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On some compound distributions with Borel summands
Affiliation:1. Institute of Biometrics and Epidemiology, German Diabetes Center at the Heinrich-Heine-Universität Düsseldorf, Auf’m Hennekamp 65, D-40225 Düsseldorf, Germany;2. Mathematical Institute, Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, D-40225 Düsseldorf, Germany;1. Department of Statistics, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom;2. Department of Applied Finance and Actuarial Studies, Faculty of Business and Economics, Macquarie University, Sydney NSW 2109, Australia;3. Department of Finance, School of Economics & Wang Yanan Institute for Studies in Economics, Xiamen University, Xiamen, Fujian 361005, China;4. MOE Key Laboratory of Econometrics, Xiamen University, Xiamen, Fujian 361005, China;1. Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 203, Barrackpore Trunk Road, Kolkata 700108, India;2. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, United States;1. Department of Mathematics, University of California, Irvine CA 92697, USA;2. Department of Ecology and Evolutionary Biology, University of California, Irvine CA 92697, USA;1. School of Statistics, Qufu Normal University, Qufu 273165, China;2. School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China
Abstract: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.
Keywords:Generalized Poisson distribution  Delaporte distribution  Claim number distribution  Lagrangian probability distribution  Recursive evaluation  Aggregate claim distribution  Panjer recursion  Multinomial Abel identity
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