On some compound distributions with Borel summands. (English) Zbl 1320.60041
Summary: The generalized Poisson distribution is well known to be a compound Poisson distribution with Borel summands. As a generalization we present closed formulas for compound Bartlett and Delaporte distributions with Borel summands and a recursive structure for certain compound shifted Delaporte mixtures with Borel summands. Our models are introduced in an actuarial context as claim number distributions and are derived only with probabilistic arguments and elementary combinatorial identities. In the actuarial context related compound distributions are of importance as models for the total size of insurance claims for which we present simple recursion formulas of Panjer type.
MSC:
| 60E05 | Probability distributions: general theory |
| 05A19 | Combinatorial identities, bijective combinatorics |
| 62P05 | Applications of statistics to actuarial sciences and financial mathematics |
| 91B30 | Risk theory, insurance (MSC2010) |
Keywords:
generalized Poisson distribution; delaporte distribution; claim number distribution; Lagrangian probability distribution; recursive evaluation; aggregate claim distribution; Panjer recursion; multinomial Abel identityReferences:
| [1] | Albrecht, P., Laplace transforms, Mellin transforms and mixed Poisson process, Scand. Actuar. J., 58-64 (1984) · Zbl 0544.60081 |
| [2] | Ambagaspitiya, R. S.; Balakrishnan, N., On the compound generalized Poisson distribution, ASTIN Bull., 24, 255-263 (1994) |
| [3] | Asmussen, S.; Albrecher, H., Ruin Probabilities (2010), World Scientific: World Scientific New Jersey · Zbl 1247.91080 |
| [4] | Bartlett, M. S., Distributions associated with cell populations, Biometrika, 56, 391-400 (1969) · Zbl 0179.25402 |
| [5] | Chakraborty, S., On some new \(\alpha \)-modified binomial and Poisson distributions and their applications, Comm. Statist. Theory Methods, 37, 1755-1769 (2008) · Zbl 1318.62035 |
| [6] | Consul, P. C., Generalized Poisson Distributions (1989), Dekker: Dekker New York · Zbl 0691.62015 |
| [7] | Consul, P. C.; Famoye, F., Lagrangian Probability Distributions (2006), Birkhäuser: Birkhäuser Boston · Zbl 1072.60004 |
| [8] | Consul, P. C.; Jain, G. C., A generalization of the Poisson distribution, Technometrics, 15, 791-799 (1973) · Zbl 0271.60020 |
| [9] | Delaporte, P., Quelques problémes de statistique mathematique posés par l’assurance automobile et le bonus pour non sinistre, Bull. Trimest. Inst. Actuaires Fr., 227, 87-102 (1959) |
| [10] | Dempster, A. P., Generalized \(D_n^+\) statistics, Ann. Math. Statist., 30, 593-597 (1959) · Zbl 0122.37003 |
| [11] | Dickson, D. C.M., A review of Panjer’s recursion formula and its applications, Br. Actuar. J., 1, 107-124 (1995) |
| [12] | Embrechts, P.; Frei, M., Panjer recursion versus FFT for compound distributions, Math. Methods Oper. Res., 69, 497-508 (2009) · Zbl 1205.91081 |
| [13] | Feller, W., An Introduction to Probability Theory and its Applications, Vol. I (1968), Wiley: Wiley New York · Zbl 0155.23101 |
| [14] | Finner, H.; Roters, M., On the false discovery rate and expected type I errors, Biom. J., 43, 985-1005 (2001) · Zbl 0989.62061 |
| [15] | Gerber, H. U., When does the surplus reach a given target?, Insurance Math. Econom., 9, 115-119 (1990) · Zbl 0731.62153 |
| [16] | Goovaerts, M. J.; Kaas, R., Evaluating compound generalized Poisson distributions recursively, ASTIN Bull., 21, 193-198 (1991) |
