Estimation of mean squared error of model-based small area estimators

Test (2011) 20:367–388DOI 10.1007/s11749-010-0206-2

O R I G I NA L PA P E R

Estimation of mean squared error of model-based smallarea estimators

Gauri Sankar Datta · Tatsuya Kubokawa ·Isabel Molina · J.N.K. Rao

Received: 18 June 2009 / Accepted: 27 June 2010 / Published online: 20 July 2010© Sociedad de Estadística e Investigación Operativa 2010

Abstract Estimation of small area means under a basic area level model is studied,using an empirical Bayes (best) estimator or a weighted estimator with fixed weights.Mean squared errors (MSEs) of the estimators and nearly unbiased (or exactly unbi-ased) estimators of MSE are derived under three different approaches: design based(approach 1), unconditional model based (approach 2) and conditional model based(approach 3). Performance of MSE estimators under the three approaches with re-spect to relative bias and coefficient of variation is also studied, using a simulationexperiment.

Keywords Empirical best prediction · Fay–Herriot model · Mean squared error ·Weighted estimators

Mathematics Subject Classification (2000) 62D05 · 62J07

Communicated by Domingo Morales.

G.S. DattaDepartment of Statistics, University of Georgia, Athens, GA 30602-1952, USAe-mail: [email protected]

T. KubokawaFaculty of Economics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japane-mail: [email protected]

I. MolinaDepartment of Statistics, University Carlos III de Madrid, C/Madrid 126, 28903 Getafe (Madrid),Spaine-mail: [email protected]

J.N.K. Rao (�)School of Mathematics and Statistics, Carleton University, Ottawa ON K1S 5B6, Canadae-mail: [email protected]

mailto:[email protected]




368 G.S. Datta et al.

1 Introduction

Reliable small area statistics are needed for formulating policies and programs, allo-cation of funds, marketing decisions and so on. Customary direct area-specific esti-mators for small areas provide unacceptably large coefficient of variation (CV) due tosmall sample sizes. Therefore, it becomes necessary to employ indirect small area es-timators that make use of the sample data from related areas through linking models,and thus increase the “effective” sample size in the small areas. Such estimators canprovide significantly smaller CV than direct estimators, provided the linking modelsare valid. Indirect estimation for small areas, based on explicit linking models, hasreceived much attention in recent years, including mean squared error (MSE) estima-tion. MSE estimators are used as measures of variability associated with the indirectestimators, and CV of an estimator is estimated as (mse)1/2/(estimator), where msedenotes an estimator of MSE of the estimator. We refer the reader to Rao (2003) fora detailed account of model-based small area estimation.

In this paper, we focus on a basic area level model, known as the Fay–Herriotmodel (Fay and Herriot 1979). Suppose we have m sampled areas and θi denotesthe ith area mean for a variable of interest y. Let Yi be a design-unbiased directestimator of θi with E(Yi |θi) = θi and Var(Yi |θi) = Di , where the sampling vari-ance, Di , is assumed to be known. In practice, Di is ascertained from external sourcesor by smoothing estimated sampling variances using a generalized variance function(GVF) method (Fay and Herriot 1979). We assume that a p-vector of area level co-variates, xi , linearly associated with θi is available for each area i. Under this setup,the basic area level model, assuming normality, may be written as

(i) Yi |θiind∼ N(θi,Di), i = 1, . . . ,m;

(ii) θiind∼ N(x′

iβ,A), i = 1, . . . ,m.

(1)

Component (ii) of (1) is the linking model and component (i) is the sampling model.Marginally,

Yiind∼ N(x′

iβ,Di + A), i = 1, . . . ,m. (2)

In matrix notation, (2) may be written as Y ∼ N{Xβ,Σ(A)}, where Y =(Y1, . . . , Ym)′, X = (x1, . . . ,xm)′ and Σ(A) = diag(A + D1, . . . ,A + Dm).

The best (or Bayes) estimator of θi under squared error loss is given by

θBi = E(θi |Yi) = Yi − Di

A + Di

(Yi − x′iβ)

= {1 − Bi(A)

}Yi + Bi(A)x′

iβ, (3)

noting that θi |Yiind∼ N{θB

i , g1i (A)}, where

g1i (A) = Di

{1 − Bi(A)

}(4)

Estimation of mean squared error of model-based small area estimators 369

and Bi(A) = Di/(A+Di). Expression (3) shows that θBi is a convex combination of

the direct estimator Yi and the regression synthetic estimator x′iβ . It also follows that

MSE(θBi

) = E(θBi − θi

)2 = E{V (θi |Yi)

} = E{g1i (A)

} = g1i (A),

under model (1), showing that a large reduction in MSE over MSE(Yi) =E[E{(Yi − θi)

2|θi}] = E(Di) = Di is obtained when 1 − Bi(A) = A/(A + Di) issmall.

In practice, the model parameters β and A are unknown. For a given A, the maxi-mum likelihood (ML) estimator of β is obtained from (2) as

β(A) = {X′Σ−1(A)X

}−1X′Σ−1(A)Y

={

m∑

j=1

(A + Dj)−1xj x′

j

}−1 m∑

j=1

(A + Dj)−1xjYj . (5)

Note that β(A) is also the weighted least squares (WLS) estimator of β without nor-

mality assumption under the regression model Yi = x′iβ + ui with ui

ind∼ (0,A + Di).Substituting β(A) for β into (3), we get the first-step empirical best (or empirical

Bayes) estimator, θiEB

(A) of θi , which is also equal to the best linear unbiased pre-diction (BLUP) estimator of θi without normality assumption.

Several estimators, A of A, have been proposed in the literature including twomoment estimators without normality assumption, ML and restricted (or residual)ML estimator (REML); see Sect. 2. Substituting A for A into the first-step empiricalbest (EB) estimator, θEB

i (A), we get the final EB estimator θEBi :

θEBi = θEB

i (A) = {1 − Bi(A)

}Yi + Bi(A)x′

i β

= Yi − Bi(A)(Yi − x′i β), (6)

where β = β(A). The estimator (6), based on a moment estimator A without normal-ity assumption, is also the empirical BLUP (EBLUP) estimator.

