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

By Stuart A. Klugman

The debate among the proponents of "classical" and "Bayesian" statistica} tools maintains unabated. it isn't the aim of the textual content to unravel these matters yet quite to illustrate that in the realm of actuarial technology there are many difficulties which are relatively suited to Bayesian research. This has been obvious to actuaries for a very long time, however the loss of sufficient computing energy and applicable algorithms had ended in using a variety of approximations. the 2 maximum merits to the actuary of the Bayesian method are that the strategy is autonomous of the version and that period estimates are as effortless to acquire as aspect estimates. the previous characteristic signifies that as soon as one learns the best way to research one challenge, the answer to related, yet extra advanced, difficulties might be not more tough. the second takes on extra value because the actuary of at the present time is anticipated to supply proof in regards to the caliber of any estimates. whereas the examples are all actuarial in nature, the tools mentioned are acceptable to any based estimation challenge. particularly, statisticians will realize that the fundamental credibility challenge has a similar environment because the random results version from research of variance.

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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. If the posterior mode is on a boundary some arbitrary value will need to be selected. Then a small sample can be used to obtain an improved estimate. An appropriate number of degrees of freedom is the number of degrees of freedom that would be associated with a frequentist estimate of the elements of 9.

The goal is to find the predictive distribution for the losses in some future year. The particular data used here contained losses on comprehensive general liability policies and was supplied by the Insurance Services Office. The losses were tabulated by accident year (A Y) with data from AY 1973 through AY 1986. Only those losses settled at lag 1 (that is, in the same year) are being studied here. To get the pure premium, data from all lags would have to be considered. These issues and more details about the data are available in Klugman (1989).

It was developed by Tierney and Kadane {1986) and is based on a Taylor series expansion using the information matrix. 31) 11"*(01 z) Here E is the negative inverse Hessian of ln1r*(OI z) evaluated at O, the value of 8 that maximizes ln1T*(OI z). E* and O* are the corresponding values for the function Ing( O)+ ln1r*(BI z). 20). This process is easy to program and is relatively fast for finding one integral. 31) must be found from scratch for each integral. This is in contrast to the methods in Sections B and C in which many integrals can be done simultaneously.

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