# zbMATH — the first resource for mathematics

Lundberg approximations for compound distributions with insurance applications. (English) Zbl 0962.62099
Lecture Notes in Statistics. 156. New York, NY: Springer. x, 250 p. (2000).
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 bounds in this chapter impose little or no restriction on the distributions of the summand terms, and the superiority of these Lundberg bounds over simpler and well known Markov-type inequalities is discussed as well.
Simpler bounds than the general results in Ch. 4 may be obtained by imposition of restrictions on the distributions of the summand terms, and this is the subject matter of Ch. 5, Bounds based on reliability classifications. Exponential tails are the simplest as well as the most natural (due in part to the connection with the Lundberg asymptotics) and these are considered in some detail. In cases where the exponential bounds are inappropriate or inapplicable, moment-based Pareto tails are often useful. Finally, in some cases, particularly for the medium-tailed subexponential type models considered by P. Embrechts and Ch.M. Goldie [see Ann. Probab. 9, 468-481 (1981; Zbl 0459.60017); J. Aust. Math. Soc., Ser. A 29, 243-256 (1980; Zbl 0425.60011)], products of exponential and other tails may be employed. These parametric choices for the bounding tail are considered in Ch. 6, Parametric bounds.
The remaining chapters deal with applications of these results as well as related concepts in various insurance and applied probability contexts. Compound geometric distributions and negative binomial distributions are considered in Ch. 7, Compound geometric and related distributions, and the associated tails are particularly well suited both for modeling as well as for employment of the bounds discussed earlier. An approximation due to H.C. Tijms in a ruin or queueing theoretic context is discussed and generalized in Ch. 8, Tijms approximations, where a close connection to the Lundberg bounds is established. Many quantities of interest in insurance and other applied probability models are known to satisfy defective renewal equations. The close connections to compound geometric distributions allows for the application of the results from earlier chapters to these situations. Ch. 9, Defective renewal equations, includes an important general equation in insurance modeling due to H.U. Gerber and E.S.W. Shiu [see Insur. Math. Econ. 21, No. 2, 129-137 129-137 (1997; Zbl 0894.90047)]. In Ch. 10, The severity of ruin, a mixture representation is given which allows for the derivation of various useful results. Finally, the renewal risk model of Sparre Andersen is discussed in Ch. 11, Renewal risk processes.
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. The material is self-contained, but an introductory course in insurance risk theory is beneficial to prospective readers.

##### MSC:
 62P05 Applications of statistics to actuarial sciences and financial mathematics 62-02 Research exposition (monographs, survey articles) pertaining to statistics 91B30 Risk theory, insurance (MSC2010) 91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance