Abstract
A compound distribution is the distribution of a random sum, which consists of a random number N of independent identically distributed summands, independent of N. Buchmann and Grübel (Ann Stat 31:1054–1074, 2003) considered decompounding a compound Poisson distribution, i.e. given observations on a random sum when N has a Poisson distribution, they constructed a nonparametric plug-in estimator of the underlying summand distribution. This approach is extended here to that of general (but known) distributions for N. Asymptotic normality of the proposed estimator is established, and bootstrap methods are used to provide confidence bounds. Finally, practical implementation is discussed, and tested on simulated data. In particular we show how recursion formulae can be inverted for the Panjer class in general, as well as for an example drawn from the Willmot class.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Asmussen S. (2003). Applied probability and queues (2 edn). Springer, New York
Bingham N.H. and Pitts S.M. (1999a). Nonparametric estimation for the M/G/∞ queue. Annals of the Institute of Statistical Mathematics 51: 71–97
Bingham N.H. and Pitts S.M. (1999b). Nonparametric inference from M/G/1 busy periods. Communications in Statistics and Stochastic Models 15: 247–272
Buchmann B. and Grübel R. (2003). Decompounding: An estimation problem for Poisson random sums. Annals of Statistics 31: 1054–1074
Buchmann B. and Grübel R. (2004). Decompounding Poisson random sums: Recursively truncated estimates in the discrete case. Annals of the Institute of Statistical Mathematics 56: 743–756
Bultheel A. and Martínez-Sulbaran H. (2006). Recent developments in the theory of the fractional fourier and linear canonical transforms. Bulletin of the Belgian mathematical society—Simon Stevin 13: 971–1005
Copson E.T. (1935). An introduction to the theory of functions of a complex variable. The Clarendon Press, Oxford
Delaporte P. (1959). Quelques problèmes de statistiques mathématiques posés par l’assurance automobile et le bonus pour non sinistre. Bulletin Trimestriel de l’Institut des Actuaires Français 227: 87–102
Dickson D.C.M. (2005). Insurance risk and ruin. Cambridge University Press, Cambridge
Gill R. (1989). Non- and semi-parametric maximum likelihood estimators and the von Mises method (Part I). Scandinavian Journal of Statistics 16: 97–128
Gorenflo R. and Hofmann B. (1994). On autoconvolution and regularization. Inverse Problems 10: 353–373
Grandell J. (1997). Mixed poisson processes. Chapman and Hall, London
Grübel R. and Pitts S.M. (1993). Nonparametric estimation in renewal theory I: the empirical renewal function. Annal of Statistics 21: 1431–1451
Hall P. and Park J. (2004). Nonparametric inference about service time distribution from indirect measurements. Journal of the Royal Statistical Society, Series B 66: 861–875
Hansen M.B. and Pitts S.M. (2006). Nonparametric inference from the M/G/1 workload. Bernoulli 12: 737–759
Henrici P. (1974). Applied and Computational Complex Analysis, Vol. 1: Power series—integration— conformal mapping—location of zeros. Wiley, New York
Hohn, N., Veitch, D. (2003). Inverting sampled traffic. In Proceedings of the 3rd ACM SIGCOMM conference on Internet measurement, Miami Beach, USA (pp. 222–223). New York: ACM Press.
Johnson N.L., Kemp A.W. and Kotz S. (2005). Univariate discrete distributions (3 edn). Wiley, New York
Neyman J. and Scott E.L. (1958). A statistical approach to problems of cosmology. Journal of the Royal Statistical Society, Series B 20: 1–43
Panjer H.H. (1981). Recursive evaluation of a family of compound distributions. Astin Bulletin 12: 22–26
Pollard D. (1984). Convergence of stochastic processes. Springer, New York
Redheffer R.M. (1962). Reversion of power series. American Mathematical Monthly 69: 423–425
Ruohonen M. (1988). On a model for the claim number process. Astin Bulletin 18: 57–68
Schröter K. (1990). On a family of counting distributions and recursions for related compound distributions. Scandinavian Actuarial Journal 304: 161–175
Vaart A.W. (1998). Asymptotic Statistics. Cambridge University Press, Cambridge
Wellner J.A. and Vaart A.W. (1996). Weak convergence and empirical processes. Springer, New York
Gugushvili S., Spreij P. and Es B. (2007). A kernel type nonparametric density estimator for decompounding. Bernoulli 13: 672–694
Jongbloed G., Es B. and Zuijlen M. (1998). Isotonic inverse estimators for nonparametric deconvolution. Annals of Statistics 26: 2395–2406
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Bøgsted, M., Pitts, S.M. Decompounding random sums: a nonparametric approach. Ann Inst Stat Math 62, 855–872 (2010). https://doi.org/10.1007/s10463-008-0200-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-008-0200-6