Discussion of W.R. Bell’s paper Partha Lahiri JPSM University of Maryland, College Park Partha Lahiri (JPSM, University of Maryland, College Park) SAE 2011, August 12, 2011 1 / 15
The Fay-Herriot Model: An Area Level Normal Model For i = 1 , · · · , m , assume Level 1: (Sampling Model) Y i = θ i + e i ; Level 2: (Linking Model) θ i = x ′ i β + v i . Equivalently, Y i = x ′ i β + v i + e i , i = 1 , · · · , m . Y i : direct estimate for area i ; D i : known sampling variance of y i . x i : a p × 1 column vector of known auxiliary variables; X ′ = ( x 1 , · · · , x m ) , and Σ( A ) = diag { A + D j ; j = 1 , ..., m } . { e i } and { v i } are independent with e i ∼ N [ 0 , D i ] and v i ∼ N [ 0 , A ] , Lower dimensional ( p + 1) or Hyperparameters: β and A Higher dimensional ( m ) or small area means: θ i Main Objective: Estimation of small area means Partha Lahiri (JPSM, University of Maryland, College Park) SAE 2011, August 12, 2011 2 / 15
The BP, BLUP and EBLUP The BP (Bayes estimator) of θ i ˆ θ i ( Y i ; A ) = B i Y i + ( 1 − B i ) x ′ i β, A where B i = D i + A , i = 1 , . . . , m . The BLUP of θ i ˆ i ˆ θ i ( Y i ; A ) = B i Y i + ( 1 − B i ) x ′ β ( A ) , where ˆ β ( A ) = [ X ′ Σ − 1 ( A ) X ] − 1 X ′ Σ − 1 ( A ) Y and B i = A D i + A , i = 1 , . . . , m . An EBLUP of θ i θ i ( Y i ; ˆ ˆ A ) = ˆ B i Y i + ( 1 − ˆ i ˆ β (ˆ A ) =: ˆ B i ) x ′ θ i , ˆ where ˆ A , i = 1 , . . . , m , and ˆ A B i = A is a consistent estimator of A . D i +ˆ Partha Lahiri (JPSM, University of Maryland, College Park) SAE 2011, August 12, 2011 3 / 15
Components of the FH Model Level 1 Components 1. Normality 2. y i are unbiased 3. D i are known 4. Independence Level 2 Components 1. Normality 2. Linear mean function 3. Homoscedasticity 4. Independence Partha Lahiri (JPSM, University of Maryland, College Park) SAE 2011, August 12, 2011 4 / 15
Bell’s paper: Impact of Level 1 Misspecification of D i Bell’s paper considered impact of Level 1 misspecification of D i Impact is more on MSE estimation than point estimation Underestimation for D i ’s are more severe than underestimation Partha Lahiri (JPSM, University of Maryland, College Park) SAE 2011, August 12, 2011 5 / 15
Impact of Non-normality in Both Levels: A Non-Normal Area Level Model { e i } and { v i } are uncorrelated with e i ∼ [ 0 , D i , κ ei ] and v i ∼ [ 0 , A , κ v ] , [ µ, σ 2 , κ ] representing a probability distribution with mean µ , variance σ 2 and kurtosis κ. Let Φ = Diag { κ ej ; j = 1 , · · · , m } . κ = µ 4 /σ 4 − 3 , where µ 4 is the the fourth central moment of the distribution respectively. We assume that [ β, A , κ v ] is unknown, but [ D i , κ ei ] is known. Partha Lahiri (JPSM, University of Maryland, College Park) SAE 2011, August 12, 2011 6 / 15
Approximation to MSPE MSPE (ˆ θ i ) = E (ˆ θ i − θ i ) 2 , where the expectation is taken over the joint distribution of Y and θ under the non-normal Fay-Herriot model. We decompose the MSPE of EBLUP ˆ θ i as MSPE [ˆ θ i ( Y i , ˆ MSPE [ˆ θ i ( Y i , A )] + E [ˆ θ i ( Y i , ˆ A ) − ˆ θ i ( Y i , A )] 2 A )] = + 2 E [ˆ θ i ( Y i , ˆ A ) − ˆ θ i ( Y i , A )][ˆ θ i ( Y i , A ) − θ i ] . where ˆ θ i ( Y i , A ) is the BLUP of θ i . Partha Lahiri (JPSM, University of Maryland, College Park) SAE 2011, August 12, 2011 7 / 15
Approximation to MSPE A second-order expansion to MSPE of EBLUP ˆ θ i is given by AMSPE i = g 1 i ( A ) + g 2 i ( A ) + g 3 i ( A , κ v ) + 2 g 4 i ( A , κ v ) D 2 AD i ( A + D i ) 2 var [ˆ i = + β ( A )] A + D i D 2 2 AD 2 ( A + D i ) 3 var (ˆ i m ( A + D i ) 3 [ D i κ ei − A κ v ] c (ˆ i + A ) + A ; A ) D 2 ( A + D i ) 3 η (ˆ i = AMSPE i , N + A ; A , κ v ) + 2 g 4 i ( A , κ v ) , . where AMSPE i , N is the normality-based MSPE approximation as given in Prasad and Rao (1990) and Datta, Rao and Smith (2005). Partha Lahiri (JPSM, University of Maryland, College Park) SAE 2011, August 12, 2011 8 / 15
