The use of reduced variables eliminates the above nonhomogeneous term as it

The use of reduced variables eliminates the above

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The use of reduced variables eliminates the above non–homogeneous term as it multiplies a 0–expectation term. Using the fact that a scalar random variable trivially equals its trace and also that Tr( A B ) = Tr( B A ) we finally have E braceleftbig ( X 0 j ) prime c prime 1 Σ j bracketleftbig β 0 j ) - Z j ˆ β 0 j bracketrightbigbracerightbig = E braceleftbig Tr bracketleftbig ( X 0 j ) prime c prime 1 Σ j ( β 0 j ) - Z j ˆ β 0 j ) bracketrightbigbracerightbig = E braceleftbig Tr bracketleftbig c prime 1 Σ j ( β 0 j ) - Z j ˆ β 0 j ) ( X 0 j ) prime bracketrightbigbracerightbig = Tr bracketleftbig c prime 1 Σ j E braceleftbig( β 0 j ) - Z j ˆ β 0 j ) ( X 0 j ) prime bracerightbigbracketrightbig . This last expression is seen to be equal to 0 for Z j given in (6.5), since: E braceleftbig( β 0 j ) - Z j ˆ β 0 j ) ( X 0 j ) prime bracerightbig = E bracketleftbig β 0 j ) ( X 0 j ) prime bracketrightbig - Z j E bracketleftbig ˆ β 0 j ( X 0 j ) prime bracketrightbig = Cov bracketleftbig β 0 j ) , X 0 j bracketrightbig - Z j Cov[ ˆ β 0 j , X 0 j ] = Cov[ β j ) , X j ] - Z j Cov[ ˆ β j , X j ] = A Y prime - Z j ( A + s 2 U j ) Y prime = 0 . a50
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66 CHAPTER 6. CREDIBILITY REGRESSION MODELS Lemma 6.3. To estimate b and Z j = A ( A + s 2 U j ) - 1 in (6.5), use ˆ s 2 = 1 k 1 ( n - p ) k summationdisplay j =1 [ X j - Y ˆ β j bracketrightbig prime V - 1 j [ X j - Y ˆ β j ] , ˆ A = 1 ( k - 1) k summationdisplay j =1 Z j [ ˆ β j - ˆ b ] [ ˆ β j - ˆ b ] prime , where ˆ b = ( k summationdisplay j =1 Z j ) - 1 k summationdisplay j =1 Z j ˆ β j , which are unbiased pseudo–estimators of the parameters s 2 , A and b , respec- tively. *Proof: Left as an exercise. Theorem 5.2.4 on page 183 of Goovaerts et al. (1990) gives a proof of the unbiasedness of ˆ b . Other suggested estimators as well formulas for unequal numbers of periods, n j , are also given. a50 6.3 A Numerical Illustration We now compare the estimates obtained in the previous chapters for Hachemeis- ter’s data set to the premiums obtained with a regression credibility model. We use Hachemeister’s standard design matrix Y = 1 12 1 11 . . . . . . 1 1 and a diagonal weight matrix V - 1 j = diag( w j 1 , . . . , w j 12 ). Other design matri- ces could be used, but in this way the predicted value of next years’ credibility premium, that is for t = 0 is simply obtained from the regression intercept and there is no need to pre–multiply by any vector of y j elements. These premiums were obtained with initial diagonal matrices Z j , equal to the identity, which, together with ˆ s 2 , produced the first iteration estimates of b . With it the first estimate of A was obtained, yielding a second estimated value of Z j . The stopping rule for this iterative procedure was that the
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6.3. A NUMERICAL ILLUSTRATION 67 Table 6.1: Hachemeister’s Regression Premiums Individual Regression Estimators Class j = 1 2 3 4 5 Intercept 2,470 1,621 2,096 1,538 1,676 Slope -62 -17 -43 -28 -12 Credibility Adjusted Regression Estimators Intercept 2,437 1,651 2,073 1,507 1,759 Slope -57 -21 -41 -15 -26 largest difference between two successive estimates, of either the intercept or the slope, be at most 0 . 0001. Convergence occurred after 36 iterations.
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