Nonparametric Estimation of the Finite-Time Survival Probability with Zero Initial Capital in the Classical Risk Model

Abstract

In a classical risk model with zero initial capital and unknown claim-size distribution, we consider the statistical problem of estimating uniformly in t the (unknown) finite-time survival probability φ 0(t) at time t, given a sample of claim sizes. We construct an empirical estimator of the function φ 0(·) based on the sample of claim sizes, and using a functional approach we establish asymptotic statistical properties of our estimator with respect to supremum norm. We also consider numerical evaluation of finite-time survival probabilities and their empirical counterparts using the fast Fourier transform algorithm, and we carry out small-scale simulation studies of the behaviour of our estimator.