Lundberg Approximations for Compound Distributions with Insurance Applications

TLDR: In this article, the authors consider the problem of estimating the residual lifetime distribution and its mean for a set of classes of distributions and derive bounds on the ratio of discrete tail probabilities.

Abstract: 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.