Lundberg approximations for compound distributions with insurance applications
type: Item
instance of: scholarly article
title: Lundberg approximations for compound distributions with insurance applications
zbMATH Open document ID: 0962.62099
author: Gordon E. Willmot
author: X. Sheldon Lin
published in: Lecture Notes in Statistics
publication date: 2000-12-19T00:00:00Z
review text: This monograph discusses Lundberg approximations for compound distributions with special emphasis on applications in insurance risk modeling. These distributions are somewhat awkward from an analytic standpoint, but play a central role in insurance and other areas of applied probability modeling such as queueing theory. The book represents the authors' summary of research that has been done in recent years on approximations and bounds. The basic technique employed in the derivation of many bounds is inductive, an approach that is motivated by arguments used by Sparre Andersen in connection with renewal risk models in insurance. The bounds themselves are motivated by the classical Lundberg exponential bounds which apply to ruin probabilities, and the connection to compound distributions is through the interpretation of the ruin probability as the tail probability of a compound geometric distribution. The results depend quite heavily on monotonicity ideas which are summarized in notions from reliability theory. As such, Ch. 2, Reliability background, is devoted to a discussion of those ideas which are relevant in the main part of the book. Ch. 3, Mixed Poisson distributions, considers properties of mixed Poisson distributions, an important modeling component of many insurance models. In Ch. 4, Compound distributions, basic properties of compound tails are considered. In general, closed form expressions for tails are not available, but a few examples are provided for which analytic results are obtainable. A particularly useful model for the incidence of claims in the context of claims inflation as well as `incurred but not reported claims' is the mixed Poisson process, which is also briefly discussed. Lundberg asymptotics for compound tails are presented. The main upper and lower bounds on compound tails are introduced. The derivation of these results involves mathematical induction, which is seen to be a simple yet powerful approach for this type of problem. The bound…
Mathematics Subject Classification ID: 62P05
Mathematics Subject Classification ID: 62-02
Mathematics Subject Classification ID: 91B30
Mathematics Subject Classification ID: 91-02
zbMATH DE Number: 1544810
zbMATH Keywords: insurance risk modeling
zbMATH Keywords: reliability
zbMATH Keywords: bounds on ratios of discrete tail probabilities
zbMATH Keywords: mixed Poisson distributions
zbMATH Keywords: asymptotic tail behaviour
zbMATH Keywords: compound distributions
zbMATH Keywords: Wald-type martingale approach
zbMATH Keywords: bounds based on reliability classifications
zbMATH Keywords: parametric bounds
zbMATH Keywords: compound geometric distributions
zbMATH Keywords: compound negative binomial distributions
zbMATH Keywords: Lundberg approximations
zbMATH Keywords: Tijms approximations
zbMATH Keywords: time of ruin
zbMATH Keywords: severity of ruin
zbMATH Keywords: mixture representations of conditional distributions
zbMATH Keywords: Erlang mixtures
zbMATH Keywords: delayed renewal risk processes
zbMATH Keywords: equilibrium renewal risk processes
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1544810