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Read e-book online An introduction to non-life insurance mathematics PDF

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B) Find expressions for EX and Var X when N is Poisson distributed with parameter 0, that is, on-{} Pn = Pr(N=n) = 1iT e . 1B. s. for all n, then the structure distribution U has to be either continuous with density of the form r( a+f1 oa--1(1-0)~1 u( ())-r(a)l' ) (0<0<1; a,{J>O) or concentrated at one point. 5 We consider an insurance portfolio with N independent policies. Policy i has risk parameter 0-EIR, and its claim amount in year j is Xij We assume that Xil'xi ,... are inlependent and identically distributed with mean m(Oi) and vari2 ance 'If...

N x. n for a good empirical credibility estimator. One way to utilise more rating criteria, but at the same time have more data available for the estimation of structural parameters, is to introduce a hierarchical model. To apply such a model, we should be able to assume that the different J= 1 J The expected loss of l n+l is We introduce the following normed versions of 1n ).. and·'· . t = IPo + n IPo and can now express K 1 1 ; + A() as We see that K and 7/Jo n+ 1 are independent of the "average" risk level of the rating class.

J It is assumed that the We consider first multiplicative rating models, that is, (i=l, ... ,I; j=l, ... ,J) We shall look at some different methods for determination of the factors a. s. ' minimises J By the least squares method one Q1 =E. I ~= 1 E. ). 'J 'J ' r b) Show how the als and fljs can be determined by the least squares method, and show that this method produces maximum likelihood estimators when the accumulated claim amount of a policy in cell (i,J) is normally distributed with mean and variance u2.

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