4Credibility Estimators

4Credibility Estimators - 3 Credibility Estimators ] We...

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3 Credibility Estimators We have seen that the Bayes premium ] μ ( % )= E [ μ ( % ) | X ] is the best possi- ble estimator in the class of all estimator functions. In general, however, this estimator cannot be expressed in a closed analytical form and can only be cal- culated by numerical procedures. Therefore it does not ful f l the requirement of simplicity. Moreover, to calculate ] μ ( % ) , one has to specify the conditional distributions as well as the a priori distribution, which, in practice, can often neither be inferred from data nor guessed by intuition. The basic idea underlying credibility is to force the required simplicity of the estimator by restricting the class of allowable estimator functions to those which are linear in the observations X =( X 1 ,X 2 ,...,X n ) 0 .Inotherwords , we look for the best estimator in the class of all linear estimator functions . “Best” is to be understood in the Bayesian sense and the optimality crite- rion is again quadratic loss. Credibility estimators are therefore linear Bayes estimators. In the previous chapters X stands for the observations of the individual risk and μ ( % ) for the individual premium. Also, until now, we have essentially worked under the rather simple Assumption 1.1, which in the language of Bayesian statistics (see Subsection 1.2.4) means that the components of X are, conditional on % = & , independent and identically distributed. In this chapter we will f rst continue within this framework, but later (see Subsection 3.2.1 and further) we shall pass to a more general interpretation and we shall de f ne the credibility estimator in a general set-up. We will also see that the credibility estimators can be understood as orthogonal projections in the Hilbert space of square integrable random variables, and we will prove some general characteristics and properties.
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56 3 Credibility Estimators 3.1 Credibility Estimators in a Simple Context 3.1.1 The Credibility Premium in a Simple Credibility Model We consider the following simple credibility model: Model Assumptions 3.1 (simple credibility model) i) The random variables X j ( j =1 ,...,n ) are, conditional on % = & , inde- pendent with the same distribution function F & with the conditional mo- ments μ ( & )= E [ X j | % = & ] , > 2 ( & )=Var[ X j | % = & ] . ii) % is a random variable with distribution U ( & ) . In this model we have P ind = μ ( % E [ X n +1 | % ] , P coll = μ 0 = Z % μ ( & ) dU ( & ) . Our aim is again to f nd an estimator for the individual premium μ ( % ) , but now we concentrate on estimators, which are linear in the observations. We will denote the best estimator within this class by P cred or [ [ μ ( % ) , which we are going to derive now. By de f nition, [ [ μ ( % ) has to be of the form [ [ μ ( % b a 0 + n X j =1 b a j X j , where the real coe ! cients b a 0 , b a 1 ,..., b a n need to solve E 5 9 7 3 C μ ( % ) # b a 0 # n X j =1 b a j X j 4 D 2 6 : 8 =m i n a 0 ,a 1 ,...,a n 5 R E 5 9 7 3 C μ ( % ) # a 0 # n X j =1 a j X j 4 D 2 6 : 8 .
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4Credibility Estimators - 3 Credibility Estimators ] We...

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