Bayes and EB benchmarkig for SAE. Dissertation 2012.pdf

Note that λ ij need not be the same as w ij

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Note that Λ ij need not be the same as W ij . Similarly, Ω i need not be the same as Γ i . Also, p is given. 47
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Proof. By standard results, the problem reduces to minimization of g = summationdisplay i summationdisplay j ( ˆ θ ij ˆ θ B ij ) T Λ ij ( ˆ θ ij ˆ θ B ij ) + summationdisplay i ( e i ¯ ˆ θ B iw ) T Ω i ( e i ¯ ˆ θ B iw ) 2 summationdisplay i λ 1i T ( summationdisplay j W ij ˆ θ ij e i ) 2 λ 2 T ( summationdisplay i Γ i e i p ), where λ 1i and λ 2 are the Lagrange multipliers. We solve 0 = g ˆ θ ij = 2Λ ij ( ˆ θ ij ˆ θ B ij ) 2 W ij λ 1i i , j (4.1.16) 0 = g e i = 2Ω i ( e i ¯ ˆ θ B iw ) + 2 λ 1i i λ 2 i . (4.1.17) From ( 4.1.16 ), ˆ θ ij = ˆ θ B ij + Λ 1 ij W ij λ 1i = e i = ¯ ˆ θ B iw + s i λ 1i ⇐⇒ λ 1i = s 1 i ( e i ¯ ˆ θ B iw ) i . (4.1.18) From ( 4.1.17 ), e i = ¯ ˆ θ B iw + Ω 1 i Γ i λ 2 Ω 1 i λ 1i ⇐⇒ e i = ¯ ˆ θ B iw + Ω 1 i Γ i λ 2 Ω 1 i s 1 i ( e i ¯ ˆ θ B iw ) ⇐⇒ Ω 1 i Γ i λ 2 = ( I + Ω 1 i s 1 i )( e i ¯ ˆ θ B iw ) = e i = ¯ ˆ θ B iw + (Ω i + s 1 i ) 1 Γ i λ 2 (4.1.19) Applying constraint (ii), we find p = ¯ ˆ θ B w + i Γ i i + s 1 i ) 1 Γ i λ 2 = ¯ ˆ θ B w + R λ 2 , where R := i Γ i i + s 1 i ) 1 Γ i . This implies that λ 2 = R 1 ( p ¯ ˆ θ B w ). From ( 4.1.19 ), e i = ¯ ˆ θ B iw + (Ω i + s 1 i ) 1 Γ i R 1 ( p ¯ ˆ θ B w ). (4.1.20) Combining ( 4.1.18 ) and ( 4.1.20 ), ˆ θ ij = ˆ θ B ij + Λ 1 ij W ij λ 1i ⇐⇒ ˆ θ ij = ˆ θ B ij + Λ 1 ij W ij s 1 i ( e i ¯ ˆ θ B iw ) = ˆ θ ij = ˆ θ B ij + Λ 1 ij W ij s 1 i i + s 1 i ) 1 Γ i R 1 ( p ¯ ˆ θ B w ). (4.1.21) 48
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The result follows from ( 4.1.20 ) and ( 4.1.21 ). 4.2 An Example This section considers small area/domain estimation of the proportion of persons without health insurance for several domains of the Asian subpopulation. Our goal is to benchmark the aggregated probabilities that a person does not have health insurance to the corresponding domain values. We also benchmark the domain estimates to match the overall population estimates. The small domains were constructed on the basis of age, sex, race, and the region where each person lives. The National Health Interview Survey (NHIS) data provides the individual-level binary response data as well as the individual-level covariates ( Ghosh, Kim, Sinha, Maiti, Katzoff and Parsons ( 2009 )). We have information on the main response variable of interest, whether or not a person has health insurance. More information on this data is described in Ghosh, Kim, Sinha, Maiti, Katzoff and Parsons ( 2009 ). Moreover, when targeting specific subpopulations crossclassified by demo- graphic characteristics, direct estimates are usually accompanied with large standard errors and coefficients of variation. Hence, a procedure such as the one proposed in Section 2 is appropriate.
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  • Spring '16
  • Yessi
  • The Land, Estimation theory, Mean squared error, Bayes estimator, Empirical Bayes method

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