| [17] | Haight, F. A.; Breuer, M. A., The Borel-Tanner distribution, Biometrika, 47, 143-150 (1960) · Zbl 0117.14001 |
| [18] | Jain, G. C., A linear function Poisson distribution, Biom. Z., 17, 501-506 (1975) · Zbl 0322.60016 |
| [19] | Janardan, K. G., Weighted Lagrange distributions and their characterizations, SIAM J. Appl. Math., 47, 411-415 (1987) · Zbl 0623.62008 |
| [20] | Joe, H.; Zhu, R., Generalized Poisson distributions: the property of mixture of Poisson and comparison with negative binomial distribution, Biom. J., 47, 219-229 (2005) · Zbl 1442.62431 |
| [21] | Johnson, N. L.; Kemp, A. W.; Kotz, S., Univariate Discrete Distributions (2005), Wiley: Wiley Hoboken · Zbl 1092.62010 |
| [22] | Kupper, J., Wahrscheinlichkeitstheoretische Modelle in der Schadenversicherung I: Die Schadenzahl, Bl. DGVFM, 5, 451-503 (1960) · Zbl 0108.33002 |
| [23] | Lerner, B.; Lone, A.; Rao, M., On generalized Poisson distributions, Probab. Math. Statist., 17, 377-385 (1997) · Zbl 0902.60015 |
| [24] | Lüders, R., Die Statistik der seltenen Ereignisse, Biometrika, 26, 108-128 (1934) · JFM 60.1180.05 |
| [25] | Medhi, J.; Borah, M., On generalized four-parameter Charlier distribution, J. Statist. Plann. Inference, 14, 69-77 (1986) · Zbl 0601.60012 |
| [26] | Mikosch, T., Non-Life Insurance Mathematics (2009), Springer: Springer Berlin · Zbl 1166.91002 |
| [27] | Pakes, A. G., Lambert’s W, infinite divisibility and Poisson mixtures, J. Math. Anal. Appl., 378, 480-492 (2011) · Zbl 1223.60016 |
| [28] | Panjer, H. H., Recursive evaluation of a family of compound distributions, ASTIN Bull., 12, 22-26 (1981) |
| [29] | Panjer, H. H., Discrete parametric distributions, (Teugels, J. L.; Sundt, B., Encyclopedia of Actuarial Science, Vol. I (2006), Wiley: Wiley New York) |
| [30] | Philipson, C., The theory of confluent hypergeometric functions and its application to Poisson processes, Skand. Aktuarietidskr., 43, 136-162 (1960) · Zbl 0107.35203 |
| [31] | Pólya, G.; Szegő, G., Problems and Theorems in Analysis I (1972), Springer: Springer Berlin · Zbl 0236.00003 |
| [32] | Riordan, J., Combinatorial Identities (1968), Wiley: Wiley New York · Zbl 0194.00502 |
| [33] | Ruohonen, M., On a model for the claim number process, ASTIN Bull., 18, 57-68 (1988) |
| [34] | Scheer, M., Controlling the number of false rejections in multiple hypotheses testing (2012), Univ. Düsseldorf, Available at http://docserv.uni-duesseldorf.de/servlets/DocumentServlet?id=23691 |
| [35] | Schröter, K. J., On a family of counting distributions and recursion for related compound distributions, Scand. Actuar. J., 1990, 161-175 (1990) · Zbl 0746.62102 |
| [36] | Schröter, K. J., Verfahren zur Approximation der Gesamtschadenverteilung (1995), VVW: VVW Karlsruhe |
| [37] | Steliga, K.; Szynal, D., On elementary characterizations of the \(\alpha \)-modified Poisson distribution, Probab. Math. Statist., 32, 215-225 (2012) · Zbl 1261.60021 |
| [38] | Sundt, B., On some extensions of Panjer’s class of counting distributions, ASTIN Bull., 22, 61-80 (1992) |
| [39] | Willmot, G. E., Limiting tail behaviour of some discrete compound distributions, Insurance Math. Econom., 8, 175-185 (1989) · Zbl 0686.62092 |
| [40] | Willmot, G. E.; Sundt, B., On evaluation of the Delaporte distribution and related distributions, Scand. Actuar. J., 101-113 (1989) · Zbl 0721.62019 |
| [41] | Wimmer, G.; Altmann, G., Thesaurus of Univariate Discrete Probability Distributions (1999), Stamm: Stamm Essen |
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