We also study weighted estimators of the form

θwi = (1 − wi)Yi + wix′

i β, (7)

where wi is a fixed weight (0 ≤ wi ≤ 1) ascertained from past knowledge or simplychosen as wi = 1/2, say. The regression synthetic estimator θSY

i = x′i β is a special

case with wi = 1.The main purpose of this paper is to study the estimation of MSE of θEB

i andθwi under three different approaches. Under approach 1, we obtain an exactly un-

biased estimator of the conditional MSE given the vector of small area meansθ = (θ1, . . . , θm)′. It is also called design-based MSE because it assumes only thesampling model (i) of (1). For the estimators θEB

i and θwi , the conditional MSEs

given the small area means are defined as MSE1(θEBi ) = E{(θEB

i − θi)2|θ} and


MSE1(θwi ) = E{(θw

i − θi)2|θ}. The design-based approach 1 is appealing to the sur-

vey practitioners, but the exactly unbiased estimator of the conditional MSE can beunstable. Under approach 2, we study the estimation of unconditional MSE undermodel (1) with components (i) and (ii), where the unconditional MSEs of θEB

i andθwi are given by MSE2(θ

EBi ) = E(θEB

i − θi)2 and MSE2(θ

wi ) = E(θw

i − θi)2 respec-

tively, and the expectation refers to (1). No closed-form expression for the uncondi-tional MSE exists except in special cases, and hence approximately (or nearly) unbi-ased estimators of MSE2(θ

EBi ) under approach 2 have been studied in the literature

(see Rao 2003, Chap. 7). A third approach (Fuller 1989) uses the conditional MSEgiven the estimator Yi from the ith area as a measure of variability of θEB

i , keepingYj (j �= i) as random. In this case, the goal is to obtain nearly unbiased estimators ofthe conditional MSE, MSE3(θ

EBi ) = E{(θEB

i − θi)2|Yi}, and similarly for θw

i . Thisconditional approach may be appealing because it conditions on the data in the areaof interest.

In Sect. 2, we provide a list of the different estimators of A proposed in the lit-erature. Section 3 deals with unbiased estimation of design-based MSE of θEB

i andθwi under approach 1, while Sect. 4 develops nearly unbiased estimators under ap-

proach 2. Approach 3 is used in Sect. 5 to obtain nearly unbiased estimators of MSEof θEB

i and θwi , conditionally given Yi . Simulation results on the performance of the

MSE estimators under the three approaches are reported in Sect. 6. The three MSEestimators are applied to real data in Sect. 7. Finally, Sect. 8 gives some concludingremarks.

2 Estimation of A

In this section, we give the formulae for four estimators of A, based on two momentmethods, ML and REML. A simple moment estimator of A, due to Prasad and Rao(1990), is given by APR = max(0,A∗

PR) where

A∗PR = (m − p)−1

{m∑

j=1

(Yj − x′j βLS)2 −

m∑

j=1

Dj

{1 − x′

j (X′X)−1xj

}}

, (8)

where X = (x1, . . . ,xm)′ and βLS is the ordinary least squares (OLS) estimator givenby

βLS = (X′X)−1X′Y =(

m∑

j=1

xj x′j

)−1 m∑

j=1

xj Yj .

We refer to this estimator as the ANOVA estimator. Another moment estimator, dueto Fay and Herriot (1979), is given by AFH = max(0,A∗

FH) with A∗FH obtained itera-

tively as the solution of the following nonlinear equation in A:

m∑

j=1

(A + Dj)−1{Yj − x′

j β(A)}2 = m − p. (9)


One could use A(0) = APR or A(0) = 0 as the starting value for the iteration. Conver-gence of the iteration is generally rapid, requiring less than 10 iterations. At step a

of the iteration, the solution A(a), is constrained to be nonnegative for a = 1,2, . . . .Rao (2003, p. 118) gives details of the iterative solution.

The ML estimator of A is given by AML = max(0,A∗ML), where A∗

ML is obtainediteratively as the solution of the following nonlinear equation in A:

m∑

j=1

(A + Dj)−2{Yj − x′

j β(A)}2 =

m∑

j=1

(A + Dj)−1. (10)

Method of scoring may be used for the iteration (Rao 2003, p. 119).For REML estimation, we solve the following nonlinear equation in A to obtain

A∗REML:

m∑

j=1

{Yj − x′j β(A)}2

(A + Dj)2=

m∑

j=1

(A + Dj)−1 −

m∑

j=1

x′j {

∑mk=1(A + Dk)

−1xkx′k}−1xj

(A + Dj)2.

(11)

The REML estimator is given by AREML = max(0,A∗REML). Again, the method of

scoring may be used for the iteration (Rao 2003, p. 119).We have studied MSE estimation for θEB

i and θwi under the three different ap-

proaches, using the four different estimators of A. But we report the results here onlyunder AREML and AFH because in simulation studies the associated estimators of θi

have performed better than the estimators based on APR and AML.

3 Estimation of MSE: approach 1

Rivest and Belmonte (2000) considered general estimators of θi of the form θi =Yi + hi(Y), where hi(Y) depends on the shrinking strategy used. The estimators θEB

i

and θwi are special cases of θi with hi(Y) = −Bi(A){Yi − x′

i β(A)} =: hEBi (Y) and

hi(Y) = −wi{Yi − x′i β(A)} =: hw

i (Y), respectively. The design-based MSE of θi isgiven by

MSE1(θi ) = E{(θi − θi)

2|θ} = Di + 2E{(Yi − θi)hi(Y)|θ} + E

{h2

i (Y)|θ}. (12)

Applying the well-known Stein identity for normal random variables, the cross-product term on the right-hand side of (12) may be written as

E{(Yi − θi)hi(Y)|θ} = DiE

{∂hi(Y)/∂Yi |θ

}. (13)

It now follows from (12) and (13) that an exactly unbiased estimator of MSE1(θi) isgiven by

mse1(θi) = Di + 2Di

{∂hi(Y)/∂Yi

} + h2i (Y). (14)


In the case of θEBi we have

∂hEBi (Y)

∂Yi

= −Bi(A)

{1 − x′

i

∂β(A)

∂Yi

}+ D−1

i

{Bi(A)

}2{Yi − x′

i β(A)} ∂A

∂Yi

. (15)

Similarly,

∂hwi (Y)

∂Yi

= −wi

{1 − x′

i

∂β(A)

∂Yi

}. (16)

Now noting that β(A) satisfies∑

j (A + Dj)−1xjYj = {∑j (A + Dj)

−1xj x′j }β(A),

we get

∂β(A)

∂Yi

={∑

j

xj x′j

A + Dj

}−1[ xi

A + Di

−∑

j

Yj − x′j β(A)

(A + Dj)2xj

∂A

∂Yi

]. (17)

It now follows from (15)–(17) that it remains to evaluate ∂A/∂Yi = ∂A∗/∂Yi ×I (A∗ > 0) for a specified method of estimating A. As noted in Sect. 2, we reportthe results only under AFH and AREML.

