Lundberg Approximations for Compound Distributions with Insurance Applications

1. Front Matter

   Pages i-x

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2. Introduction

   • Gordon E. Willmot, X. Sheldon Lin

   Pages 1-5

3. Reliability background

   • Gordon E. Willmot, X. Sheldon Lin

   Pages 7-36

4. Mixed Poisson distributions

   • Gordon E. Willmot, X. Sheldon Lin

   Pages 37-49

5. Compound distributions

   • Gordon E. Willmot, X. Sheldon Lin

   Pages 51-80

6. Bounds based on reliability classifications

   • Gordon E. Willmot, X. Sheldon Lin

   Pages 81-91

7. Parametric Bounds

   • Gordon E. Willmot, X. Sheldon Lin

   Pages 93-105

8. Compound geometric and related distributions

   • Gordon E. Willmot, X. Sheldon Lin

   Pages 107-140

9. Tijms approximations

   • Gordon E. Willmot, X. Sheldon Lin

   Pages 141-149

10. Defective renewal equations

    • Gordon E. Willmot, X. Sheldon Lin

    Pages 151-181

11. The severity of ruin

    • Gordon E. Willmot, X. Sheldon Lin

    Pages 183-208

12. Renewal risk processes

    • Gordon E. Willmot, X. Sheldon Lin

    Pages 209-234

13. Back Matter

    Pages 235-252

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These notes represent our summary of much of the recent research that has been done in recent years on approximations and bounds that have been developed for compound distributions and related quantities which are of interest in insurance and other areas of application in applied probability. The basic technique employed in the derivation of many bounds is induc­ tive, an approach that is motivated by arguments used by Sparre-Andersen (1957) in connection with a renewal risk model in insurance. This technique is both simple and powerful, and yields quite general results. The bounds themselves are motivated by the classical Lundberg exponential bounds which apply to ruin probabilities, and the connection to compound dis­ tributions is through the interpretation of the ruin probability as the tail probability of a compound geometric distribution. The initial exponential bounds were given in Willmot and Lin (1994), followed by the nonexpo­ nential generalization in Willmot (1994). Other related work on approximations for compound distributions and applications to various problems in insurance in particular and applied probability in general is also discussed in subsequent chapters. The results obtained or the arguments employed in these situations are similar to those for the compound distributions, and thus we felt it useful to include them in the notes. In many cases we have included exact results, since these are useful in conjunction with the bounds and approximations developed.

• Binomial distribution
• G/G/1 queue
• Martingale
• Poisson distribution
• binomial
• modeling
• queueing theory
• statistics
• quantitative finance

• Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Canada

  Gordon E. Willmot

• Department of Statistics and Actuarial Science, University of Iowa, Iowa City, USA

  X. Sheldon Lin

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