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} equipment keeps unabated. it's not the aim of the textual content to solve these matters yet fairly to illustrate that in the realm of actuarial technological know-how 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 sufficient computing strength and acceptable algorithms had ended in using a number of approximations. the 2 maximum benefits to the actuary of the Bayesian technique are that the tactic is autonomous 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 to study one challenge, the answer to comparable, yet extra complicated, difficulties should be not more tricky. the second takes on further importance because the actuary of at the present time is anticipated to supply facts about the caliber of any estimates. whereas the examples are all actuarial in nature, the equipment mentioned are acceptable to any based estimation challenge. particularly, statisticians will realize that the fundamental credibility challenge has an analogous atmosphere because the random results version from research of variance.

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18) to find new approximations, say p 1 and E 1 . Continue this process until convergence is obtained. The steps are as follows: 1. Let l'o and E 0 be the current estimates of the mean and covariance of7r* (O). 2. Write E 0 = H 0 D 0 H 10 where D 0 is diagonal and H 0 is lower triangular with ones on the diagonal. 3. Let di be the jth diagonal element of D0 and obtain mi hiexp(x~). 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).

An Example The next task is to see how well the Gauss-Hermite method works on the Weibull example. To give this method a full workout, no attempt will be made to reduce the dimension of the problem. 21) With the recommended transformation a= ln(a), t = ln(r), we have t t 1r*( a,t) = exp( an+ tn )IIxi - 1 exp( -eaExi ). 2. While not the perfect concentric circles of the standard bivariate normal distribution, this plot does indicate that Gauss-Hermite integration is likely to produce a satisfactory answer.

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. Finding them is the subject of the next subsection. Bayesian Statistics in Actuarial Science 24 3. Estimating the Mean and Covariance5 Naylor and Smith (1982) suggest an iterative approach to finding p and E. 18) to find new approximations, say p 1 and E 1 .

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