Download An introduction to non-life insurance mathematics by by Bjørn Sundt. PDF

By by Bjørn Sundt.

ISBN-10: 3884873989

ISBN-13: 9783884873984

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Example text

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. afJj By the Bailey-Simon method one minimises c) Show how we can determine the als and fljs by the Bailey-Simon method.

Tn. 1 d) How would you estimate the value of e? 3 Let N be the number of claims incurred in an insurance portfolio during a given year, and let Yi be the amount of the ith of these claims. We assume that the Y/s are mutually independent and identically distributed and independent of N. We introduce the accumulated claim amount Var X= ENVar Y + E2 YVar N. 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.

H) * =~E. 1 N E. n P. (X . ~X . )2 1 1 1n ).. = }:2 J= z= "\n-~, P N - E. ,. ~- ~J ~J -2 N . _ 1 P. (X. -X) -(N-1)tp) ~- ~· E[m(E>i,H)jH) = v(H); Var(X .. ; Ev(8i,H) = tp; E(Xi)E>i,H) = m(8i,H); ~· ~· are unbiased estimators of 1fJ and>-.. Would you estimate).. by>-.? This exercise is based on Biihlmann & Straub {1970). Sundt {1983) generalises the estimators of the structural parameters to the case where the ceded portfolios do not need to be observed for the same number of years. +1 ·= p)..

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