Asymptotic statistics in insurance risk theory
type: Item
instance of: scholarly article
title: Asymptotic statistics in insurance risk theory
DOI: 10.1007/978-981-16-9284-0
author: Yasutaka Shimizu
published in: SpringerBriefs in Statistics
publication date: 2022-02-09T00:00:00Z
review text: The author presents a family of results establishing the consistency and asymptotic normality of estimators for the ruin probability and the Gerber-Shiu function for the Lévy surplus model. The book focuses on theory, numerical examples illustrating the workings of the results in practice are not given. In Chapter 1, ruin probability and the Gerber-Shiu function are introduced and calculated for the diffusion perturbation of the Cramér-Lundberg model \[ X_{t} = u + c\cdot t + \sigma \cdot W_{t} - \sum_{i=1}^{N_{t}}U_{i} \] where \(u\) is the initial reserve, \(c\) is the premium rate, \(N_{t}\) is a Poisson process with intensity \(\lambda\), \(\sigma \in \mathbb{R}\), \(W_{t}\) is a Wiener process, and \(U_{i}\) are the independent, identically distributed insurance claim sizes. The main results presented in the book are for the case where the ruin probability \[ \psi(u) = \mathbb{P}[\inf\{t > 0\colon X_{t} < 0\} < \infty|X_{0} = u] \] satisfies the Lundberg approximation \(\psi(u) \sim e^{-\gamma \cdot u}\) as \(u \rightarrow \infty\) where \(\gamma\) is the adjustment coefficient, i.e.\ the positive root of \[ \lambda \cdot (m_{U}(\gamma)-1) - c \cdot \gamma + \frac{\sigma^{2}}{2}\cdot \gamma^{2} = 0 \] with \(m_{U}\) denoting the moment generating function of \(U_{i}\). Chapter 2 introduces Lévi processes, and extends the results of Chapter 1 on analytic formulas for the ruin probability and the Gerber-Shiu function to the Lévy surplus model \[ X_{t} = u + c\cdot t + Z_{t} \] where \(Z_{t}\) is a Lévy process. Chapter 3 discusses the concepts of statistical inference which are used later in the book: e.g., parametric and nonparametric estimators, weak and strong consistency, asymptotic normality, asymptotic confidence bounds, \(M\)- and \(Z\)-estimators, and the Delta Method. In Chapter 4 it is shown -- under some regularity conditions -- that an asymptotically normal parametric estimator for the claim distribution can be used to construct an asymptotically norm…
reviewed by: Tamás Mátrai
Mathematics Subject Classification ID: 62-02
Mathematics Subject Classification ID: 60G51
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 62G20
Mathematics Subject Classification ID: 62P05
Mathematics Subject Classification ID: 91-02
Mathematics Subject Classification ID: 91G05
Mathematics Subject Classification ID: 91G70
zbMATH DE Number: 7471659
zbMATH Keywords: Cramér-Lundberg model
zbMATH Keywords: Lévy surplus model
zbMATH Keywords: ruin probability
zbMATH Keywords: Gerber-Shiu function
zbMATH Keywords: defective renewal equation
zbMATH Keywords: Lévy process
zbMATH Keywords: statistical inference
zbMATH Keywords: parametric estimator
zbMATH Keywords: nonparametric estimator
zbMATH Keywords: empirical estimator
zbMATH Keywords: consistent estimator
zbMATH Keywords: asymptotic normality
zbMATH Keywords: asymptotic confidence bound
zbMATH Keywords: Delta Method
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-981-16-9284-0
OpenAlex ID: W4207074184
description: scientific article; zbMATH DE number 7471659