Nonparametric estimation of the expected discounted penalty function in the compound Poisson model

: We propose a nonparametric estimator of the expected dis- counted penalty function in the compound Poisson risk model. We use a projection estimator on the Laguerre basis and we compute the co- efficients using Plancherel theorem. We provide an upper bound on the MISE of our estimator, and we show it achieves parametric rates of conver- gence 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.