Section on Survey Research Methods – JSM 2012
Benchmarked Small Area Prediction
Emily J.Berg1, Wayne A.Fuller1, Andreea L.Erciulescu1
1
Center for Survey Statistics and Methodology, Iowa State University, Department of
Statistics, Ames, IA 50010
Abstract
Small area estimation often involves constructing predictions with an estimated model
followed by a benchmarking step. In the benchmarking operation the predictions are
modified so that weighted sums satisfy constraints. The most common constraint is the
constraint that a weighted sum of the predictions is equal to the same weighted sum of the
original observations. Augmented models as a method of imposing the constraint are
investigated for both linear and nonlinear models. Variance estimators for benchmarked
predictors are presented.
Key Words: Restricted estimation, Calibration, Mixed linear models, Best linear
unbiased prediction, Self-calibrated.
1. Introduction
Small area predictors based on random models are used to improve the mean squared
error (MSE) for areas whose direct estimates have large MSEs. In situations where the
direct estimates are from survey samples, the random model predictors are often
constrained so that the weighted sum of the predictions is equal to the same weighted
sum of the direct estimates. See You and Rao (2002) and You and Rao (2003) for
examples.
Two reasons are commonly given for imposing the constraint. If there are previously
released estimated totals, it is desirable for the small area estimates to sum to those totals.
Second, it is felt that imposing the constraint will reduce the bias associated with an
imperfect model (see Pfeffermann and Barnard (1991)).
Wang, Fuller and Qu (2008) contains a review of some benchmarking methods and
defines a class of benchmarking procedures that minimize a criterion. Bell, Datta, and
Ghosh (2012) discuss theoretical properties of a class of benchmarked predictors for the
linear model with known variances. They extend the Wang and Fuller (2008) estimators
to enforce a vector of restrictions (instead of a single restriction) and derive optimal
benchmarked estimators using a quadratic loss function where the weight matrix is not
necessarily a diagonal matrix. Berg (2010) and Berg and Fuller (2009) consider multiple
benchmarking restrictions in the context of a nonlinear model. In the application of Berg
(2010) and Berg and Fuller (2009), the benchmarking restrictions are defined by the row
and column sums of a two-way table. Berg (2010) applies a vector modification of the
augmented model predictor of Wang, Fuller and Qu (2008) to enforce the restrictions,
while Berg and Fuller (2009) use raking. Nandram and Sayit (2011) develop a Bayesian
procedure to incorporate a constraint in the context of a beta-binomial model. Montanari,
Ranalli, and Vcarelli (2009) impose the benchmarking restriction by adding the constraint
to a penalized likelihood for a logistic mixed model.
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2. Linear Model
We consider the linear small area model
yi   i  ei
(1)
 i  x i β  ui
for i  1, 2, , m, where m is the number of small areas,  i is the small area mean, x i
is a known fixed vector, u i is the random small area effect, and ei is the sampling error.
We assume ei is independent of u j for all i and j and that
 ei    0   ei2
  ~   , 
 u i    0   0
0 
.
 ui2  
(2)
2
2
To introduce prediction subject to a constraint, assume  ui and  ei , i  1, 2, , m, are
known and that ( yi , x i ), i  1, 2, , m, are observed. Then the best linear unbiased
predictor (BLUP) of  i of (1) is
ˆi  xi βˆ   i ( yi  xi βˆ ),
(3)
where
1
1
βˆ  (XΣaa
X) 1 XΣaa
y,
(4)
 i  ( ui2   ei2 ) 1 ui2 ,
Σ aa
is
a
diagonal
matrix
2
2
with  ui   ei
as
the
(5)
ith
diagonal
element,
and β  ( 1 ,  2 ,...,  k ) . X is the m  k matrix with x i as ith
row. The variance of the prediction error is
y  ( y1 , y2 ,..., ym )
'
'
V {ˆi  i }   i ei2  (1   i ) 2 xiV {βˆ}x i .
(6)
The restriction that a linear function of the predictors be equal to the same linear function
of the original observations can be written
m

