Abstract
In this paper, we study of Pólya urn model containing balls of (m+1) different labels under a general replacement scheme, which is characterized by an (m+1) × (m+1) addition matrix of integers without constraints on the values of these (m+1)2 integers other than non-negativity. LetX 1,X 2,...,X n be trials obtained by the Pólya urn scheme (with possible outcomes: “O”, “1”,...“m”). We consider the multivariate distributions of the numbers of occurrences of runs of different types arising from the various enumeration schemes and give a recursive formula of the probability generating function. Some closed form expressions are derived as special cases, which have potential applications to various areas. Our methods for the derivation of the multivariate run-related distribution are very simple and suitable for numerical and symbolic calculations by means of computer algebra systems. The results presented here develop a general workable framework for the study of Pólya urn models. Our attempts are very useful for understanding non-classic urn models. Finally, numerical examples are also given in order to illustrate the feasibility of our results.
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References
Aki, S. and Hirano, K. (1988). Some characteristics of the binomial distribution of orderk and related distributions.Statistical Theory and Data Analysis II, Proceedings of the Second Pacific Area Statistical Conference (eds. K. Matusita), 211–222, North-Holland, Amsterdam.
Aki, S. and Hirano, K. (2000). Numbers of success-runs of specified length until certain stopping time rules and generalized binomial distributions of orderk, Annals of the Institute of Statistical Mathematics,52, 767–777.
Balakrishnan, N. and Koutras, M. V. (2002),Runs and Scand with Applications, John Wiley, New York.
Boutsikas, M. V. and Koutras, M. V. (2002). On a class of multiple failure mode systems,Naval Research Logistics,49, 167–185.
Eggenberger, F. and Pólya, G. (1923). Über die Statistik verketetter Vorgänge,Zeitschrift fur Angewandte Mathematik und Mechanik,1, 279–289.
Feller, W. (1968).An Introduction to Probability Theory and Its Applications, Vol. I, 3rd ed., Wiley, New York.
Friedman, B. (1949). A simple urn model,Communications on Pure and Applied Mathematics,2, 59–70.
Gibbons, J. D. (1971).Nonparametric Statistical Inference, McGraw-Hill, New York.
Goldstein, L. (1990). Poisson approximation and DNA sequence matching,Communications in Statistics Theory and Methods,19(11), 4167–4179.
Inoue, K. (2003). Generalized Pólya urn models and related distributions,Proceedings of the Symposium, Research Institute for Mathematical Science, Kyoto University,1308, 29–38.
Inoue, K. and Aki, S. (2001), Pólya urn models under general replacement schemes,Journal of the Japan Statistical Society,31, 193–205.
Inoue, K. and Aki, S. (2003). Generalized binomial and negative binomial distributions of orderk by thel-overlapping enumeration scheme.Annals of the Institute of Statistical Mathematics,55, 153–167.
Johnson, N. L. and Kotz, S. (1977).Urn Models and Their Applications, Wiley, New York.
Johnson, N. L., Kotz, S. and Balakrishnan, N. (1997).Discrete Multivariate Distributions, Wiley, New York.
Kotz, S. and Balakrishnan, N. (1997). Advances in urn models during the past two decades,Advances in Combinatorial Methods and Applications to Probability and Statistics (ed. N. Balakrishnan), 203–257, Birkhäuser, Boston.
Kotz, S., Mahmound, H. and Robert, P. (2000). On generalized Pólya urn models,Statistics and Probability Letters,49, 163–173.
Ling, K. D. (1988). On binomial distributions of orderk, Statistics and Probability Letters,6, 247–250.
Ling, K. D. (1993). Sooner and later waiting time distributions for frequency quota defined on a Pólya-Eggenberger urn model,Soochow Journal of Mathematics,19, 139–151.
Mood, A. M. (1940). The distribution theory of runs,Annals of Mathematical Statistics,11, 367–392.
Sen, K. and Jain, R. (1997). A multivariate generalized Pólya-Eggenberger probability model-first passage approach,Communications in Statistics Theory and Methods,26, 871–884.
Shur, W. (1984). The negative contagion reflection of the Pólya-Eggenberger distribution,Communications in Statistics Theory and Methods,13, 877–885.
Tripsiannis, G. A., Philippou, A. N., and Papathanasiou, A. A. (2002). Multivariate generalized Pólya distribution of orderk, Communications in Statistics Theory and Methods,31, 1899–1912.
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This research was partially supported by the ISM Cooperative Research Program (2003-ISM·CRP-2007).
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Inoue, K., Aki, S. A generalized Pólya urn model and related multivariate distributions. Ann Inst Stat Math 57, 49–59 (2005). https://doi.org/10.1007/BF02506878
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DOI: https://doi.org/10.1007/BF02506878