By Stuart A. Klugman

ISBN-10: 9048157900

ISBN-13: 9789048157907

ISBN-10: 9401708452

ISBN-13: 9789401708456

The debate among the proponents of "classical" and "Bayesian" statistica} tools keeps unabated. it's not the aim of the textual content to solve these matters yet really to illustrate that in the realm of actuarial technology there are various difficulties which are really fitted to Bayesian research. This has been obvious to actuaries for a very long time, however the loss of enough computing strength and acceptable algorithms had ended in using numerous approximations. the 2 maximum merits to the actuary of the Bayesian technique are that the tactic is self reliant of the version and that period estimates are as effortless to acquire as aspect estimates. the previous characteristic implies that as soon as one learns tips on how to learn one challenge, the answer to related, yet extra complicated, difficulties may be not more tough. the second takes on additional importance because the actuary of this day is predicted to supply proof about the caliber of any estimates. whereas the examples are all actuarial in nature, the equipment mentioned are appropriate to any established estimation challenge. specifically, statisticians will realize that the fundamental credibility challenge has an identical environment because the random results version from research of variance.

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**Extra resources for Bayesian Statistics in Actuarial Science: with Emphasis on Credibility**

**Example text**

Suppose the model for a single observation is the density f(x 1 0). Then the mie of O, iJ, has an approximate normal distribution with mean O and covariance E where E is the inverse of the information matrix. J 1 1 O)]. 8) This expression will usually involve the unknown parameter O and so an approximation will be required. The usual approach is to just insert the mie. 20). 8) is available directly from the method of scoring (Hogg and Klugman, 1984). The point of aU of this is that the normal density f(O 1 O,E) can be used as the model in place of the original density.

4. 18) to evaluate the following integrals. = Let M be the result when f(O) 11"*(0). r*(O). Let Mii be the result when f(O) = 0;8 j7r*{O). 5. Then let (p 1 )i = MJM and {E1 )ij = M;j/M- MiMlM 2 • 6. Let l'o = p 1 and E 0 the values do not change. = E 1. 18). The posterior mean and covariance matrix of Oare already available as l'o and E 0 • f(O) = g(0)7r*(O) The one problem that remains is the preliminary estimation of the mean and covariance. The easiest choice for the mean is the mode of the posterior density.

It is assumed that the reader has access to a high quality uniform(O,l) random number generator. Computational Aspects of Bayesian Analysis 31 will have the desired distribution where H is any matrix 7 such that HH'=E. The next task is to select the parameters of the t distribution. As mentioned above the pdf should have a shape that is similar to that of f(8). Reasonable approximations for the mean and covariance matrix of the posterior distribution were given in Section B and can also be used here.

### Bayesian Statistics in Actuarial Science: with Emphasis on Credibility by Stuart A. Klugman

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