i 1
m
iˆi   i yi
(7)
i 1
or as
m
  (1  
i 1
i
i
)( yi  xi βˆ )  0,
where i are fixed coefficients. Wang, Fuller and Qu (2008) showed that the linear
predictor that satisfies (7) and minimizes
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m
Q(θˆ  )   i E (ˆi   i ) 2
(8)
i 1
can be expressed as

m
m


j 1
j 1

ˆi  ˆi  ˆ i    j y j    jˆ j ,
(9)
where
1
 m 1 2  1
ˆ
 i     j  j  i i .
 j 1



The predictor (9) can be written
ˆi  ˆi  i1i ˆaug ,
(10)
where
ˆ
aug
 m

    j1 2j 
 j 1

1
m

j 1
j
 j ( y j  ˆ j )
1
j
(11)
and ˆi is defined in (3). We have written (11) to illustrate that ˆaug can be viewed as a
generalized least squares coefficient, where  j 1 j is the explanatory variable,  j is the
weight and y j  ˆ j is the dependent variable.
Battese, Harter and Fuller (1988),
Pfeffermann and Barnard (1991), and Isaki, Tsay and Fuller (2000) suggested predictors
that are equivalent to (10) with i1  ( ei2   ui2 )1 ei2  ui2 ,
i1  i1 cov(ˆi ,  mj1 jˆ j ),
1
2
2
and i  ( ei   ui ) , respectively.
The predictor can also be written
ˆi  xi βˆ  zi ˆaug   i [ yi  xi βˆ  zi ˆaug ]
(12)
where

m

ˆaug    z i i1 z i 
 i 1

1 m

i 1
z i i1 ( yi  xi βˆ ),
(13)
 i  i1 (1   i ) 2 and zi   i (1   i )i . The alternative expression for (10) given in
1
(12) uses a generalized least squares coefficient with weight  i , explanatory variable
zi , and dependent variable ( yi  xi βˆ ). The form (12) can be viewed as a predictor for
an augmented model where the original vector of explanatory variables is augmented by
zi   i (1   i )ωi . The estimated coefficient for zi is not the usual regression
coefficient because zi arises from a constraint that is not part of the original model. The
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predictor (12) is unbiased under model (1), but would be biased under a model that
contained zi as an explanatory vector.
Remark Simple ratio adjustment is not a linear predictor, but can be written as predictor
(10),

ˆ ˆ ˆ
ˆratio
,i   i   i  aug
1
1
where i  ˆii and
ˆaug
m ˆ 
    ii 
 i 1

1 m
 i ( yi  ˆi ).
(14)
i 1
□□
If there are r constraints, we write the vector constraint as
m

i 1
ω i ( yi  ˆi )  0,
(15)
where ωi  (1i , 2i , , ri ). The predictor that minimizes (8) subject to the vector
constraint (15) is
ˆi  ˆi  i1ωi βˆ aug ,
(16)
where
βˆ aug  ( W Φ 1 W ) 1 W ( y  θˆ )
(17)
 ( W Φ W ) W ( Ι  Γ)aˆ ,
1
1
θˆ  (ˆ1 ,..., ˆm )' , the ith element of â is aˆi  yi  xi βˆ , W is the matrix with ωi as ith
row, Φ is a diagonal matrix with i as ith diagonal element, and Γ is a diagonal matrix
with  i as the ith diagonal element.
The prediction error for predictor (16) under model (1) is
ˆi  xi β  ui  (1   i )xi (βˆ  β)  i1ωi βˆ aug   i (ui  ei )  ui ,
(18)
where the error in β̂ is
1
1
βˆ  β  (XΣaa
X) 1 XΣaa
a,
(19)
and ai  ui  ei is ith element of a. Under model (1), E (βˆ aug )  0, and
βˆ aug  ( W Φ 1 W) 1 W (I  Γ) aˆ
(20)
 ( W Φ 1 W) 1 W (I  Γ)(I  X( X Σ aa1 X) 1 X Σ aa1 )a.
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It follows that β̂ and β̂ aug are uncorrelated. Because ui   i (ui  ei ) is uncorrelated

