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1992 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Bayesian Statistics in Actuarial Science

with Emphasis on Credibility

verfasst von: Stuart A. Klugman

Verlag: Springer Netherlands

Buchreihe : Catastrophe Modeling

Enthalten in: Professional Book Archive

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Über dieses Buch

The debate between the proponents of "classical" and "Bayesian" statistica} methods continues unabated. It is not the purpose of the text to resolve those issues but rather to demonstrate that within the realm of actuarial science there are a number of problems that are particularly suited for Bayesian analysis. This has been apparent to actuaries for a long time, but the lack of adequate computing power and appropriate algorithms had led to the use of various approximations. The two greatest advantages to the actuary of the Bayesian approach are that the method is independent of the model and that interval estimates are as easy to obtain as point estimates. The former attribute means that once one learns how to analyze one problem, the solution to similar, but more complex, problems will be no more difficult. The second one takes on added significance as the actuary of today is expected to provide evidence concerning the quality of any estimates. While the examples are all actuarial in nature, the methods discussed are applicable to any structured estimation problem. In particular, statisticians will recognize that the basic credibility problem has the same setting as the random effects model from analysis of variance.

Inhaltsverzeichnis

Frontmatter
1. Introduction
Abstract
In the minds of most statisticians there are (at least) two mutually exclusive approaches to data analysis. The “classical” or “frequentist” theory consisting of confidence intervals and hypothesis tests is certainly the most widely used and takes up the vast majority, if not all, of the typical statistics text. On the other hand, “Bayesian” statistics, a mode of inference based on Bayes’ Theorem, has attracted a small group of passionate supporters. The debate continues with papers such as “Why Isn’t Everyone a Bayesian?” (Efron, 1986) drawing numerous comments and letters.
Stuart A. Klugman
2. Bayesian Statistical Analysis
Abstract
As discussed in Chapter 1, it is not the intention of this monograph to provide a convincing philosophical justification for the Bayesian approach. Excellent discussions of these matters can be found, for example, in Berger (1985, Chapters 1 and 4) and Lindley (1983). In the latter paper the Bayesian paradigm is described in its simplest form. Of interest is a quantity θ whose value is unknown. What is known is a probability distribution π(θ) that expresses our current relative opinion as to the likelihood that various possible values of θ are the true value. For additional discussion of the merits of expressing uncertainty by probability see Lindley (1982 and 1987). This is called the prior distribution as it represents the state of our knowledge prior to conducting the experiment.
Stuart A. Klugman
3. Computational Aspects of Bayesian Analysis
Abstract
To complete a Bayesian analysis it is often necessary to perform integrations and/or maximizations with respect to functions of many variables. In this Chapter, five approaches will be presented for solving these problems. They all have advantages and disadvantages. Often, but not always, the ones that take the smallest amount of time will be the least accurate. Programs for implementing these procedures are given in the Appendix.
Stuart A. Klugman
4. Prediction with Parameter Uncertainty
Abstract
The model is the same one that introduced the Bayesian paradigm in Chapter 2. Observations have been obtained from a random variable with known general form, but unknown parameters. Of interest is the value of a future observation whose distribution also depends on these parameters. Of course, this is the traditional actuarial problem. The observations are the benefits paid in the past to policyholders and we desire to predict the payments that will be made in the future.
Stuart A. Klugman
5. The Credibility Problem
Abstract
The problem is estimation of the amount or number of claims to be paid on a particular insurance policy in a future coverage period. This is a random quantity whose ultimate value will be affected by a number of factors: the individual characteristics of the insured, the characteristics of a larger group to which the insured belongs, external factors (mostly economic quantities), and the random nature of the insured event. Recognizing that no amount of information will allow us to exactly predict future claims, we settle for either the probability distribution of this amount or properties of this distribution such as the mean and variance. Of greatest interest is the mean, which (under squared error loss) would be our best guess as to what the future claims might be. For the most part we will ignore the economic variables, or equivalently, assume they are accounted for outside the credibility analysis.
Stuart A. Klugman
6. The Hierarchical Bayesian Approach
Abstract
It is essential at the outset to be clear about what is and what is not a Bayesian approach. In particular, none of the credibility methods being used at this time qualify as true Bayesian analyses. The requirements as introduced in Chapter 2 are few — a prior probability distribution that is determined before the data are collected and a model probability distribution. What we need to do for the credibility problem is identify just where these two items come in.
Stuart A. Klugman
7. The Hierarchical Normal Linear Model
Abstract
In this Chapter one more restriction to the normal model of Chapter 6 will be imposed: linearity in the parameters. Within this model most all standard situations involving severity, pure premiums, or loss ratios can be handled. The only reasonable case that cannot be handled is the Poisson model for frequency. This will be covered in Chapter 9.
Stuart A. Klugman
8. Examples
Abstract
In this Chapter a number of data sets will be introduced. Then the credibility models from the previous Chapter will be analyzed.
Stuart A. Klugman
9. Modifications to the Hierarchical Normal Linear Model
Abstract
All the modifications discussed in this Chapter have to do with relaxing the assumption of normality. The first two cover specific distribution choices—lognormal and Poisson. In all cases the normal distribution is retained for the second level. This can usually be accomplished by careful parametrization of the first level parameters. The third modification presented is a general method for dealing with non-normal distributions.
Stuart A. Klugman
Backmatter
Metadaten
Titel
Bayesian Statistics in Actuarial Science
verfasst von
Stuart A. Klugman
Copyright-Jahr
1992
Verlag
Springer Netherlands
Electronic ISBN
978-94-017-0845-6
Print ISBN
978-90-481-5790-7
DOI
https://doi.org/10.1007/978-94-017-0845-6