X. Sheldon Lin

Lundberg Approximations for Compound Distributions with Insurance Applications

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.

LUNDBERG INEQUALITIES FOR RENEWAL EQUATIONS

Sharp upper and lower bounds are derived for the solution of renewal equations. These include as special cases exponential inequalities, some of which have been derived for specific renewal equations. Together with the well-known Cramér-Lundberg asymptotic estimate, these bounds give additional information about the behaviour of the solution. Nonexponential bounds, which are of use in connection with defective renewal equations, are also obtained. The results are then applied in examples involving the severity of insurance ruin, age-dependent branching processes, and a generalized type II Geiger counter.

Collective Risk Models

This article reviews models for aggregate claims arising from an insurance portfolio when policy replacements are allowed. It presents various models and approaches to the evaluation of aggregate claims. Keywords: aggregate claims; claim frequency; (a,b,1) class; claim severity; compound distribution; mixed Erlang; parametric approximation