with u j  e j for all i and j, the variance of ˆi   i is
V {ˆi   i }  V {ˆi   i }  i2 ωiV {βˆ aug }ω i ,
(21)
where
1
V {βˆ aug }  ( WΦ1W) 1 W(I  Γ)[ Σaa  X( XΣaa
X) 1 X](I  Γ)W( WΦ1W) 1.
(22)
1
Note that only WΦ W of V {βˆ aug } is a function of i .
In the typical case  ui2 are unknown. Retain model (1), assume  ui2   u2 is unknown,
2
2
assume the  ei are known, and assume (ei , ui ) is normally distributed. Let ~u be an
2
estimator of  u . Then the estimated best linear unbiased predictor (EBLUP) of  i is
~
~
~
i  x i βi  ~i ( yi  x i βi ),
(23)
where
~
~ 1 1 ~ 1
β  ( X Σ aa
X ) X  Σ aa y,
~i  (~u2   ei2 ) 1~u2 ,
~
~
2
2
and the ith element of the diagonal matrix Σ aa is ~u   ei . Defining βaug by (7) with
~
~
~
β replacing β̂ and i replacing i , where i may be a function of ~u2 , let
~
~
~
~
i   i  i 1ωi βaug.
(24)
~
2
Assume that the covariance between ~u and β is o(m 0.5 ). Then the covariance
~
~
~

between  i and βaug is o(m 0.5 ) and an estimator of the variance of  i   i of (24) is
~
~
~
~
Vˆ{i   i }  Vˆ{i  i }  i 2 ωiVˆ{βaug}ωi ,
~
~
where Vˆ{ i   i } is an estimator of the variance of  i   i ,
~
~ 1 1
 ,
Vˆ{βˆ aug }  MW ( Σaa  X( XΣaa
X) X)MW
~
ˆ),
MW  ( WΦ1W) 1 W(I  Γ
~
~
ˆ  diag (ˆi ).
Φ  diag (i ) and Γ
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3. Nonlinear Models
Consider the nonlinear small area model
yi   i  ei
(26)
 i  g ( x i , β )  ui ,
where the first derivatives of the function g (x i , β) with respect to β are continuous,
E{(ei , ui )}  (0, 0), E{ei | u j }  0 for all i and j, and E{(ei2 , ui2 )}  diag{ ei2 ,  ui2 }.
Assume that the estimator of β satisfies an equation of the form
m

i 1
g (x i , βˆ ) 2
( ui   ei2 ) 1[ yi  g (x i , βˆ )]  0,
β
where g (xi , βˆ ) / β is the partial derivative of g (x i , β) with respect to β evaluated at
β̂. Assume ( ui2 ,  ei2 ), i  1, 2, , m, are known or are functions of a finite vector of
parameters. By standard approximation methods, and given regularity conditions,
βˆ  β  (H Σ aa1 H) 1 H Σ aa1 a  o p (m 0.5 )
: Ma  o p (m 0.5 ),
(27)
where H is the matrix with h i  g (x i , β) / β as ith row, Σ aa is a diagonal matrix
2
2
with  ui   ei as ith row and the ith element of a is ai  yi  g (x i , β). Let the
predictor be of the form (3) with g (x i , βˆ ) replacing x i βˆ ,
ˆi  g (x i , βˆ )   i [ yi  g (x i , βˆ )].
(28)
The approximate variance of the prediction error for known  i is
V {ˆi  i }   i ei2  (1   i ) 2 hiV {βˆ}hi .
(29)
1
One can impose the constraint (15) on the prediction by adding the term i ωi βˆ la to the
predictor to obtain
ˆla,i  ˆi  i1ωi βˆ la
(30)
where ˆi is defined in (28),
m

