By Stuart A. Klugman
The debate among the proponents of "classical" and "Bayesian" statistica} equipment 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 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 energy and acceptable algorithms had ended in using a variety of approximations. the 2 maximum benefits to the actuary of the Bayesian strategy are that the tactic is self sufficient 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 tips to learn one challenge, the answer to comparable, yet extra complicated, difficulties can be not more tricky. the second takes on additional importance because the actuary of at the present time is predicted to supply facts about the caliber of any estimates. whereas the examples are all actuarial in nature, the equipment mentioned are acceptable to any established estimation challenge. specifically, statisticians will realize that the elemental credibility challenge has a similar environment 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
Next, make the transformation The Jacobian is 2k/ 2l E 11 / 2 = 2k/ 2 (dc ·dk) 1 12 where di is the ith diagonal element of D. 17) where x 1 = (xi ,.. ,xi ). It is assumed that the number of points, n, used for the approxfmatioâ is the same for each dimension. The same set of h's and x's are used in each dimension. 1) that our goal is to integrate functions of the form g( 8)1r*( 9). 15) indicates that the integration will be exact. This suggests that p and E be taken as the mean vector and covariance matrix of the posterior distribution.
Prediction with Parameter Uncertainty 53 randomly about a long-term value, but there is no dependence on the previous ycar. For comparison, the analyses that follow will also be done with model one, an inflation model with unchanging parameter values. 46 With no variation in the model, the current values at time fourteen are identica} to the estimated long-term averages. 98 It is not surprising that the estimates of the long-term average are similar for the two models. The major difference is in the covariances.
We have (with k being the dimension of x) J f(x)dx = J f(x)(21r)k/ 2 E 112exp[(x-Jl) E- 1 (x-Jl)/2](21r)-kl 2 1 X 1E l 1 l- 1 / 2 exp[-(x-Jl) 1E- 1(x-Jl)/2]dx = J g(x)(21r)-k/ 21E l- 1 / 2 exp[-(x-Jl) E- 1 (x-Jl)/2]dx. 15) Now let E = H DH 1 where D is diagonal and H is lower triangular with ones on the diagonal. If E is positive definite the factorization can be done Computational Aspects of Bayesian Analysis 23 by using the Choleski method. Next, make the transformation The Jacobian is 2k/ 2l E 11 / 2 = 2k/ 2 (dc ·dk) 1 12 where di is the ith diagonal element of D.