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**Loss models. From data to decisions. With assistance from: Gary G. Venter.**
*(English)*
Zbl 0905.62104

Wiley Series in Probability and Mathematical Statistics, Applied Section. New York, NY: Wiley. xiii, 644 p. (1998).

In this comprehensive textbook, some fundamental parts of risk theory are presented in 6 chapters and not less than 9 (mostly short) appendices. As the title suggests, emphasis is laid on bridging the gap between the stochastic models of risk theory and real-world data from insurance (mostly non-life, but sometimes also life, in particular group-life).

The contents of the book and its organization can be described appropriately by the chapter titles: 1. Introduction – A Model-Based Approach to Actuarial Science; 2. Loss Distributions – Models for the Amount of a Single Payment; 3. Frequency Distributions – Models for the Number of Payments; 4. Aggregate Loss Models; 5. Credibility Theory; 6. Long-Term Models.

The mathematical setting is not always completely rigorous which is mostly due to the fact that the authors do without introducing probability spaces, probability measures etc. This leads to formulations in definitions like “The distribution of \(N\) does not depend in any way on the values of \(X_1,X_2, \dots\)” (p. 291) and “A continuous-time process is denoted by \(\{X_t; t\geq 0\}\). If there are random elements, it is sufficient to specify the joint distribution of \((X_{t_1}, \dots, X_{t_n})\) for all \(t_1, \dots, t_n\) and \(n\)” (p. 512), e.g. To most sections of the book, numerous exercises and practical case studies are added.

The contents of the book and its organization can be described appropriately by the chapter titles: 1. Introduction – A Model-Based Approach to Actuarial Science; 2. Loss Distributions – Models for the Amount of a Single Payment; 3. Frequency Distributions – Models for the Number of Payments; 4. Aggregate Loss Models; 5. Credibility Theory; 6. Long-Term Models.

The mathematical setting is not always completely rigorous which is mostly due to the fact that the authors do without introducing probability spaces, probability measures etc. This leads to formulations in definitions like “The distribution of \(N\) does not depend in any way on the values of \(X_1,X_2, \dots\)” (p. 291) and “A continuous-time process is denoted by \(\{X_t; t\geq 0\}\). If there are random elements, it is sufficient to specify the joint distribution of \((X_{t_1}, \dots, X_{t_n})\) for all \(t_1, \dots, t_n\) and \(n\)” (p. 512), e.g. To most sections of the book, numerous exercises and practical case studies are added.

Reviewer: W.R.Heilmann (Karlsruhe)