βˆ la    ωii1ωi 
 i 1

4301
1 m

i 1
ωi ( yi  ˆi ),
Section on Survey Research Methods – JSM 2012
yi  ˆi  (1   i )aˆi , and aˆi  yi  g (x i , βˆ ). The predictor (30) is easy to construct,
but the predictor is not constrained to fall in the parameter space for models with
restricted parameter space.
If g (x i , βˆ ) and ˆi satisfy the parameter restrictions for all i, the original model can be
replaced with the augmented model
yi  g{(xi , zi ), (β, βaug )}  ui  ei .
(31)
By analogy to estimator (10), the model is estimated in two steps. The first step is the
estimation of β by the basic estimation procedure. In the second step
m

i 1
[ yi  g{(xi , zi ), (βˆ , β aug )}]2
1
i
(32)
is minimized with respect to β aug , where z i is chosen such that
ωi (1   i ) i 
m

i 1
g{(x i , zi ), (βˆ , βˆ aug )}
β aug
,
ωi (1   i )[ yi  g{(xi , zi ), (βˆ , βˆ aug )}]  0,
(33)
and the predictor
ˆi  g{(xi , zi ), (βˆ , βˆ aug )}   i ( yi  g{(xi , zi ), (βˆ , βˆ aug )})
(34)
is benchmarked. By a first order Taylor expansion of g{(xi , zi ), (βˆ , βˆ aug )} and using
the fact that the true value of β aug is the zero vector,
m

i 1
ω i (1   i )[ yi  g{(xi , z i ), (βˆ , 0)}]  h aug (z i , 0)βˆ aug ]  o p (m 0.5 )
where h aug ,i  h aug (z i , 0)  g{(xi , z i ), (β, 0)} / β aug . Thus, an approximation for
the variance of β̂ aug satisfying (33) is
V {βˆ aug}  ( Z' ΨZ) 1 Z' (I  HM) Σaa (I  HM)' Z( Z' ΨZ) 1 ,
(35)
where Ψ is a diagonal matrix with  i as the ith diagonal element, Z is the m  r matrix
with z i as the ith row and H, Σ aa and M are defined in (27).
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Under model (26), the true value of β aug is 0 and
βˆ aug  (Haug Ψ 1H aug ) 1 Haug Ψ 1 (I  HM )a  o p (m 0.5 ),
where the ith row of H aug is haug ,i  g{(xi , zi ), (β, βaug )} / βaug and β̂ aug is the
value that minimizes (32). To the level of approximation of (35), β̂ aug is uncorrelated
with β̂. Therefore
V {ˆi   i }  V {ˆi   i }  (1   i ) 2 haug,iV {βˆ aug}h aug,i  o(m 1 )
(36)
where ˆi is defined by (28). If one has an estimator of V {ˆi   i }, an estimator of
V {ˆi   i } is constructed by adding an estimator of (1   i )2 haug,iV {βˆ aug}h aug,i to the
estimator of V {ˆ   } .
i
i
4. Example
4.1 Simulation Model
We present an example of estimation for a nonlinear small area model. The model was
initially developed to obtain predictors of proportions using direct estimators from the
Canadian Labour Force Survey (LFS) and covariates from the previous Canadian Census
of Population. See Hidiroglou and Patak (2009) and Berg and Fuller (2012) for details
about the data and the model.
Letting pˆ i , pC ,i , and ni be the direct estimator, Census proportion, and sample size,
respectively, for area i, we consider the model,
̂
where
(37)
 i  g ( x i , β )  ui ,
g (x i , β)  [1  exp(  0  1 xi )]1 exp(  0  1 xi ),


xi  log pC ,i (1  pC ,i ) 1 ,
x i  (1, xi )' and
  0   n 1 2
 ei 
  ~ IND   ,  i ei
0
 ui 
 0

