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

By Stuart A. Klugman

The debate among the proponents of "classical" and "Bayesian" statistica} tools keeps unabated. it's not the aim of the textual content to unravel these matters yet fairly 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 enough computing strength and acceptable algorithms had ended in using a variety 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 procure as aspect estimates. the previous characteristic implies that as soon as one learns the right way to research one challenge, the answer to comparable, yet extra complicated, difficulties might be not more tricky. the second takes on additional importance because the actuary of this present day 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 fundamental credibility challenge has an identical environment because the random results version from research of variance.

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3) This prior is based on an assumption that true values follow a polynomial of degree z- 1 with some room for variation. To get the prior for O assume that K- 1 exists, in which case O has a multivariate normal distribution with mean O and variance 0' 2 ( K' K) - 1 • Of course, this inverse does not exist, which is an indication that this is not a proper prior. One way to eliminate this problem is suggested by Gersch and Kitagawa (1988). Their idea is to fiii in additional rows at the top to make K a square matrix.

26) If there are any unknown parameters it is customary to estimate them by maximum likelihood 9 . 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.

This is precisely the empirica! Bayes style approximation from Chapter 3. In this Section an example concerning the evaluation of reserves for a term insurance policy will be discussed. The policy we will study is a ten year term insurance issued to an individual who is 40 years old. We are particularly interested in the reserve that should be established at age 45, should the policyholder be alive and stiU insured. To enable us to concentrate on the statistica} issues it will be assumed that there are no expenses and that the interest rate is known and fixed at 6%.

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