Abstract
A new type of mixtures of discrete probability distributions is presented. A family of discrete averaged mixed distributions is introduced. Its subclass of averaged mixed logarithmic distributions is analyzed. Probabilistic characterizations and connections with other types of mixing are derived. We show also some examples of the analyzed distributions found in literature.
Dedicated to Professor Anatolij Dvurečenskij on the occasion of his 65th birthday
(Communicated by Sylvia Pulmannová)
Supported by the grant No. 2/0047/15 of the Grant Agency VEGA (G. Wimmer, J. Mačutek) and grant No. APVV-15-0295 of the Grant Agency APVV (G. Wimmer).
References
[1] Bunge, M.: Treatise on Basic Philosophy. Epistemology and Methodology II: Understanding the World, Reidel, Dordrecht, 1983.10.1007/978-94-015-6921-7Search in Google Scholar
[2] Cane, V.: Mathematical models for neural networks. In: Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability 4, Math. Sci. Publ., Berkeley, CA, 1967, pp. 21–26.Search in Google Scholar
[3] Elderton, W. P.—Johnson, N. L.: Systems of Frequency Curves, Cambridge University Press, Cambridge, 2008.Search in Google Scholar
[4] Gross, D.—Harris, C. M.: Fundamentals of Queueing Theory, Wiley, New York, 1985.Search in Google Scholar
[5] Holgate, P.: A modified geometric distribution arising in trapping studies, Acta Theriologica 9 (1964), 353–356.10.4098/AT.arch.64-36Search in Google Scholar
[6] Holgate, P.: Contribution to the mathematics of animal trapping, Biometrics 22 (1966), 925–936.10.2307/2528082Search in Google Scholar
[7] Jackson, R. R. P.—Nichols, D. G.: Some equilibrium results for the queueing process 𝓁k/M/1, J. R. Stat. Soc. Ser. B Stat. Methodol. 18 (1956), 275–279.10.1111/j.2517-6161.1956.tb00234.xSearch in Google Scholar
[8] Johnson, N. L.—Kemp, A. W.—Kotz, S.: Univariate Discrete Distributions, Wiley, Hoboken, NJ, 2005.10.1002/0471715816Search in Google Scholar
[9] Kemp, A. W.: Convolutions involving binomial pseudovariables, Sankhyāá 41 (1979), 232–243.Search in Google Scholar
[10] Kobayashi, H.: Stochastic modelling: Queueing models. In: Probability Theory and Computer Science (G. Louchard, G. Latouche, eds.), Academic Press, London, 1983, pp. 53–121.Search in Google Scholar
[11] Kopp-Schneider, R. A.: Birth-death process with piecewise constant rates, Statist. Probab. Lett. 13 (1992), 121–127.10.1016/0167-7152(92)90086-KSearch in Google Scholar
[12] Lotka, A. J.: Théorie analytique des associations biologiques II. Acualités scientifiques et industrielles 780, 1939.Search in Google Scholar
[13] Lwin, T.: A modified power series distribution, Ann. Inst. Statist. Math. (Tokyo) 33 (1981), 361–374.10.1007/BF02480947Search in Google Scholar
[14] Manly, B. F. J.—Seber, G. A. F.: Animal life tables from capture-recapture data, Biometrics 29 (1973), 487–500.10.2307/2529172Search in Google Scholar
[15] Osawa, H.: Reversibility of Markov chains with applications to storage models, J. Appl. Probab. 22 (1985), 123–137.10.2307/3213752Search in Google Scholar
[16] Panaretos, J.: On Moran’s property of the Poisson distribution, Biom. J. 25 (1983), 69–76.10.1002/bimj.19830250108Search in Google Scholar
[17] Plunkett, I. G.—Jain, G. C.: Three generalized negative binomial distributions, Biometrische Zeitschrift 17 (1975), 286–302.10.1002/bimj.19750170503Search in Google Scholar
[18] Prabhu, N. V.: Queues and Inventories, Wiley, New York, 1965.Search in Google Scholar
[19] Rubin, H.—Vere-Jones, D.: Domains of attraction for the subcritical Galton-Watson branching process, J. Appl. Probab. 5 (1968), 216–219.10.2307/3212089Search in Google Scholar
[20] Rubinovitch, M.: The slow server problem: A queue with stalling, J. Appl. Probab. 22 (1985), 879–892.10.2307/3213955Search in Google Scholar
[21] Tin, P.: A queueing system with Markov-dependent arrivals, J. Appl. Probab. 22 (1985), 668–677.10.1017/S0021900200029417Search in Google Scholar
[22] Vadzinskii, R. N.: Spravochnik po veroyatnostnym raspredeleniyam, Nauka, Sankt-Peterburg, 2001.Search in Google Scholar
[23] Wimmer, G.—Altmann, G.: The Multiple Poisson distribution, its characteristics and variety of forms, Biom. J. 38 (1996), 995–1011.10.1002/bimj.4710380811Search in Google Scholar
[24] Wimmer, G.—Altmann, G.: Thesaurus of univariate discrete probability distributions, Stamm, Essen, 1999.Search in Google Scholar
[25] Wimmer, G.—Altmann, G.: Unified derivation of some linguistic laws. In: Quantitative Linguistics. An International Handbook (R. Köohler, G. Altmann, R. G. Piotrowski, eds.), de Gruyter, Berlin-New York, 2005, pp. 791-807.Search in Google Scholar
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