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A kernel type nonparametric density estimator for decompounding. (English) Zbl 1129.62030
Summary: Given a sample from a discretely observed compound Poisson process, we consider estimation of the density of the jump sizes. We propose a kernel type nonparametric density estimator and study its asymptotic properties. An order bound for the bias and an asymptotic expansion of the variance of the estimator are given. Pointwise weak consistency and asymptotic normality are established. The results show that, asymptotically, the estimator behaves very much like an ordinary kernel estimator.

##### MSC:
 62G07 Density estimation 62M09 Non-Markovian processes: estimation 65C60 Computational problems in statistics (MSC2010) 62G20 Asymptotic properties of nonparametric inference
##### Keywords:
asymptotic normality; consistency; kernel estimation
KernSmooth
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##### References:
 [1] Buchmann, B. and Grübel, R. (2003). Decompounding: an estimation problem for Poisson random sums. Ann. Statist. 31 1054-1074. · Zbl 1105.62309 [2] Buchmann, B. and Grübel, R. (2004). Decompounding Poisson random sums: recursively truncated estimates in the discrete case. Ann. Inst. Statist. Math. 56 743-756. · Zbl 1078.62020 [3] Carr, P. and Madan, D.B. (1998). Option valuation using the Fast Fourier Transform. J. Comput. Finance 2 61-73. [4] Chung, K.L. (1974). A Course in Probability Theory . New York: Academic Press. · Zbl 0345.60003 [5] Devroye, L. (1991). Exponential inequalities in nonparametric estimation. In G. Roussas (ed.), Nonparametric Functional Estimation and Related Topics , pp. 31-44. Dordrecht: Kluwer. · Zbl 0739.62025 [6] Devroye, L. and Györfi, L. (1985). Nonparametric Density Estimation: The $$L_1$$ View . New York: Wiley. · Zbl 0546.62015 [7] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events . Berlin: Springer. · Zbl 0873.62116 [8] Finkelestein, M., Tucker, H.G. and Veeh, J.A. (1997). Extinguishing the distinguished logarithm problems. Proc. Amer. Math. Soc. 127 2773-2777. · Zbl 0930.60009 [9] Hansen, M.B. and Pitts, S.M. (2006). Decompounding random sums: a nonparametric approach. Report AALBORG-R-2006-20, Aalborg University. [10] Liu, M.C. and Taylor, R.L. (1989). A consistent nonparametric density estimator for the deconvolution problem. Canad. J. Statist. 17 427-438. · Zbl 0694.62017 [11] Prabhu, N.U. (1980). Stochastic Storage Processes: Queues, Insurance Risk and Dams . New York: Springer. · Zbl 0453.60094 [12] Prakasa Rao, B.L.S. (1983). Nonparametric Functional Estimation . Orlando: Academic Press. · Zbl 0542.62025 [13] Schwartz, L. (1966). Mathematics for the Physical Sciences . Paris: Hermann. · Zbl 0151.34001 [14] Serfling, R.J. (1980). Approximation Theorems of Mathematical Statistics . New York: Wiley. · Zbl 0538.62002 [15] Stefanski, L.A. and Caroll, R.J. (1990). Deconvoluting kernel density estimators. Statistics 21 169-184. · Zbl 0697.62035 [16] Tsybakov, A. (2004). Introduction à l’estimation non-parametrique . Berlin: Springer. [17] van Es, B., Gugushvili, S. and Spreij, P. (2006). A kernel type nonparametric density estimator for decompounding. Available at http://arxiv.org/abs/math.ST/0505355. · Zbl 1129.62030 [18] Wand, M.P. (1998). Finite sample performance of deconvolving density estimators. Statist. Probab. Lett. 37 131-139. · Zbl 0886.62048 [19] Wand, M.P. and Jones, M.C. (1995). Kernel Smoothing . London: Chapman & Hall. · Zbl 0854.62043
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