, 221 Burwood Highway, Burwood 3125, Melbourne, Victoria, Australia

    Charles J. Corrado

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  1. Charles J. Corrado
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Correspondence to Charles J. Corrado.

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Corrado, C.J. The exact distribution of the maximum, minimum and the range of Multinomial/Dirichlet and Multivariate Hypergeometric frequencies. Stat Comput 21, 349–359 (2011). https://doi.org/10.1007/s11222-010-9174-3

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  • DOI: https://doi.org/10.1007/s11222-010-9174-3

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