X jW X WW 2 k 1 ˆ s 2 bracketrightbig w 2 j w 2 j are unbiased pseudoestimators

X jw x ww 2 k 1 ˆ s 2 bracketrightbig w 2 j w 2 j

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( ¯ X jW - ¯ X WW ) 2 - ( k - 1) ˆ s 2 bracketrightbig [ w 2 .. - j w 2 j. ] , are unbiased pseudo–estimators for m , s 2 and a , respectively. *Proof: [Dubey and Gisler (1981)] The first assertion E ( ˆ m ) = m is trivial; notice that ¯ X WW would have been a more natural choice here. However one can show that V ( ¯ X WW ) > V ( ¯ X ZW ) and that is why the latter is preferred. In fact it can be shown that V ( j α j ¯ X jW ) is minimal over all constants α j such that j α j = 1 for α j = Z j Z . . Now, we have that k ( n - 1) E s 2 ) = summationdisplay j summationdisplay r w jr E ( X jr - m + m - ¯ X jW ) 2 = summationdisplay j summationdisplay r w jr bracketleftbig V ( X jr ) + V ( ¯ X jW ) - 2 Cov( X jr , ¯ X jW ) bracketrightbig = k ( n - 1) s 2 , where the last equality follows from Lemma 4.2-(2) and (3). Similarly for the proof of the unbiasedness of ˆ a . a50 Remark 4.5. Since ˆ m = X ZW depends on the values of the Z j , which themselves depend on the parameters a and s 2 , they are called pseudo– estimators. One approach suggested is to first compute ˆ s 2 , with it compute ˆ a , to finally obtain ˆ ˆ m = ¯ X ˆ ZW . However, the unbiasedness of ˆ m is lost this way. De Vylder (1984) suggested an alternate pseudo–estimator ˜ a = 1 ( k - 1) summationdisplay j Z j ( ¯ X jW - ¯ X ZW ) 2 ,
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4.2. THE B ¨ UHLMANN–STRAUB MODEL 47 which he claimed to be unbiased. Unfortunately his proof was incorrect and ˜ a is biased. If unbiassedness is important, then ˆ a should be preferred. Remark 4.6. ˆ m = ¯ X ZW is chosen over other estimators because its vari- ance is minimal among unbiased estimators (BLUE). But now notice that V ( ¯ X ˆ ZW ) > V ( ¯ X ZW ) and the BLUE property may no longer apply to the empirical credibility estimator. Corollary 4.2. The homogeneous linear credibility estimator of μ j ), for j = 1 , . . . , k , is given by (1 - Z j ) ¯ X ZW + Z j ¯ X jW , 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 unbiasedness constraint E ( M a j ) = m . *Proof: Left as an exercise. Example 4.2. [Hachemeister’s Data Set (...revisited)] Now we can use the full data set given in Hachemeister (1975), with the number of claims as the natural weights. We apply here B¨uhlmann–Straub’s model to this data. In the coming chapters we will compare the estimates obtained with those produced with different models. Table 4.4: Hachemeister’s Data Set Average Claims per Period (Number of Claims per Period) 1,738 (7,861) 1,364 (1,622) 1,759 (1,147) 1,223 (407) 1,456 (2,902) 1,642 (9,251) 1,408 (1,742) 1,685 (1,357) 1,146 (396) 1,499 (3,172) 1,794 (8,706) 1,597 (1,523) 1,479 (1,329) 1,010 (348) 1,609 (3,046) 2,051 (8,575) 1,444 (1,515) 1,763 (1,204) 1,257 (341) 1,741 (3,068) 2,079 (7,917) 1,342 (1,622) 1,674 (998) 1,426 (315) 1,482 (2,693) 2,234 (8,263) 1,675 (1,602) 2,103 (1,077) 1,532 (328) 1,572 (2,910) 2,032 (9,456) 1,470 (1,964) 1,502 (1,277) 1,953 (352) 1,606 (3,275) 2,035 (8,003) 1,448 (1,515) 1,622 (1,218) 1,123 (331) 1,735 (2,697) 2,115 (7,365) 1,464 (1,527) 1,828 (896) 1,343 (287) 1,607 (2,663) 2,262 (7,832) 1,831 (1,748) 2,155 (1,003) 1,243 (384) 1,573 (3,017) 2,267 (7,849) 1,612 (1,654) 2,233 (1,108) 1,762 (321) 1,613 (3,242) 2,517 (9,077) 1,471 (1,861) 2,059 (1,121) 1,306 (342) 1,690 (3,425)
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48 CHAPTER 4. B ¨ UHLMANN’S CREDIBILITY MODELS Table 4.5: Weighted Averages of Hachemeister’s Data Class j = 1 2 3 4 5 w j. 100,155 19,895 13,735 4,152 36,110 in percentage 57.5 11.4 7.9 2.4 20.7 ¯ X jW 2,061 1,511 1,806 1,353 1,600 The following tables summarize the estimation results for each risk class.
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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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