We first derive ∂A∗FH/∂Yi . We rewrite (9) as

F{Y, β(A),A

} =m∑

j=1

(A + Dj)−1{Yj − x′

j β(A)}2 − (m − p).

Then F {Y, β(A∗FH),A∗

FH} = 0. Now, using the implicit function theorem, we observethat

m∑

j=1

{Yj − x′j β(A∗

FH)}2

(A∗FH + Dj)2

∂A∗FH

∂Yi

+ 2m∑

j=1


FH)}(A∗

FH + Dj)x′j

∂β(A∗FH)

∂Yi

= 2{Yi − x′

i β(A∗FH)}

(A∗FH + Di)

.

Since∑m

j=1(A∗FH + Dj)

−1{Yj − x′j β(A∗

FH)}xj = 0, we get

∂A∗FH

∂Yi

= 2(A∗FH + Di)

−1{Yi − x′i β(A∗

FH)}∑m

j=1(A∗FH + Dj)−2{Yj − x′

j β(A∗FH)}2

. (18)

We now turn to the derivation of ∂A∗REML/∂Yi , where A∗

REML satisfies (11), i.e.,

m∑

j=1


REML)}2

(A∗REML + Dj)2

−m∑

j=1

(A∗REML + Dj)

−1

+m∑

j=1

x′j {

∑mk=1(A

∗REML + Dk)

−1xkx′k}−1xj

(A∗REML + Dj)2

= 0. (19)


Proceeding along the lines of the derivation of (18), it follows that

∂A∗REML

∂Yi

= t1(A∗REML)

t2(A∗REML) − h′(A∗

REML), (20)

where

t1(A∗REML) = (A∗

REML + Di)−1

[

(A∗REML + Di)

−1{Yi − x′i β(A∗

REML)}

−m∑

j=1


REML)}(A∗

REML + Dj)2x′j

{m∑

j=1

(A∗REML + Dj)

−1xj x′j

}−1

xi

]

,

(21)

t2(A∗REML) =

m∑

j=1


REML)}2

(A∗REML + Dj)3

− 1

2

m∑

j=1

(A∗REML + Dj)

−2

−{

m∑

j=1


REML)}(A∗

REML + Dj)2x′j

}{m∑

j=1

(A∗REML + Dj)

−1xj x′j

}−1

×m∑

j=1


REML)}(A∗

REML + Dj)2xj , (22)

and

h′(A) = 1

2

m∑

j=1

x′j (X

′Σ−1(A)X)−1(X′Σ−2(A)X)(X′Σ−1(A)X)−1xj

(A + Dj)2

−m∑

j=1

x′j (X

′Σ−1(A)X)−1xj

(A + Dj)3, (23)

where X′Σ−2(A)X = ∑mj=1(A + Dj)

−2xj x′j .


An estimator, mse2(θEBi ), under approach 2 is nearly unbiased for MSE2(θ

EBi ) if

E{mse2(θEBi )} = MSE2(θ

EBi )+o(1/m), for large m. Similarly, a nearly unbiased es-

timator mse2(θwi ) of MSE2(θ

wi ) is defined. The estimators mse2(θ

EBi ) and mse2(θ

wi )

using AREML and AFH are obtained in Sects. 4.1 and 4.2 respectively.


4.1 Estimation of MSE2(θEBi )

Define

g2i (A) = {Bi(A)

}2x′i

{m∑

j=1

(A + Dj)−1xj x′

j

}−1

xi

=: {Bi(A)

}2hii (24)

and

g3i (A) = {Bi(A)

}2(A + Di)

−1V (A), (25)

where V (A) is the asymptotic variance of A (as m → ∞). Note that g3i (A) dependson the choice of A. Using the REML estimator AREML, a nearly unbiased estimatorof MSE2(θ

EBi ) is given by

mse2,REML(θEBi

) = g1i (AREML) + g2i (AREML) + 2g3i (AREML), (26)

where g1i (A) is given by (4) and

V (AREML) = 2∑m

j=1(A + Dj)−2; (27)

see Datta and Lahiri (2000).For the Fay–Herriot (FH) estimator AFH, we need the following expression for its

bias to terms of order O(m−1):

bFH(A) = 2[m

∑mj=1(A + Dj)

−2 − {∑mj=1(A + Dj)

−1}2]

{∑mj=1(A + Dj)−1

}3. (28)

Note that the bias of AREML is zero if terms of order o(m−1) are ignored.A nearly unbiased estimator of MSE(θEB

i ) using AFH is given by

mse2,FH(θEBi

) = g1i (AFH) + g2i (AFH) + 2g3i (AFH) − bFH(AFH){Bi(AFH)

}2, (29)

where

V (AFH) = 2m{∑m

j=1(A + Dj)−1}2

; (30)

see Datta et al. (2005).


4.2 Estimation of MSE2(θwi )

In this section, we obtain nearly unbiased estimators of MSE2(θwi ) under approach 2,

using AREML and AFH. Under normality, it can be shown that

MSE2(θwi

) = E{θi

EB(A) − θi

}2 + E{θwi − θi

EB(A)

}2

= g1i (A) + g2i (A) + g3wi(A), (31)

where

g3wi(A) = E{θwi − θi

EB(A)

}2

= E[Bi(A)

{Yi − x′

i β(A)} − wi

{Yi − x′

i β(A)}]2

= {Bi(A) − wi

}2E

{Yi − x′

i β(A)}2

− 2wi

{Bi(A) − wi

}E

[{Yi − x′

i β(A)}x′i β

(1)(A)(A − A)}]

+ o(m−1). (32)

In (32), we used the Taylor expansion β(A) = β(A) + (A − A)β(1)

(A) + op(m−1),

where β(1)

(A) = ∂β(A)/∂A = Op(m−1/2) and A − A = Op(m−1/2) for the twoestimators AREML and AFH. In Appendix A.1 we prove that

E[{

Yi − x′i β(A)

}x′i β

(1)(A)(A − A)

] = o(m−1). (33)

