By Stuart A. Klugman
The debate among the proponents of "classical" and "Bayesian" statistica} equipment keeps unabated. it's not the aim of the textual content to solve these matters yet relatively to illustrate that in the realm of actuarial technology there are many difficulties which are relatively fitted to Bayesian research. This has been obvious to actuaries for a very long time, however the loss of sufficient computing strength and acceptable algorithms had ended in using quite a few approximations. the 2 maximum merits to the actuary of the Bayesian process are that the strategy is self reliant of the version and that period estimates are as effortless to procure as aspect estimates. the previous characteristic implies that as soon as one learns tips on how to study one challenge, the answer to related, yet extra advanced, difficulties may be not more tough. the second takes on additional value because the actuary of this present day is predicted to supply proof about the caliber of any estimates. whereas the examples are all actuarial in nature, the tools mentioned are appropriate to any dependent estimation challenge. specifically, statisticians will realize that the elemental credibility challenge has a similar atmosphere because the random results version from research of variance.
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Extra info for Bayesian Statistics in Actuarial Science: with Emphasis on Credibility
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.
Examples will be given in later chapters in which the empirical Bayes approach does indeed obviate the need for approximate integration. F. SUMMARY Of the methods presented so far, adaptive integration is clearly the best for one dimension while the Gauss-Hermite approach is best for two to five or six dimensions. For higher dimensions the method of choice is Monte-Carlo integration. The method presented here is capable of significant refinement, but does seem to be a good all-purpose choice. The Tierney-Kadane approach is valuable when just one or two integrals are needed and knowledge of the accuracy of the result is not criticat.
27) If a proper prior is being used (Ca positive definite) the summations can start at a= 1. For a noninformative prior use 'PrJ =O and Ca= mi where m is a very large number. The value of a must be large enough to allow the process to reach a state where there is enough information to estimate the parameters. In most cases a = 2 is appropriate as the model will involve just one lag. That is, two observations will be necessary to get initial estimates. 20). This will make Kt =O and so the variance will increase (due to the passage of time) but not be reduced (due to the lack of data).