Convergence and asymptotic variance of bootstrapped finite-time ruin probabilities with partly shifted risk processes

Abstract

Abstract. The classical risk model is considered and a sensitivity analysis of finite-time ruin probabilities is carried out. We prove the weak convergence of a sequence of empirical finite-time ruin probabilities. So-called partly shifted risk processes are introduced, and used to derive an explicit expression of the asymptotic variance of the considered estimator. This provides a clear representation of the influence function associated with finite time ruin probabilities, giving a useful tool to quantify estimation risk according to new regulations.

Asymptotic behavior of the finite-time expected time-integrated negative part of some risk processes and optimal reserve allocation

“…The dependent case has been studied by [7,8]. Sensitivity analysis has been carried out by [9][10][11].…”

Exponential martingale for compound Poisson process with latent variable and its applications

“…In this section, we assume ξ ∼ U(0, 1). Process (17) is usually called partly shifted risk process ( [8]). The part xI {U≤t} can be explained as follows: at a random time U , the insurance company receives x unit extra claims.…”

“…Xiong and Yang ([11]) consider the ruin probability of the Cramér-Lundberg model with investment in an asset with large volatility. Loisel, Mazza and Rullière ( [8]) have discussed the partly shifted risk process, they give the finite time ruin probability for the process by the ballot theorem. In their consideration, the Poisson process N t is homogeneous, which is a particular case of our consideration with P (ξ = 1) = 1, μ(u) = μ being a constant.…”

The chain rule for functionals with applications to functions of moments

“…Other recent application areas of von Mises derivatives include: least squares support vector regression filtering methods, bootstrapping, functional principal components analysis, linearization and composite estimation, dimension reduction, quantile regressions, machine learning, cusum statistics, methods of sieves and penalization, change point estimation, Hadamard differentiability, change-of-variance function, measuring and testing dependence by correlation of distances, empirical finite-time ruin probabilities (Loisel et al, 2009), nonparametric maximum likelihood estimators (Nickl, 2007), estimating mean dimensionality of analysis of variance decompositions, monotonicity of information in the Central Limit Theorem, generalizations of the Anderson-Darling statistic, M -estimation, U -statistics (Volodko, 2011), information criteria in model selection, goodness-of-fit tests for kernel regression, empirical Bayes estimation, and estimation of Kendall's tau.…”