# Nonparametric estimation in random sum models

## DOI:

https://doi.org/10.6092/issn.1973-2201/3549## Abstract

Let *X*1,*X*2,…,*X**N *be independent, identically distributed, non-negative, integervalued random variables and let *N *be a non-negative, integer-valued random variable independent of *X*1,*X*2,…,*X**N *. In this paper, we consider two nonparametric estimation problems for the random sum variable. The first is the estimation of the means of *X**i *and *N *based on the second-moment assumptions on distributions of *X**i *and *N *. The second is the nonparametric estimation of the distribution of *X**i *given a parametric model for the distribution of *N *. Some asymptotic properties of the proposed estimators are discussed.

## References

H. S. BAKOUCH, M. M. RISTIĆ (2009), Zero truncated Poisson integer-valued AR(1) model, “Metrika”, doi 10.1007/s00184-009-0252-5.

G. BUCHMANN, R. GRÜBEL (2003), Decompounding: an estimation problem for Poisson random sums, “Annals of Statistics”, 31, pp. 1054-1074.

G. BUCHMANN, R. GRÜBEL (2004), Decompounding Poisson random sums: Recursively truncated estimates in the discrete case, “Annals of the Institute of Statistical Mathematics”, 56, pp. 743-756.

J. CAI, G. E. WILLMOT (2005), Monotonicity and aging properties of random sums, “Statistics & Probability Letters”, 73, pp. 381-392.

C. A. CHARALAMBIDES (2005), Combinatorial methods in discrete distributions, New York: Wiley.

C. CHATFIELD, C. M. THEOBALD (1973), Mixtures and random sums, “The Statistician”, 22, pp. 281-287.

S. A. KLUGMAN, H. H. PANJER, G. E. WILLMOT (2004), Loss models: from data to decisions, 2nd ed., Wiley-Interscience.

O. LUNDBERG (1964), On random processes and their application to sickness and accident statistics, 2nd ed., Almqvist and Wiksell (Uppsala).

P. MCCULLAGH, J. A. NELDER (1989), Generalized linear models, 2nd ed., Chapman & Hall/CRC.

J. N. MEDHI (1972), Stochastic processes, 3rd ed., John Wiley: New York.

H. H. PANJER (2006), Operational risk: modeling analytics, New York: Wiley.

E. PEKÖZ, S. M. ROSS (2004), Compound random variables, “Probability in the Engineering and Informational Sciences”, 18, pp. 473-484.

S. M. PITTS (1994), Nonparametric estimation of compound distributions with applications in insurance, “Annals of the Institute of Statistical Mathematics”, 46, pp. 537-555.

M. M. RISTIĆ, H. S. BAKOUCH, A. S. NASTIĆ (2009), A new geometric first-order integer-valued autoregressive (NGINAR(1)) process, “Journal of Statistical Planning and Inference”, 139, pp. 2218-2226.

M. SAHINOGLU (1992), Compound Poisson software reliability model, “IEEE Transactions on Software Engineering”, 18, pp. 624-630.

T. A. SEVERINI (2005), Elements of distribution theory, Cambridge University Press: Cambridge.

M. WEBA (2007), Optimal prediction of compound mixed Poisson processes, “Journal of Statistical Planning and Inference”, 137, pp. 1332-1342.

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*Statistica*,

*69*(1), 73–88. https://doi.org/10.6092/issn.1973-2201/3549

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