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2022 Nonparametric estimation of the expected discounted penalty function in the compound Poisson model
Florian Dussap
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Electron. J. Statist. 16(1): 2124-2174 (2022). DOI: 10.1214/22-EJS2003

Abstract

We propose a nonparametric estimator of the expected discounted penalty function in the compound Poisson risk model. We use a projection estimator on the Laguerre basis and we compute the coefficients using Plancherel theorem. We provide an upper bound on the MISE of our estimator, and we show it achieves parametric rates of convergence on Sobolev–Laguerre spaces without needing a bias-variance compromise. Moreover, we compare our estimator with the Laguerre deconvolution method. We compute an upper bound of the MISE of the Laguerre deconvolution estimator and we compare it on Sobolev–Laguerre spaces with our estimator. Finally, we compare these estimators on simulated data.

Funding Statement

This work was supported by a grant from Région Île-de-France.

Acknowledgments

I want to thank Fabienne Comte and Céline Duval for their helpful advice and their support of my work. I also want to thank fedja for their help with Lemma 3.4 regarding the primitives of the Laguerre functions, and Emmanuel Rio for his hints for Lemma 7.1.

Citation

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Florian Dussap. "Nonparametric estimation of the expected discounted penalty function in the compound Poisson model." Electron. J. Statist. 16 (1) 2124 - 2174, 2022. https://doi.org/10.1214/22-EJS2003

Information

Received: 1 April 2021; Published: 2022
First available in Project Euclid: 29 March 2022

Digital Object Identifier: 10.1214/22-EJS2003

Subjects:
Primary: 62G05 , 62P05
Secondary: 91G70

Keywords: classical risk model , Fourier inversion , Gerber-Shiu function , Laguerre series expansion , nonparametric estimator , rate of estimation

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Vol.16 • No. 1 • 2022
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