By Stuart A. Klugman
The debate among the proponents of "classical" and "Bayesian" statistica} tools maintains unabated. it isn't the aim of the textual content to solve these matters yet relatively to illustrate that in the realm of actuarial technological know-how there are various difficulties which are fairly fitted to Bayesian research. This has been obvious to actuaries for a very long time, however the loss of enough computing strength and applicable algorithms had ended in using a number of approximations. the 2 maximum benefits to the actuary of the Bayesian strategy are that the strategy is self sufficient 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 find out how to study one challenge, the answer to related, yet extra complicated, difficulties can be not more tough. the second takes on further value because the actuary of this day is anticipated to supply facts about the caliber of any estimates. whereas the examples are all actuarial in nature, the equipment mentioned are appropriate to any established estimation challenge. particularly, statisticians will realize that the fundamental credibility challenge has an analogous surroundings because the random results version from research of variance.
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Extra resources for Bayesian Statistics in Actuarial Science: with Emphasis on Credibility
At time t the posterior distribution of the parameter l't is the normal distribution with mean Pt and covariance Ct. Let n be the last year for which data are available and suppose we want to make a prediction about year n + r. 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.
19) expresses the variance of this prediction. 17). The unconditional variance of this observation is v;- 1 , which consists of two components. 20) is the variance of the changing parameters and the second is the variance of the la test observation. 23) to effect a compromise between the previous estimate and the information from the current observation (as is typical of a Bayes estimate). 22), the subtraction indicating that the observation at time t serves to reduce our uncertain t y. At time t the posterior distribution of the parameter l't is the normal distribution with mean Pt and covariance Ct.
Put small constants in the lower triangle of these extra rows. This creates a proper prior. Letting the constants go to zero creates the answer we are about to obtain. We now have normal distributions for the model and for the prior. 4) where V = diag( vv .. , v5 ). 5) which is a normal distribution with mean 1J = EV- 1 0 and covariance matrix E. The posterior mean is the standard Whittaker estimate. The advantage of the Bayesian approach is that we also have the covariance. The usual Whittaker approach is to select a value for 0' 2 that produces attractive results.