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A discrete multivariate distribution resulting from the law of small numbers

Published online by Cambridge University Press:  14 July 2016

Nobuaki Hoshino*
Affiliation:
Kanazawa University
*
Postal address: Faculty of Economics, Kanazawa University, Kakuma-machi, Kanazawa-shi, Ishikawa, 920-1192, Japan. Email address: [email protected]
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Abstract

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In the present article we derive a new discrete multivariate distribution using a limiting argument that is essentially the same as the law of small numbers. The distribution derived belongs to an exponential family, and randomly partitions positive integers. The facts shown about the distribution are useful in many fields of application involved with count data. The derivation parallels that of the Ewens distribution from the gamma distribution, and the new distribution is produced from the inverse Gaussian distribution. The method employed is regarded as the discretization of an infinitely divisible distribution over nonnegative real numbers.

Type
Research Article
Copyright
© Applied Probability Trust 2006 

References

Anscombe, F. J. (1950). Sampling theory of the negative binomial and logarithmic series distributions. Biometrika 37, 358382.Google Scholar
Aoki, M. (2000). Cluster size distributions of economic agents of many types in a market. J. Math. Anal. Appl. 249, 3252.Google Scholar
Bunge, J. and Fitzpatrick, M. (1993). Estimating the number of species: a review. J. Amer. Statist. Assoc. 88, 364373.Google Scholar
Charalambides, C. A. and Singh, J. (1988). A review of the Stirling numbers, their generalizations and statistical applications. Commun. Statist. Theory Meth. 17, 25332595.Google Scholar
Engen, S. (1974). On species frequency models. Biometrika 61, 263270.Google Scholar
Engen, S. (1978). Stochastic Abundance Models. Chapman and Hall, London.Google Scholar
Ewens, W. J. (1972). The sampling theory of selectively neutral alleles. Theoret. Pop. Biol. 3, 87112.Google Scholar
Good, I. J. (1953). The population frequencies of species and the estimation of population parameters. Biometrika 40, 237264.Google Scholar
Hoshino, N. (2001). Applying Pitman's sampling formula to microdata disclosure risk assessment. J. Official Statist. 17, 499520.Google Scholar
Hoshino, N. (2003). Random clustering based on the conditional inverse Gaussian–Poisson distribution. J. Japan Statist. Soc. 33, 105117.Google Scholar
Hoshino, N. (2005). Engen's extended negative binomial model revisited. Ann. Inst. Statist. Math. 57, 369387.Google Scholar
Hoshino, N. and Takemura, A. (1998). Relationship between logarithmic series model and other superpopulation models useful for microdata disclosure risk assessment. J. Japan Statist. Soc. 28, 125134.CrossRefGoogle Scholar
Ismail, M. E. H. (1977). Integral representations and complete monotonicity of various quotients of Bessel functions. Canad. J. Math. 29, 11981207.Google Scholar
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1997). Discrete Multivariate Distributions. John Wiley, New York.Google Scholar
Jørgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution (Lecture Notes Statist. 9). Springer, New York.Google Scholar
Kingman, J. F. C. (1978). Random partitions in population genetics. Proc. R. Soc. London A 361, 120.Google Scholar
Lehmann, E. L. (1991). Theory of Point Estimation. Wadsworth and Brooks, Pacific Grove, CA.Google Scholar
Pitman, J. (1995). Exchangeable and partially exchangeable random partitions. Prob. Theory Relat. Fields 102, 145158.Google Scholar
Pitman, J. (2003). Poisson–Kingman partitions. In Science and Statistics: A Festschrift for Terry Speed (IMS Lecture Notes Monogr. Ser. 40), Institute of Mathematical Statistics, Hayward, CA, pp. 134.Google Scholar
Seshadri, V. (1999). The Inverse Gaussian Distribution (Lecture Notes Statist. 137). Springer, New York.Google Scholar
Sibuya, M. (1991). A cluster-number distribution and its application to the analysis of homonyms. Japanese J. Appl. Statist. 20, 139153 (in Japanese).Google Scholar
Sibuya, M. (1993). A random clustering process. Ann. Inst. Statist. Math. 45, 459465.Google Scholar
South, R. and Edelsten, H. M. (1939). The Moths of the British Isles. Frederick Warne, London.Google Scholar
Steutel, F. W. and van Harn, K. (2004). Infinite Divisibility of Probability Distributions on the Real Line. Marcel Dekker, New York.Google Scholar
Watson, G. N. (1944). A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press.Google Scholar
Yamato, H., Sibuya, M. and Nomachi, T. (2001). Ordered sample from two-parameter GEM distribution. Statist. Prob. Lett. 55, 1927.Google Scholar