By Stuart A. Klugman
The debate among the proponents of "classical" and "Bayesian" statistica} equipment maintains unabated. it's not the aim of the textual content to unravel these matters yet quite to illustrate that in the realm of actuarial technological know-how there are various difficulties which are quite 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 resulted in using numerous approximations. the 2 maximum benefits to the actuary of the Bayesian procedure are that the tactic is self sustaining of the version and that period estimates are as effortless to procure as element estimates. the previous characteristic implies that as soon as one learns find out how to examine one challenge, the answer to comparable, yet extra advanced, difficulties might be not more tough. the second takes on additional value because the actuary of at the present time is anticipated to supply facts 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. specifically, statisticians will realize that the elemental credibility challenge has an analogous environment because the random results version from research of variance.
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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. If 8 is multivariate use the smallest degrees of freedom associated with its elements. 1) will be desired. They will differ only in the choice of g(8). It is not practicat to generate a new sequence of random numbers for each one and so the following scheme will be the most efficient. • ,On from this pdf, and then obtain hi = 1r*(8i 1z)/h(Oi), i = 1, ...
One way to condense the problem is to use the asymptotic properties of maximum likelihood estimators. 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).
20 Bayesian Statistics in Actuarial Science 1. Adaptive Gaussian Integration Gaussian integration is a standard method for approximating an integral over a bounded interval using a specified number of points. The adaptations are designed to first achieve a pre-specified level of accuracy over a bounded interval and then extend the integration to an unbounded interval. This method is given in most all introductory numerica! , Burden and Faires, 1989). It is only practical for one dimensional integrals, but is outstanding for that case.