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Unformatted text preview: Second Edition 717 and the right hand side is maximized at T = , with maximizing value Var qi Wi j qj 1 2 [1  2 ] 1 = VarW . 2 ) 1 + [1  2 ]2 (1/j 1  2 j Bloch and Moses (1988) define as the solution to bmax /bmin = 1+ , 1 where bi /bj are the ratio of the normalized weights which, in the present notation, is bi /bj = (1 + ti )/(1 + tj ). The right hand side is maximized by taking ti as large as possible and tj as small as possible, and setting ti = 1 and tj = 1 (the extremes) yields the Bloch and Moses (1988) solution. b. bi = 1/k
2 (1/i ) 2 j 1/j 2 i k = 2 1/j . j Thus, bmax =
2 max k 2 1/j j and bmin = 2 min k 2 1/j j 2 2 and B = bmax /bmin = max /min . Solving B = (1 + )/(1  ) yields = (B  1)/(B + 1). Substituting this into Tukey's inequality yields 2 2 (B + 1)2 ((max /min ) + 1)2 Var W = . 2 2 Var W 4B 4(max /min ) 7.44 Xi is a complete sufficient statistic for when Xi n(, 1). X 2  1/n is a function of 2  1/n is the unique best unbiased estimator of its i Xi . Therefore, by Theorem 7.3.23, X expectation. 1 1 1 1 E X 2 = Var X + (E X)2  = + 2  = 2 . n n n n
i Therefore, X 2  1/n is the UMVUE of 2 . We will calculate Var X 2 1/n = Var(X 2 ) = E(X 4 )  [E(X 2 )]2 , where X n (, 1/n) , but first we derive some general formulas that will also be useful in later exercises. Let Y n(, 2 ). Then here are formulas for E Y 4 and Var Y 2 . E Y 4 = E[Y 3 (Y  + )] = E Y 3 (Y  ) + E Y 3 = E Y 3 (Y  ) + E Y 3 . E Y 3 (Y ) = 2 E(3Y 2 ) = 2 3 2 +2 = 3 4 + 32 2 . (Stein's Lemma) E Y 3 Var Y Thus, 1 Var X 2  n = Var X 2 = 2 1 1 42 + 42 > . 2 n n n
2 = 3 2 + 3 =
4 2 2 = 32 2 + 4 .
4 2 2 2 (Example 3.6.6) = 2 + 4 .
4 2 2 3 + 6 +  ( + ) ...
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 Spring '12
 Dr.Hackney
 Statistics

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