Hence, noting that

E{Yi − x′

i β(A)}2 = E

[(Yi − x′

iβ) − x′i

{β(A) − β

}]2

= (A + Di) − x′i

(X′Σ−1(A)X

)−1xi , (34)

we get from (32), (33) and (34) that

g3wi(A) = {Bi(A) − wi

}2[(A + Di) − x′

i

(X′Σ−1(A)X

)−1xi

] + o(m−1). (35)

To obtain a nearly unbiased estimator of MSE2(θwi ), we rewrite (31), using (35),

as

MSE2(θwi

) = [g1i (A) + {

Bi(A) − wi

}2(A + Di)

]

+ [g2i (A) − {

Bi(A) − wi

}2x′i

(X′Σ−1(A)X

)−1xi

] + o(m−1)

=: h1i (A) + h2i (A) + o(m−1). (36)

Now, E[h2i (A)] = h2i (A)+o(m−1) for the two estimators AREML and AFH. Further,we can express h1i (A) as

h1i (A) = Di(1 − wi)2 + Aw2

i . (37)


Now noting that the bias of AREML is zero to the order o(m−1), we get a nearlyunbiased estimator of MSE2(θ

wi ) under AREML as

mse2,REML(θwi

) = h1i (AREML) + h2i (AREML). (38)

On the other hand, we have bFH(A), given by (28), as the bias of AFH to O(m−1).Hence, a nearly unbiased estimator of MSE2(θ

wi ) under AFH is given by

mse2,FH(θwi

) = h1i (AFH) + h2i (AFH) − bFH(AFH)w2i . (39)


In Sect. 5.2, we derive nearly unbiased estimators of the conditional MSE given byMSE3(θ

EBi ) = E{(θEB

i − θi)2|Yi}. An MSE estimator, mse3(θ

EBi ), under approach 3

is nearly unbiased for MSE3(θEBi ) if E{mse3(θ

EBi )|Yi} = MSE3(θ

EBi ) + op(1/m)

for large m. Nearly unbiased estimators, mse3(θwi ), of the conditional MSE of θw

i ,MSE3(θ

wi ) are derived in Sect. 5.4. The estimators mse3(θ

EBi ) and mse3(θ

wi ) are ob-

tained by first deriving second order approximations of the conditional MSEs of θEBi

and θwi (Sects. 5.1 and 5.3).

5.1 Second order approximation of MSE3(θEBi )

We write MSE3(θEBi ) as

MSE3(θEBi

) = gi(A|Yi) + g3i (A|Yi) + 2g4i (A|Yi), (40)

where

gi(A|Yi) = E[{

θEBi (A) − θi

}2|Yi

],

g3i (A|Yi) = E[{

θEBi − θEB

i (A)}2|Yi

],

and

g4i (A|Yi) = E[{

θEBi (A) − θi

}{θEBi − θEB

i (A)}|Yi

],

by using θEBi − θi = {θEB

i − θEBi (A)} + {θEB

i (A) − θi}. Note that θEBi = θEB

i (A).

We now evaluate the three components of (40). First, we write θEBi (A) − θi =

{θEBi (A) − θB

i } + {θBi − θi}, where θB

i = E(θi |Yi). Using this decomposition, weget

gi(A|Yi) = g1i (A) + g2i (A|Yi), (41)

where g1i (A) is given by (4), and

g2i (A|Yi) = E[{

θEBi (A) − θB

i

}2|Yi

]

= g2i (A) + op

(m−1), (42)


where g2i (A) is given by (24); see Appendix A.2. Also,

g3i (A|Yi) = D2i

(A + Di)4(Yi − x′

i β)2V (A) + op

(m−1)

= {(Yi − x′

iβ)2(A + Di)−1}g3i (A) + op

(m−1), (43)

where A is an estimator of A; see Appendix A.3. Further,

g4i (A|Yi) = op

(m−1); (44)

see Appendix A.4. Substituting (41), (43) and (42) into (40) gives a second orderapproximation to MSE3(θ

EBi ):

MSE3(θEBi

)

= g1i (A) + g2i (A) + {(Yi − x′

iβ)2(A + Di)−1}g3i (A) + op

(m−1). (45)

Expression (45) is valid for the two estimators AREML and AFH.

5.2 Estimation of MSE3(θEBi )

Arguments similar to those used in Sect. 4.2 show that

E[g1i (A)|Yi

]

= g1i (A) + g(1)1i (A)E(A − A|Yi) + 1

2g

(2)1i (A)E

{(A − A)2|Yi

} + op

(m−1)

= g1i (A) + g(1)1i (A)h(A,Yi − x′

iβ) − g3i (A) + op

(m−1), (46)

where g(1)1i (A) = {Bi(A)}2 is the first derivative of g1i (A), g

(2)1i (A) = −2{Bi(A)}2 ×

(A + Di)−1 is the second derivative of g1i (A), and the function h(A,Yi − x′

iβ) =E{(A − A)|Yi} is the conditional bias of A given Yi , which is function only of A andYi − x′

iβ . Similarly,

E{g2i (A)|Yi

} = g2i (A) + op

(m−1) (47)

and

E

[{(Yi − x′

i β)2

A + Di

g3i (A)

}∣∣∣∣Yi

]= g3i (A|Yi) + op

(m−1), (48)

where β = β(A). It now follows from (45), (46), (47) and (48) that a nearly unbiasedestimator of MSE3(θ

EBi ) is given by

mse3(θEBi

) = g1i (A) + g2i (A) − {Bi(A)

}2h(A,Yi − x′

i β),

+{

(Yi − x′i β)2

A + Di

+ 1

}g3i (A), (49)


where h(A,Yi − x′i β) is the estimated conditional bias obtained by substituting A for

A and β for β into h(A,Yi − x′iβ). The conditional biases for the estimators AREML

and AFH are respectively given by

h(A,Yi − x′iβ)REML

={

(A + Di)

m∑

j=1

(A + Dj)−2

}−1{(Yi − x′

iβ)2

A + Di

− 1

}+ op

(m−1) (50)

and

h(A,Yi − x′iβ)FH

= bFH(A) +{

m∑

j=1

(A + Dj)−1

}−1{(Yi − x′

iβ)2

A + Di

− 1

}+ op

(m−1). (51)

Proofs of (50) and (51) are available from the authors. Letting A = AREML andA = AFH in (49) and using (50) and (51), we obtain mse3(θ

EBi ) under the estima-

tors AREML and AFH. Note from (50) that the conditional bias of AREML, ignoringterms of order op(m−1), is not zero unlike the corresponding unconditional zero biasignoring terms of order o(m−1).

