Bayes and EB benchmarkig for SAE. Dissertation 2012.pdf

Hence r 4 is o m 1 we now show that r 5 o m 1 by

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Hence, R 4 is o ( m 1 ). We now show that R 5 = o ( m 1 ). By calculations similar to those in expression ( 5.2.9 ), summationdisplay j w j B j E bracketleftBigg 2 ˆ θ B i ( σ 2 u ) 2 σ 2 u σ 2 u ) 2 ( ˆ θ j x j T ˜ β ) bracketrightBigg = summationdisplay j w j B j E bracketleftBigg 2 ˆ θ B i ( σ 2 u ) 2 σ 2 u σ 2 u ) 2 ( ˆ θ j x j T ˜ β ) bracketrightBigg + o ( m r ). Recall that E bracketleftbigg braceleftBig j w j B j ( ˆ θ j x j T ˜ β ) bracerightBig 2 bracketrightbigg = O ( m 1 ) by ( 5.2.2 ). Now consider summationdisplay j w j B j E bracketleftBigg 2 ˆ θ B i ( σ 2 u ) 2 σ 2 u σ 2 u ) 2 ( ˆ θ j x j T ˜ β ) bracketrightBigg E 1 4 braceleftBigg 2 ˆ θ B i ( σ 2 u ) 2 bracerightBigg 4 E 1 4 bracketleftbig σ 2 u σ 2 u ) 8 bracketrightbig E 1 2 braceleftBigg summationdisplay j w j B j ( ˆ θ j x j T ˜ β ) bracerightBigg 2 E 1 4 braceleftBigg sup σ 2 u 0 2 ˆ θ B i ( σ 2 u ) 2 bracerightBigg 4 E 1 4 bracketleftbig σ 2 u σ 2 u ) 8 bracketrightbig E 1 2 braceleftBigg summationdisplay j w j B j ( ˆ θ j x j T ˜ β ) bracerightBigg 2 = O ( m 3 / 2 ) by Lemma 2 (ii), by Theorem A.5 of Prasad and Rao ( 1990 ), and by expression ( 5.2.2 ). Thus, R 5 is o ( m 1 ), and E [( ˆ θ EB i ˆ θ B i )( ¯ ˆ θ B w t )] = o ( m 1 ). For the last term in ( 5.2.1 ), we use the the Cauchy-Schwartz inequality to show E [( ¯ ˆ θ EB w ¯ ˆ θ B w )( ¯ ˆ θ B w t )] E 1 2 [( ¯ ˆ θ EB w ¯ ˆ θ B w ) 2 ] E 1 2 [( ¯ ˆ θ B w t ) 2 ] = o ( m 1 ). This concludes the proof of the theorem. 5.3 Estimator of MSE Approximation We now obtain an estimator of the MSE approximation for the Fay-Herriot model (assuming normality). Theorem 8 shows that the expectation of the MSE estimator is correct to O ( m 1 ). 66
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Lemma 1: Suppose that sup t T | h ( t ) | = O ( m 1 ) (5.3.1) for some interval T R . If ˆ σ 2 u , σ 2 u T w.p. 1, then E [ h σ 2 u )] = h ( σ 2 u ) + o ( m 1 ). Proof. Consider the expansion h σ 2 u ) = h ( σ 2 u ) + h ( σ 2 u )(ˆ σ 2 u σ 2 u ) for some σ 2 u between σ 2 u and ˆ σ 2 u . Then σ 2 u T a.s. , and h ( σ 2 u ) sup t T | h ( t ) | a.s. as well. This implies E [ h ( σ 2 u )(ˆ σ 2 u σ 2 u )] sup t T | h ( t ) | E | ˆ σ 2 u σ 2 u | = O ( m 3 / 2 ) by equation ( 5.3.1 ) and since E | ˆ σ 2 u σ 2 u | ≤ E 1 2 [(ˆ σ 2 u σ 2 u ) 2 ]. Hence, if assumption ( 5.3.1 ) holds, then E [ h σ 2 u )] = h ( σ 2 u ) + o ( m 1 ). Theorem 8. E [ g 1 i σ 2 u ) + g 2 i σ 2 u ) + 2 g 3 i σ 2 u ) + g 4 σ 2 u )] = g 1 i ( σ 2 u ) + g 2 i ( σ 2 u ) + g 3 i ( σ 2 u ) + g 4 ( σ 2 u ) + o ( m 1 ), where g 1 i ( σ 2 u ), g 2 i ( σ 2 u ), g 3 i ( σ 2 u ), and g 4 ( σ 2 u ) are defined in Theorem 1. Proof. By Theorem A.3 in Prasad and Rao ( 1990 ), E [ g 1 i σ 2 u ) + g 2 i σ 2 u ) + 2 g 3 i σ 2 u )] = g 1 i ( σ 2 u ) + g 2 i ( σ 2 u ) + g 3 i ( σ 2 u ) + o ( m 1 ). In addition, we consider E [ g 4 σ 2 u )], where g 4 ( σ 2 u ) = m i =1 w 2 i B 2 i V i m i =1 m j =1 w i w j B i B j h V ij =: g 41 ( σ 2 u ) + g 42 ( σ 2 u ).
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  • Spring '16
  • Yessi
  • The Land, Estimation theory, Mean squared error, Bayes estimator, Empirical Bayes method

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