Then averaging over all j we get 1 k k summationdisplay j 1 E bracketleftBig 1

Then averaging over all j we get 1 k k

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Then, averaging over all j we get 0 = 1 k k summationdisplay j =1 E bracketleftBig 1 ( n - 1) n summationdisplay r =1 ( X jr - ¯ X j ) 2 - σ 2 j ) | Θ j bracketrightBig = E s 2 - σ 2 (Θ) | Θ] , for any Θ , where the last equality follows from the definition of ˆ s 2 and the fact the the Θ j are iid. Hence, taking expectations on both sides 0 = E braceleftBig E bracketleftbig ˆ s 2 - σ 2 (Θ) | Θ bracketrightbig bracerightBig = E bracketleftbig ˆ s 2 - σ 2 (Θ) bracketrightbig , by the tower property. This proves that E s 2 ) = E bracketleftbig σ 2 (Θ) bracketrightbig = s 2 . Finally to prove the unbiasedness of ˆ a it is sufficient to see that V ( ¯ X j ) = a + s 2 n , which was left as an exercise in Section 2.1. a50
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40 CHAPTER 4. B ¨ UHLMANN’S CREDIBILITY MODELS Remark 4.1. Although E ( M a j ) = m and the above estimators of m , s 2 and a are unbiased, the empirical credibility estimator ˆ M a j = (1 - ˆ Z ) ˆ m + ˆ Z ¯ X j is not, since E ( ˆ M a j ) negationslash = m . Remark 4.2. Alternative estimators have been proposed, including maxi- mum likelihood estimators under full distributional assumptions. 4.1.2 The Homogeneous Linear Estimator As a corollary to Theorem 4.1 let us derive B¨uhlmann’s linear estimator in the homogeneous case, that is c 0 = 0. Corollary 4.1. The homogeneous linear credibility estimator of μ j ), for j = 1 , . . . , k , is given by (1 - Z ) ¯ X + Z ¯ X j , when Q H j = E bracketleftbig μ j ) - k summationdisplay i =1 n summationdisplay r =1 c j ir X ir bracketrightbig 2 , is minimized under the portfolio unbiasedness constraint E [ k summationdisplay i =1 n summationdisplay r =1 c j ir X ir ] = E bracketleftbig μ j ) bracketrightbig . *Proof: Consider g H j ( x 11 , . . . , x kn ) = k i =1 n r =1 c j ir x ir and minimize Q H j , with respect to the c j ir , under the above unbiasedness constraint; left as an exercise. a50 Example 4.1. (Hachemeister’s Data Set) We illustrate the estimation procedure under B¨uhlmann’s model (and of subsequent models) with a practice data set given in Hachemeister (1975). It reports claims for five different US states (our classes, here k = 5) observed over twelve periods of three months ( n = 12). The X jr are the state average claims (in US $) for private passenger bodily injury insurance, from July 1970 to June 1973. The following tables summarize the estimation results for each class. The estimated parameters used in the above premiums are: (1) the collective premium for the whole portfolio, ¯ X = $1 , 671,
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4.2. THE B ¨ UHLMANN–STRAUB MODEL 41 Table 4.2: Hachemeister’s Data Set Average Claims per Period 1,738 1,364 1,759 1,223 1,456 1,642 1,408 1,685 1,146 1,499 1,794 1,597 1,479 1,010 1,609 2,051 1,444 1,763 1,257 1,741 2,079 1,342 1,674 1,426 1,482 2,234 1,675 2,103 1,532 1,572 2,032 1,470 1,502 1,953 1,606 2,035 1,448 1,622 1,123 1,735 2,115 1,464 1,828 1,343 1,607 2,262 1,831 2,155 1,243 1,573 2,267 1,612 2,233 1,762 1,613 2,517 1,471 2,059 1,306 1,690 Table 4.3: B¨uhlmanns Premiums for Hachemeister’s Data Class j = 1 2 3 4 5 ¯ X j 2,064 1,511 1,822 1,360 1,599 ˆ M a j 2,044 1,519 1,814 1,376 1,602 (2) the estimated within–class variance ˆ s 2 = 46 , 000, (3) the estimated between–class variance (of conditional means) ˆ a = 72 , 300, (4) the corresponding empirical credibility factor ˆ Z = 0 . 94961.
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  • Fall '09
  • Dr.D.Dryanov
  • Trigraph, Estimation theory, Mean squared error, Bias of an estimator, Credibility Models

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