5.3 Second order approximation of MSE3(θwi )

In this section, we derive a second order approximation of the conditional MSEof the weighted estimator θw

i with fixed weights wi . We write MSE3(θwi ) =

E{(θwi − θi)

2|Yi} as

MSE3(θwi

) = E[{

θEBi (A) − θi

}2|Yi

] + E[{

θwi − θEB

i (A)}2|Yi

]

+ 2E[{

θEBi (A) − θi

}{θwi − θEB

i (A)}|Yi

]

=: gi(A|Yi) + g3wi(A|Yi) + 2g4wi(A|Yi), (52)

where gi(A|Yi) is given by (41).The second term g3wi(A|Yi) in (52) is evaluated from (32) by conditioning on Yi

and replacing o(m−1) terms by op(m−1). First, we have

E[{

Yi − x′i β(A)

}2|Yi

] = (Yi − x′iβ)2 − 2(Yi − x′

iβ)E[x′i

{β(A) − β

}|Yi

]

+ x′iE

[{β(A) − β

}{β(A) − β

}′|Yi

]xi

= (Yi − x′iβ)2 − 2

(Yi − x′iβ)2

A + Di

x′i

{X′Σ−1(A)X

}−1xi

+ x′i

{X′Σ−1(A)X

}−1xi + op

(m−1). (53)


Secondly,

E[{

Yi − x′i β(A)

}x′i β

(1)(A)(A − A)|Yi

] = op

(m−1). (54)

Hence, it follows from (34) conditional on Yi , (53) and (54), that

g3wi(A|Yi) = {Bi(A) − wi

}[(Yi − x′

iβ)2 − x′i

{X′Σ−1(A)X

}−1xi

]

+ 2{Bi(A) − wi

}x′i

{X′Σ−1(A)X

}−1xi

{1 − (Yi − x′

iβ)2

A + Di

}. (55)

Also, it can be shown that the cross-product term satisfies

g4wi(A|Yi) = Bi(A){Bi(A) − wi

}x′i

{X′Σ−1(A)X

}−1xi

{(Yi − x′

iβ)2

A + Di

− 1

}

+ op

(m−1). (56)

Note from (56) that the cross-product term is o(m−1) unconditionally. From (55) and(56), we get

g3wi(A|Yi) + 2g4wi(A|Yi) =: gwi(A|Yi − x′iβ) + op

(m−1), (57)

where

gwi(A|Yi − x′iβ) = {

Bi(A) − wi

}2[(Yi − x′

iβ)2 − x′i

{X′Σ−1(A)X

}−1xi

]

+ 2wi

{Bi(A) − wi

}x′i

{X′Σ−1(A)X

}−1xi

×{

(Yi − x′iβ)2

A + Di

− 1

}. (58)

Now combining (41) and (58), a second order approximation to the conditionalMSE(θw

i ) is given by

MSE3(θwi

) = g1i (A) + g2i (A) + gwi(A|Yi − x′iβ) + op

(m−1). (59)

5.4 Estimation of MSE3(θwi )

In this section, we obtain nearly unbiased estimators of MSE3(θwi ) under approach 3

for our two estimators of A. It follows from the results given in Sect. 5.2 that

E[g1i (A) + g2i (A) − {

Bi(A)}2

h(A,Yi − x′i β) + g3i (A)|Yi

]

=: E(I |Yi) = g1i (A) + g2i (A) + op

(m−1) (60)

and


E

[(Yi − x′

i β)2{pi(A) − p

(1)i (A)h(A;Yi − x′

i β) − 1

2p

(2)i (A)V (A)

}

+ 2Bi(A){Bi(A) − wi

}x′i

{X′Σ−1(A)X

}−1xi

+ 2wi

{Bi(A) − wi

}x′i

{X′Σ−1(A)X

}−1xi

{(Yi − x′

i β)2

A + Di

− 1

}∣∣∣∣Yi

]

=: E(II|Yi) = gwi

(A|Yi − xT

i β) + op

(m−1), (61)

where pi(A) = {Bi(A) − wi}2, p(1)i (A) and p

(2)i (A) denote respectively first and

second derivative of pi(A) evaluated at A and β = β(A). It now follows from (60)and (61) that I + II is a nearly unbiased estimator MSE3(θ

wi ) under approach 3.

Expressions for h(A,Yi − x′iβ) and V (A) for AREML are given by (50) and (27) re-

spectively, and by (51) and (30) respectively for AFH, and h(A,Yi − x′i β) is obtained

from h(A,Yi − x′iβ) by substituting A for A and β for β .

6 Simulation study

We conducted a simulation study to examine the finite sample performance of eachestimator msek(θ

EBi ) of the true MSE of the EB estimator, MSEk(θ

EBi ), under ap-

proach k, for k = 1,2,3; namely, conditional on the set of θi ’s, unconditionally, andconditional on the corresponding direct estimator Yi . The same was also done forselected weighted estimator θw

i .For the simulation study, we employed model (1) with m = 30 areas, xi = (1, xi)

′,where {xi; i = 1, . . . ,m} were generated i.i.d. from N(−1,1), β = (1,1)′ and A = 1.For the sampling variances Di , we considered the three different patterns used byDatta et al. (2005) and ranging from a more homogeneous to a more disperse pattern:(a) {0.7,0.6,0.5,0.4,0.3}, (b) {2,0.6,0.5,0.4,0.2} and (c) {4,0.6,0.5,0.4,0.1}.Each different value in these sets was taken for 6 areas.

Under each of the three different approaches, the true MSEs of the estimators wereapproximated by generating a large number of Monte Carlo samples. The three MSEestimators were also calculated and their relative biases and coefficients of variationover Monte Carlo samples were computed. In the following we describe separatelyhow the simulations corresponding to each approach were done.

Simulation under approach 1 Under approach 1, the simulations were carried out,first generating a set {θi; i = 1, . . . ,m} from (1)(ii). Then, fixing those values,R = 105 Monte Carlo samples {Y (r)

i ; i = 1, . . . ,m}, r = 1, . . . ,R, were gener-

ated from (1)(i). For each sample r , we computed the EB estimators {θEB(r)i ; i =

1, . . . ,m} using both REML and Fay–Herriot fitting methods. Then the true MSEof θEB

i under approach 1, MSE1(θEBi ), was approximated by

MSEEB1i = 1

R

R∑

r=1

(θ

EB(r)i − θi

)2.


