Nonparametric Estimation for Optimal Dividend Varrier

Abstract

In the context of traditional risk theory for an insurance company, an important problem is ``dividend strategy", that is, how the portion where the insurance company's surplus exceeds a level of barrier will be paid to the shareholders as dividends. The optimal dividend barrier is defined as the level of the barrier that maximizes the expected discounted dividends until ruin. In this paper, based on the M-estimation method, we estimate the optimal dividend barrier from a sample path of the insurance company's surplus process. To show the consistency for the estimator, the uniformly convergence of the objective function is needed and this result is known as the Glivenko-Cantelli theorem. The Glivenko-Cantelli theorem describes the entropy measuring the magnitude of the functional family and how the family is easy to converge uniformly. We first show the boundedness of the uniform entropy for a family of objective functions and the uniform convergence. This result implies that the constructed M-estimator is consistent. To show the boundedness of the uniform entropy, it is shown that the family of objective functions belongs with VC-subgraph class. Finally, we confirm the uniform convergence of the objective function and the consistency of the optimal dividend barrier estimator through simulation.