Comments The term g 3 i ( A , κ v ) is the additional uncertainty due to the estimation of the variance component A and the term 2 g 4 i ( A , κ v ) is needed to adjust for the non-normality. Under the regularity conditions, g 1 i ( A ) is the leading term [of order O ( 1 ) ] and the remaining terms are all of order O ( m − 1 ) . Note that non-normality affects both var (ˆ A ) and the cross-product term 2 E [ˆ θ i (ˆ A , Y ) − ˆ θ i ( A , Y )][ˆ θ i ( A , Y ) − θ i ] . When both { e i } and { v i } are normal, the above approximation reduces to the Prasad-Rao (1990) approximation when ˆ A = ˆ A PR and the Datta-Rao-Smith (2005) approximation when ˆ A = ˆ A FH . When the { e i } are normal and ˆ A = ˆ A PR , the MSPE approximation reduces to the Lahiri-Rao (1995) approximation. Partha Lahiri (JPSM, University of Maryland, College Park) SAE 2011, August 12, 2011 9 / 15
Estimation of small area proportion For i = 1 , · · · , m , assume ind Level 1: (Sampling Model) Y i | θ i ∼ B in ( n i , θ i ) : iid Level 2: (Linking Model) θ i ∼ Beta ( α, β ) . α 1 µ = α + β , A = γµ ( 1 − µ ) , with gamma = α + β + 1 Y i is the number of units favoring an event out of a sample size of n i . Beta [ µ, A ] denotes a Beta distribution with mean µ and variance A Let us assume µ and A are known. So we drop the small area index i in the subsequent discussion. Under complex sampling, a more reasonable Level 1 might be Bin (˜ n , θ ) , where ˜ n = n /δ and δ is the design effect. Partha Lahiri (JPSM, University of Maryland, College Park) SAE 2011, August 12, 2011 10 / 15
Estimation of small area proportion The posterior distribution of θ , under misspecified model, is given by Beta [ θ B = ( 1 − B ) p + B µ, v B = γθ B ( 1 − θ B )] , where B = α + β α + β + n Under the complex sampling model, the posterior distribution of θ θ B = ( 1 − ˜ B µ, v B = γ ˜ is given by Beta [˜ p + ˜ θ B ( 1 − ˜ θ B )] , where B )˜ ˜ α + β B = α + β +˜ n ARB = E ( θ B − ˜ θ B ) = − ( 1 − B )( 1 − 1 δ ) θ α + β + 1 ( 1 − 1 1 δ ) < | ARB | < 1 − 1 δ Partha Lahiri (JPSM, University of Maryland, College Park) SAE 2011, August 12, 2011 11 / 15
Plot of ARB vs. deff 0.4 0.3 ARB 0.2 0.1 0.0 2 4 6 8 10 deff Partha Lahiri (JPSM, University of Maryland, College Park) SAE 2011, August 12, 2011 12 / 15
Plot of ARB vs. deff (B=.25) 0.7 0.6 0.5 0.4 ARB 0.3 0.2 0.1 0.0 2 4 6 8 10 deff Partha Lahiri (JPSM, University of Maryland, College Park) SAE 2011, August 12, 2011 13 / 15
Estimation of Small Area Proportions Ref: Liu, Lahiri, Kalton (2007) Model 1 For i = 1 , · · · , m , assume ind Level 1: (Sampling Model) p i | θ i ∼ N [ θ i , θ i ( 1 − θ i ) δ i ]; h ( θ i ) ind Level 2: (Linking Model) ∼ N [ x ′ i β, A ] . Model 2 For i = 1 , · · · , m , assume ind Level 1: (Sampling Model) p i | θ i ∼ Beta [ θ i , θ i ( 1 − θ i ) δ i ]; h ( θ i ) ind Level 2: (Linking Model) ∼ N [ x ′ i β, A ] . Partha Lahiri (JPSM, University of Maryland, College Park) SAE 2011, August 12, 2011 14 / 15
Some Comments Level 1 modeling could be problematic in the presence of sizable number of zeroes for small area. h W 2 � ih θ ih ( 1 − θ ih ) / n ih δ i = Deff i = , n i θ i ( 1 − θ i ) where W ih = N ih / N i , N i = � h N ih , n i = � h n ih . θ ih is the population proportion for stratum h in area i . The design effect Deff i is a function of θ ih , which are unknown. h W 2 If θ ih ≈ θ i , then δ i ≈ � ih / n ih . For complex designs, certain approximations of design effects are given in Kish (1987), Gabler, H � ¨ a der and Lahiri (1999), Gabler, Ganninger, Lahiri (2011), Hawala and Lahiri (2010). Partha Lahiri (JPSM, University of Maryland, College Park) SAE 2011, August 12, 2011 15 / 15
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