Next, with R = 104 Monte Carlo samples independently generated in the same wayas before and for the same fixed values {θi; i = 1, . . . ,m}, we computed the estimatormse1(θ

EBi ) of the true mean squared error MSE1(θ

EBi ).

Simulation under approach 2 Under approach 2 (unconditional), we simulated R =105 sets {θ(r)

i ; i = 1, . . . ,m} and {Y (r)i ; i = 1, . . . ,m}, r = 1, . . . ,R, from (1)(i)

and (1)(ii). Again, for each simulated set r , we computed EB estimators {θEB(r)i ; i =

1, . . . ,m} using both REML and Fay–Herriot fitting methods. The true MSE of θEBi

was approximated under approach 2 by

MSEEB2i = 1

R

R∑

r=1

(θ

EB(r)i − θ

(r)i

)2.

Additionally, R = 104 new samples were independently generated and the estimatormse2(θ

EBi ) was calculated for each sample.

Simulation under approach 3 Under approach 3, when estimating in area i we mustcondition on the corresponding Yi , but note that the estimator for area i dependson the full sample (Y1, . . . , Ym) and not only on Yi . Thus, when estimating in eacharea i, a separate sample (Y

(i)1 , . . . , Y

(i)m ) needs to be generated conditional on Yi ,

i = 1, . . . ,m. For the ith area, the initial conditioning value Yi was taken as

Yi = x′iβ + Zα

√A + Di, α ∈ {0.05,0.25,0.5,0.75,0.95},

where Zα is the α-quantile of the standard normal distribution. Then we generated θi

from the distribution of θi given Yi , which is N{E(θi |Yi),V (θi |Yi)} with

E(θi |Yi) = Yi − Di

A + Di

(Yi − x′iβ) and V (θi |Yi) = ADi

A + Di

.

Since Yj for j �= i is independent of Yi and of θi , the rest of the sample val-ues Yj , j �= i, were generated independently from their marginal distribution Yj ∼N(x′

jβ,A + Dj). Following this procedure, R = 104 samples were generated and

with those samples we computed the Monte Carlo approximation of MSE3(θEBi ), de-

noted MSEEB3i . Additionally, R = 103 samples were independently generated in the

same way and the MSE estimator mse3(θEBi ) was calculated from each sample. The

same procedure was followed for each area i = 1, . . . ,m.Let mse(r)

k (θEBi ) denote the estimator obtained from simulated sample r under

approach k, for r = 1, . . . ,R and k = 1,2,3. We computed means, relative biases(RBs), mean squared errors and coefficients of variation (CVs) of these estimatorsover simulations as

mseEBki = 1

R

R∑

r=1

mse(r)k

(θEBi

), k = 1,2,3,

RBEBki = (

mseEBki − MSEEB

ki

)/MSEEB

ki , k = 1,2,3,


EEBki = 1

R

R∑

r=1

{mse(r)

k

(θEBi

) − MSEEBki

}2, k = 1,2,3,

CVEBki =

√EEB

ki /MSEEBki , k = 1,2,3.

Most of the results presented here are for pattern (b) of error variances Di , which is in-between the more uniform pattern (a) and the more dispersed pattern (c). Moreover,results for the FH and REML estimation methods were practically the same, so forbrevity we report only the results for REML method.

Table 1 lists the percent relative biases (RB) and the percent coefficients of varia-tion (CV) of the estimator msek(θ

EBi ) = mseEB

ki of MSEk(θEBi ) = MSEEB

ki under eachapproach k = 1,2,3; for α = 0.5, 0.25, 0.05 under approach 3. Table 2 reports thepercent mean and median absolute relative biases (ARB) and the percent mean CVof the MSE estimators. Results for α = 0.75 and α = 0.95 are similar to those forα = 0.25 and α = 0.05, so they are not reported here. Table 1 shows that the CVs ofmseEB

1i are very large (ranging from 13 to 393%), especially for the first six areas withlarger sampling variances Di : mean CV of 102% for approach 1 compared to about13% for approaches 2 and 3 as seen from Table 2. Therefore, mseEB

1i is not reliable,although it is unbiased under approach 1. Comparing now the MSE estimators un-der approaches 2 and 3, Table 1 shows that the RB values of mseEB

2i and mseEB3i (with

α = 0.5 or α = 0.25) are small: median ARB of 1.9% from Table 2. But for α = 0.05,the values of the conditioning variable Yi are extreme and in this case the range of RBfor mseEB

3i increases as seen from Table 1: −3.3 to 16.2%, although from Table 2 themean and median ARB are only 4.2 and 2.2% respectively. However, the CV valuesof mseEB

2i and mseEB3i are somewhat similar regardless the value of α, ranging from 5

to 27% for mseEB2i and 4 to 27% for mseEB

3i . Again, we see from Table 1 that the CVincreases for areas with larger sampling variances Di .

Turning now to the weighted estimator θwi , we compare the performance with

respect to different choices of weights wi . We chose a constant weight wi = 1/2,i = 1, . . . ,m, and a varying weight obtained from the guess A = 0.75 and using theweights given in the EB estimator, wi = Di/(A + Di), i = 1, . . . ,m. Figure 1 plotsthe MSEs of these two weighted estimators along with the MSE of the EB estimatorunder approach 2 for pattern (a) of sampling variances. Observe that the MSEs of theestimator with weights obtained from the guess A = 0.75 are very close to those ofthe EB estimator and generally even slightly smaller. Thus, a good choice of weightsmight result in a weighted estimator slightly more efficient than the EB estimatorbecause the weighted estimator avoids the uncertainty due to the estimation of A. Incontrast, the estimator obtained with the naive weights equal to 1/2 does not performvery well in comparison with the EB estimator, especially for areas with smaller Di .For the estimator θw

i with weights obtained from the guess A = 0.75, the performanceof each MSE estimator under its corresponding approach was studied similarly as forthe EB estimator and the main conclusions remain.


