Abstract
Making use of the peaks over threshold (POT) estimation method, we propose a semiparametric estimator for the renewal function of interoccurrence times of heavy‐tailed insurance claims with infinite variance. We prove that the proposed estimator is consistent and asymptotically normal, and we carry out a simulation study to compare its finite‐sample behavior with respect to the nonparametric one. Our results provide actuaries with confidence bounds for the renewal function of dangerous risks.
Highlights
Let X1, X2, . . . be independent and identically distributed iid positive random variables rvs, representing claim interoccurrence times of an insurance risk, with common distribution function df F having finite mean μ and variance σ2
The renewal theory has proved to be a powerful tool in stochastic modeling in a wide variety of applications such as reliability theory, where a renewal process is used to model the successive repairs of a failed machine see 1, risk theory, where a renewal process is used to model the successive occurrences of risks see 2, 3, inventory theory, where a renewal process is used to model the successive times between demand points see 4, manpower planning, where a renewal process is used to model the sequence of resignations from a given job see 5, and warranty analysis, where a renewal process is used to model the successive purchases of a new item following the expiry of a free-replacement warranty see 6
Statistical estimation of the renewal function has been considered in several ways
Summary
Let X1, X2, . . . be independent and identically distributed iid positive random variables rvs , representing claim interoccurrence times of an insurance risk, with common distribution function df F having finite mean μ and variance σ2. Let. 1.1 be the claim occurrence times, and define the number of claims recorded over the time interval 0, t by. The need for renewal function estimates seems more than pressing in many practical problems. Statistical estimation of the renewal function has been considered in several ways. Frees introduced two estimators based on the empirical counterparts of F and F k by suitably truncating the sum in 1.3. Zhao and Subba Rao proposed an estimation method based on the kernel estimate of the density and the renewal equation. A histogram-type estimator, resembling to the second estimator of Frees, was given by Markovich and Krieger
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