Statistical Estimation For Gerber-shiu Functions Under Some Risk Models

Risk theory provides the management strategies for insurance company on capital injection and shareholder dividend,and improves the operation capability and competitiveness.Gerber,H.U.and Shiu,E.S.W used expected discounted penalty function to do research on general problems about risk measure.Their efforts opened a new page for risk theory and their achievements became the standard of research on this field.After that,many scholars obtained the explicit formulae of risk measure under special circumstances by inverse Laplace transform,Martingale theory,renewal theory,and so on.But in fact,the distribution of the claim is not available for the insurance company.Therefore,it is importance for us to figure out the expected discounted penalty function by nonparametric methods.In this thesis,we study how to estimate the expected discounted penalty function by Laguerre series expansion method.We consider Laguerre series expansions of the expected discounted penalty function under the classical risk model,the risk model perturbed by a Brownian motion,the risk model with stochastic income and the pure-jump Lévy risk model.We show that the expected discounted function is square-integrable via the renewal equation satisfied by itself.Thus,it is feasible and reasonable to expand the expected discounted penalty function by Laguerre series.Expanding the renewal equation by Laguerre series and comparing coefficients of Laguerre basis on both sides of the equation gives a system of linear equations of Laguerre series.We write the equations in matrix form for the convenience of calculation.The matrix of coefficients is verified to be a lower-triangle,invertible Toeplitz matrix,so that the coefficients of the Laguerre series of the expected discounted penalty function can be obtained.An approximate expression of the expected discounted penalty function can be obtained by truncating the infinite series.Then,we can estimate the expected discount penalty function by the observed data of the surplus process.For different risk models,we use L²-norm to measure the errors.Using the Sobolev-Laguerre space,the convergence rate of series truncation error is given.Using Markov inequality,L’Hospital’s rule and other tools,the convergence rate of the estimated error is given.We finally obtain the convergence rate of the Laguerre series expansion method.We select the optimal truncation coefficient and find that the errors can obtain its optimal convergence rate-O_p^〔T^1/2〕.By comparing the error convergence rates of nonparametric estimation methods under different models in other papers,we find that the Laguerre series expansion method has the faster convergence rate.In the classical risk model,we study the asymptotic normality of the Laguerre series expansion method when the interest force is 0.Numerical experiments are carried out to illustrate the performance of the Laguerre series expansion method when the sample size is finite.By studying the mean value,the mean relative error and the integrated mean square error（IMSE）,we find that the Laguerre series expansion method performs better as the observe interval increases.Furthermore,the superiority of Laguerre series expansion method is illustrated by comparing with Fourier-Sinc series expansion method and Fast Fourier transform method.From analysis,we find that Laguerre series expansion method overcomes many weaknesses of nonparametric method（Inverse Laplace Transform,Fast Fourier Transform,Fourier-Sinc series expansion,etc.）,such as the complexity of computation and the slowly convergent rate.Meanwhile,Laguerre series expansion method has a general adaptability in the study of risk theory,and has a good approximation effect.And it has the advantages of simple calculation and fast convergence.