Table 1 Percent relative bias RBEBki

and percent coefficient of variation CVEBki

of the MSE estimator

msek(θEBi

) of MSEk(θEBi

) for each area i under approaches k = 1,2,3, for α = 0.5, α = 0.25 and α =0.05 in approach 3. Results obtained for Di -pattern (b) and using the REML method

Area RBEB1i

CVEB1i

RBEB2i

CVEB2i

RBEB3i

CVEB3i

RBEB3i

CVEB3i

RBEB3i

CVEB3i

α = 0.5 α = 0.5 α = 0.25 α = 0.25 α = 0.05 α = 0.05

1 0.90 393.57 −0.55 26.58 −0.36 27.31 1.97 27.34 2.47 27.02

2 −0.04 158.21 −0.68 25.15 −1.65 25.37 0.32 26.18 −1.54 25.20

3 −0.47 184.52 −1.32 24.52 1.42 25.65 4.75 23.42 14.43 30.12

4 −0.01 457.97 −0.59 26.54 −0.95 26.59 3.32 27.16 −0.12 26.49

5 0.07 365.84 −1.57 23.37 −2.60 22.80 −1.38 22.43 8.52 27.83

6 0.72 348.55 −1.62 24.98 −2.03 26.39 9.87 25.67 13.09 29.06

7 0.08 77.07 −0.26 14.74 0.78 15.64 2.18 13.96 −1.69 14.36

8 −0.24 71.40 −0.45 14.25 −3.12 14.31 2.08 15.66 −3.32 16.76

9 0.12 76.43 −1.02 14.58 0.77 15.50 1.56 12.17 2.74 13.48

10 −0.68 65.42 −0.44 12.81 0.94 11.22 −1.30 13.60 1.89 14.34

11 0.30 78.63 −0.65 14.69 −1.19 14.69 0.94 14.58 0.19 15.02

12 0.03 71.26 −1.05 14.37 −0.88 14.50 5.42 13.34 1.59 11.38

13 −0.05 55.06 −0.73 12.95 −1.19 14.96 1.05 12.21 0.94 12.84

14 −0.17 57.35 −0.31 12.84 −0.10 13.90 5.13 12.14 1.23 10.68

15 −0.34 47.87 −0.43 11.75 −1.51 10.96 7.45 10.31 10.26 12.60

16 0.20 59.86 −0.77 12.93 −0.28 13.36 3.83 13.88 −2.46 13.31

17 −0.23 62.15 −0.80 12.71 −0.02 14.59 0.72 10.61 2.19 10.21

18 −0.08 57.58 −1.52 12.53 −0.35 13.93 4.21 10.34 6.27 11.62

19 0.43 49.43 −0.15 10.91 −1.20 13.01 3.02 12.07 −3.26 12.15

20 −0.49 31.74 0.73 8.45 −0.18 5.93 0.32 3.64 16.21 16.71

21 0.21 31.08 0.83 8.70 2.45 6.39 1.36 3.86 15.56 16.14

22 −0.03 45.90 −0.69 10.82 0.94 12.07 6.86 11.28 2.22 8.57

23 0.07 45.05 −0.06 10.89 −0.17 12.71 2.95 9.87 2.34 9.73

24 0.40 39.46 1.18 8.18 −1.91 7.13 −1.84 7.23 5.49 9.88

25 0.13 18.05 0.64 5.54 −1.17 6.99 −1.47 6.48 −1.15 6.53

26 −0.07 24.68 0.54 4.61 3.19 4.62 −1.62 4.94 0.11 4.86

27 −0.48 22.00 0.67 5.60 0.58 7.26 3.64 7.05 0.06 6.03

28 0.21 24.17 −0.24 5.45 −0.77 6.68 −0.37 6.52 1.08 6.38

29 0.01 13.31 0.41 4.82 2.32 4.59 −0.46 6.24 1.53 5.51

30 −0.28 25.75 0.40 5.27 −0.89 5.38 1.61 3.69 1.48 3.59

7 Application

We applied the results of this paper to the Spanish Labour Force Survey (LFS) datafrom first quarter of year 2006. These data were used by González-Manteiga et al.(2010) to estimate the proportions of unemployed persons in each of 1000 domains,obtained by crossing the Spanish provinces with gender and 10 age groups. Sincethe differences between the MSE estimators under approaches 2 and 3 are of orderO(m−1), to illustrate the differences between these MSE estimators we selected the


Table 2 Percent mean and median absolute relative bias and percent mean coefficient of variation ofthe MSE estimators msek(θEB

i) of MSEk(θEB

i), i = 1, . . . ,m, under approaches k = 1,2,3 for α = 0.5,

α = 0.25 and α = 0.05 in approach 3. Results obtained for Di -pattern (b) and using the REML method

Approach 1 Approach 2 Approach 3 Approach 3 Approach 3

α = 0.5 α = 0.25 α = 0.05

Mean (ARB%) 0.25 0.71 1.20 2.77 4.18

Median (ARB%) 0.20 0.66 0.95 1.91 2.21

Mean (CV%) 101.98 13.38 13.81 12.93 14.28

Fig. 1 True MSEs of EB andweighted estimators withconstant weights wi = 1/2 andwith varying weights wi

obtained with the guessA = 0.75 for pattern ofvariances (a)

subset of domains with sample size not greater than 40, which are the smallest m = 61domains. Here, Yi is the ith domain direct estimator of the proportion of unemployedpeople, i = 1, . . . ,m. These direct estimators and their estimated sampling varianceswere obtained by applying formulas (5.8.3) and (5.8.5) of Särndal et al. (1992), usingthe LFS sampling weights. As explanatory variables in the Fay–Herriot model, weincluded an intercept, 5 indicators of the age–sex group and 3 indicators of educationlevel. These were considered also in González-Manteiga et al. (2010).

We fitted the Fay–Herriot model (1) using the FH method to derive EB estimatorsof the domain proportions of unemployed persons. Then, the MSE estimates of theEB estimators under approaches 1–3, msek,FH(θEB

i ), k = 1,2,3, were calculated. Theestimated MSEs under approach 1, mse1,FH(θEB

i ), were rather unstable across do-mains and took negative values for 27 of them, so they are not very reliable estimates.The estimates under approaches 2 and 3 are much more stable and always positive.Figure 2 shows the values of mse3,FH(θEB

i )/mse2,FH(θEBi ), labeled as mse3/mse2.

