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 concerns yet particularly to illustrate that in the realm of actuarial technological know-how there are various difficulties which are really suited 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 resulted in using a variety of approximations. the 2 maximum merits to the actuary of the Bayesian method are that the strategy is self sufficient of the version and that period estimates are as effortless to procure as element estimates. the previous characteristic signifies that as soon as one learns how you can research one challenge, the answer to related, yet extra advanced, difficulties should be not more tough. the second takes on additional value 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 appropriate to any established estimation challenge. particularly, statisticians will realize that the elemental credibility challenge has an identical environment because the random results version from research of variance.
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Additional info for Bayesian Statistics in Actuarial Science: with Emphasis on Credibility
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. That will be done here, with 0' 2 = 10-6 .
Years and the objective was to predict values that would be observed in future years. It is almost certainly the case that the underlying probability distribution was changing during the observation period and will continue to change in the future. This is another kind of uncertainty and should often be included when constructing models. One extremely useful model is the Kalman filter. Its main features are that it allows for the specific incorporation of time-varying parameters and that no integrations are required to obtain the posterior distribution.
Ok)' is the vector of unknown parameters. Let y be the fu ture observation and let its pdf be g(y 1 8). It is important to note that the density for the future observation need not match that of the past observations. All that is necessary is that it depend on the same parameters. = = Once a prior density, 1r(8), is obtained, it is merely a matter of executing the formulas set out in Chapter 2. 1) and the predictive density f * (y 1 z) = J g(y 1 0)1r * (O 1 x)dO = f g(y 1 O)f( z 1 0)1r( O) dO J f(x l 0)1r(O)dO .
Bayesian Statistics in Actuarial Science: with Emphasis on Credibility by Stuart A. Klugman