Lundberg Approximations for Compound Distributions with Insurance Applications

Overview

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. Consequently, the material is of interest to researchers and graduate students interested in these areas. The material is self-contained, but an introductory course in insurance risk theory is ...

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Overview

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. Consequently, the material is of interest to researchers and graduate students interested in these areas. The material is self-contained, but an introductory course in insurance risk theory is beneficial to prospective readers. Lundberg asymptotics and bounds have a long history in connection with ruin probabilities and waiting time distributions in queueing theory, and have more recently been extended to compound distributions. This connection has its roots in the compound geometric representation of the ruin probabilities and waiting time distributions. A systematic treatment of these approximations is provided, drawing heavily on monotonicity ideas from reliability theory. The results are then applied to the solution of defective renewal equations, analysis of the time and severity of insurance ruin, and renewal risk models, which may also be viewed in terms of the equilibrium waiting time distribution in the G/G/1 queue. Many known results are derived and extended so that much of the material has not appeared elsewhere in the literature. A unique feature involves the use of elementary analytic techniques which require only undergraduate mathematics as a prerequisite. New proofs of many results are given, and an extensive bibliography is provided. Gordon Willmot is Professor of Statistics and Actuarial Science at the University of Waterloo. His research interests are in insurance risk and queueing theory. He is an associate editor of the North American Actuarial Journal.

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Product Details

  • ISBN-13: 9780387951355
  • Publisher: Springer New York
  • Publication date: 10/23/2000
  • Series: Lecture Notes in Statistics Series , #156
  • Edition description: Softcover reprint of the original 1st ed. 2001
  • Edition number: 1
  • Pages: 250
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.55 (d)

Table of Contents

1 Introduction.- 2 Reliability background.- 2.1 The failure rate.- 2.2 Equilibrium distributions.- 2.3 The residual lifetime distribution and its mean.- 2.4 Other classes of distributions.- 2.5 Discrete reliability classes.- 2.6 Bounds on ratios of discrete tail probabilities.- 3 Mixed Poisson distributions.- 3.1 Tails of mixed Poisson distributions.- 3.2 The radius of convergence.- 3.3 Bounds on ratios of tail probabilities.- 3.4 Asymptotic tail behaviour of mixed Poisson distributions.- 4 Compound distributions.- 4.1 Introduction and examples.- 4.2 The general upper bound.- 4.3 The general lower bound.- 4.4 A Wald-type martingale approach.- 5 Bounds based on reliability classifications.- 5.1 First order properties.- 5.2 Bounds based on equilibrium properties.- 6 Parametric Bounds.- 6.1 Exponential bounds.- 6.2 Pareto bounds.- 6.3 Product based bounds.- 7 Compound geometric and related distributions.- 7.1 Compound modified geometric distributions.- 7.2 Discrete compound geometric distributions.- 7.3 Application to ruin probabilities.- 7.4 Compound negative binomial distributions.- 8 Tijms approximations.- 8.1 The asymptotic geometric case.- 8.2 The modified geometric distribution.- 8.3 Transform derivation of the approximation.- 9 Defective renewal equations.- 9.1 Some properties of defective renewal equations.- 9.2 The time of ruin and related quantities.- 9.3 Convolutions involving compound geometric distributions.- 10 The severity of ruin.- 10.1 The associated defective renewal equation.- 10.2 A mixture representation for the conditional distribution.- 10.3 Erlang mixtures with the same scale parameter.- 10.4 General Erlang mixtures.- 10.5 Further results.- 11 Renewal risk processes.- 11.1 General properties of the model.- 11.2 The Coxian-2 case.- 11.3 The sum of two exponentials.- 11.4 Delayed and equilibrium renewal risk processes.- Symbol Index.- Author Index.

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