Observe that most points are close to one, which means that mse3,FH(θEBi ) is sim-

ilar to mse2,FH(θEBi ), but not for all domains. For domains with i = 6 and i = 52,

mse3,FH(θEBi ) is respectively about 13 and 61% greater than mse2,FH(θEB

i ). Compar-ing mse2,FH(θEB

i ) in (29) with mse3,FH(θEBi ) in (49), it is easy to see that for a domain

i with (Yi −x′i β)2/(AFH +Di) > 1, mse3,FH(θEB

i ) is greater than mse2,FH(θEBi ) if and


Fig. 2 Estimated MSEs underapproach 3 over estimatedMSEs under approach 2 versusdomain for the Spanish LFS data

only if 2(AFH + Di)−1 > m−1 ∑m

j=1(AFH + Dj)−1. Domains i = 52 and i = 6 have

very small sampling variances Di together with large absolute residuals |Yi − x′i β|,

and this is the reason for the two larger points appearing in Fig. 2.

8 Concluding remarks

On the basis of our simulation study, we recommend the use of approach 3, with con-ditioning on the area-specific estimator Yi , for MSE estimation, since conditioningmay be deemed more realistic than the unconditional approach 2 and since the per-formances of mseEB

2i and mseEB3i seem to be similar except in the case of extreme ob-

served Yi . Approach 1, with conditioning on all the θi ’s, is essentially a design-basedapproach and hence also appealing, but the corresponding MSE estimator mseEB

1i per-formed poorly in terms of coefficient of variation relative to mseEB

2i and mseEB3i in the

simulation study. We also found that the weighted estimator for fixed weights ob-tained from a good prior guess of A performed well relative to the EB estimator of asmall area mean.

Acknowledgements This work was partially supported by a research grant to J.N.K. Rao from theNatural Sciences and Engineering Research Council of Canada, by the Spanish grants SEJ2007-64500and MTM2006-05693 and the European Collaborative Project 217565, Call identifier FP7-SSH-2007-1,to I. Molina and by the NSF grants DMS-0071642 and SES-0241651 and the NSA grant MSPF-07G-082to G.S. Datta. The authors wish to thank the referees and the associate editor for several useful commentsand suggestions.

Appendix: Proofs

We assume the following regularity conditions:

(i) Di ’s are uniformly bounded: 0 < L ≤ Di ≤ U < ∞,(ii) sup1≤i≤m hii = O(m−1), where hii = x′

i (X′X)−1xi , i = 1, . . . ,m.


A.1 Proof of (33)

Noting that β(1)

(A) = Op(m−1/2) and A − A = Op(m−1/2), we have

E{{

Yi − x′i β(A)

}x′i β

(1)(A)(A − A)

}

= E{(Yi − x′

iβ)x′i β

(1)(A)(A − A)

} + o(m−1)

= (A + Di)E

[∂

∂Yi

{x′i β

(1)(A)(A − A)

}] + o(m−1), (62)

where β(1)

(A) = ∂β(A)/∂A, and the Stein identity E[(Yi −x′iβ)h(Y)] = (A+Di) ×

E[∂h(Y)/∂Yi ] for Yiind∼ N(x′

iβ,A+Di) is used to get the second equality. The result

β(1)

(A) = Op(m−1/2) follows from the orthogonality of the parameters β and A inthe sense that E{∂2l(β,A)/∂β∂A} = 0, where l(β,A) is the log-likelihood undernormality; the proof is easy and omitted for simplicity.

Now we write

∂

∂Yi

{x′i β

(1)(A)(A − A)

} = ∂

∂Yi

{x′i β

(1)(A)

}(A − A) + x′

i β(1)

(A)∂A

∂Yi

, (63)

where

∂

∂Yi

{x′i β

(1)(A)

} = ∂

∂Ax′i

{∂

∂Yi

β(A)

}

= ∂

∂A

{hii

A + Di

}= O

(m−1),

where hii = x′i{

∑j (A + Dj)

−1xj x′j }−1xi is O(m−1). Further, we assume that

∂A/∂Yi = Op(m−1) which can be verified for the four estimators APR, AFH, AML

and AREML. Hence, the result (33) follows from (62) and (63), noting that A − A =Op(m−1/2) and β

(1)(A) = Op(m−1/2).

A.2 Proof of (42)

We have

g2i (A|Yi) = B2i (A)E

[{x′i

(β(A) − β

)}2|Yi

]

= B2i (A)hii

[1 + hii

{(Yi − x′

iβ)2

(A + Di)2− 1

A + Di

}]

= B2i (A)hii + op

(m−1) = g2i (A) + op

(m−1)

from (24).


A.3 Proof of (43)

We have from (6),

g3i (A|Yi) = E[{

Bi(A)(Yi − x′

i β(A)) − Bi(A)

(Yi − x′

i β(A))}2|Yi

]. (64)

By Taylor expansion, Bi(A) = Bi(A) + B(1)i (A∗)(A − A) and β(A) = β(A) +

β(1)

(A∗)(A − A) where |A∗ − A| ≤ |A − A| and B(1)i (A∗) is the derivative of Bi(A)

evaluated at A = A∗. Substituting for Bi(A) and β(A) into (64), we get

g3i (A|Yi) = E[{

B(1)i (A)

(Yi − x′

i β(A)) − Bi(A)x′

i β(1)

(A)}2

(A − A)2|Yi

]

+ op

(m−1)

= {B

(1)i (A)

}2(Yi − x′

i β(A))2

E{(A − A)2|Yi

} + op

(m−1), (65)

where B(1)i (A) = −Di/(A + Di)

2 and β(1)

(A) = Op(m−1/2). Now, noting thatE{(A − A)2|Yi} = V (A) + op(m−1), we get (43) from (65).

A.4 Proof of (44)

Again, using Taylor expansion of Bi(A) and β(A) around A, we get

g4i (A|Yi) = Bi(A)B(1)i (A)E

[x′i

{β(A) − β

}(Yi − x′

iβ)(A − A)|Yi

] + op

(m−1).

Let β−i (A) and A−i denote β(A) and A when (x′i , Yi) is excluded. Writing β(A) −

β = {β−i (A) − β} − {β−i (A) − β(A)} and A − A = (A−i − A) − (A−i − A) andnoting that β−i (A) − β(A) and A−i − A are Op(m−1) conditionally, we get

E[x′i

{β(A) − β

}(Yi − x′

iβ)(A − A)|Yi

]

= (Yi − x′iβ)x′

iE[{

β−i (A) − β}(A−i − A)}|Yi

] + op

(m−1)

= (Yi − x′iβ)x′

iE[{

β−i (A) − β}(A−i − A)

](66)

since β−i (A) and A−i do not depend on Yi . Now noting that the expectation term in(66) involves third order moments which are all zero due to normality, we get (44).

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Estimation of mean squared error of model-based small area estimators

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