Abstract
Nonparametric estimation of the Gerber-Shiu function is a popular topic in insurance risk theory. Zhang and Su (2018) proposed a novel method for estimating the Gerber-Shiu function in classical insurance risk model by Laguerre series expansion based on the claim number and claim sizes of sample. However, whether the estimators are asymptotically normal or not is unknown. In this paper, we give the details to verify the asymptotic normality of these estimators and present some simulation examples to support our result.
Highlights
In this paper, we consider the classical risk model NtUt = u + ct − ∑ Xi, t ≥ 0, i =1 where u ≥ 0 denotes the initial surplus level and c > 0 denotes the premium rate
We introduce some lemmas for the asymptotic normality of Laguerre coefficients
To check the asymptotic normality of the estimators, we present quantile-quantile plot (QQ-plot)
Summary
We verify asymptotically normal estimators of the Gerber-Shiu function when δ = 0. The Gerber-Shiu function has been studied by Yin and Wang [11], Shen et al [12], Yin and Yuen [13], Cai et al [14], Deng et al [15], Dong et al [16], Wang and Zhang [17], Yu et al [18], Peng et al [19] among others In all of these mentioned papers, they assumed that the parameter λ and the claim size density f are known. Laplace transform to estimate the Gerber-Shiu function in Lévy risk model; Zhang and Yang [21,22].
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