0 
.
 ui2  
The model for the variance of the random area effects is
 ui2  g (x i , β)(1  g (x i , β)),
where  is a parameter to be estimated. The simulation is built to reflect the situation for
the LFS cluster sampling design, so ni1 ei2 is the variance of a 2-stage cluster sample of
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size ni . To generate variables that remain in the natural parameter space for proportions,
we generate  i from a beta distribution and generate p̂i from a mixture of beta-binomial
distributions with parameters chosen to give the specified first and second moments. Berg
and Fuller (2012) describe the simulation procedure in detail.
We simulate proportions for ten areas, with sample sizes and parameters as in LFS. The
proportions and sample sizes for the simulation are given in Table 1, where g i denotes
g (x i , β) of (37). The MSE of the EBLUP depends on both the sample size and the
proportion. Comparing results from areas with same sample size but different proportions
provides information about the effect of the mean parameters. On the other hand,
comparing results from areas with different sample sizes but same proportion provides
information about the effect of sample size. The value of  for the simulation is 0.005,
which is similar to the estimate obtained from the LFS data. The information from four
sets of generated samples is used to estimate  so that the degrees of freedom estimating
variance parameters is similar to that of the LFS study.
Table 1: Parameters for simulation areas
Area
1
2
3
4
5
6
7
8
9
10
gi
0.20
0.75
0.50
0.20
0.75
0.75
0.50
0.20
0.75
0.50
Sample size
16
16
16
30
30
60
60
204
204
204
4.2 Model Estimation and Benchmarking Prediction
In practice, estimation requires an estimator of the variance of ei . Because a direct
estimator of  ei2 may have a large variance or may be correlated with the error in the
direct estimator, we consider a working model for  ei2 . Under a working assumption that
the sampling variance is proportional to a multinomial variance, an estimator of the
variance of ni0.5 ei is,
̂
̂ g (x i , βˆ 0 ) (
g (x i , βˆ 0 ) )
where β̂0 is an initial estimator of β that does not depend on an estimate of  ei2 . Berg
and Fuller (2012) gives details on the estimation of the working model.
The model (37) is nonlinear in the parameters and is estimated iteratively. Each iteration
involves two steps. The first step is the estimation of β by minimizing the quadratic form
m
Q(β)   [ pˆ i  g (x i , β)]2 (ni1ˆ ei2 ,w  ˆ ui2 ) 1 .
i 1
The second step estimates  ui2 with estimated GLS and the residuals from (39).
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Let the predictor be of the form (28) replacing yi with p̂ i and using the working model
estimator of  i , to obtain
ˆi  g (x i , βˆ )  ˆi [ pˆ i  g (x i , βˆ )],
ˆi  (ni1ˆ ei2 ,w  ˆ ui2 ) 1ˆ ui2 ,
(40)
where β̂ and ˆ ui2 are the estimators of β the variance of ui , respectively, obtained from
the iterative estimation procedure.

A predictor ˆi is benchmarked if
10

i 1
10
iˆi    i pˆ i .
(41)
i 1
We consider three different benchmarking methods:
2 1
- Ratio adjustment (raking), where  i  (1  ˆi ) i ˆi ;
- Augmented model (31), using the weight  i  (1  ˆi ) 2 (ni1ˆ ei2  ˆ ui2 ) 1 ni1ˆ ei2 ˆ ui2 ,
proposed by Battese, Harter and Fuller (1988) (Aug BHF), where ni1ˆ ei2 is the direct
estimator of the variance of ei .
- Augmented model (31) , where  i  (1  ˆi ) 2 i1 (Aug  1 ).
Appendix 1 describes the implementation of the augmented benchmarking procedure for
the simulation.
4.3 Results
We evaluate the performance of estimators constructed under the three benchmarking
methods for two simulation sets. In one set  i increases as ni increases and in the other
 i decreases as ni increases.
The results for the set of parameters where the weights  i increase as the sample size
increases, are given in Table 2. The  i are 0.01 for areas with sample size 16 and  i are
0.25 for areas of sample size 204. The direct estimators with large sample size have more
weight in the restriction. Also, for areas with large sample sizes (and large weights), the
ˆi of (40) is close to one, so the EBLUP is close to the direct estimator. As a
consequence, the variance of the difference between the weighted sum of the direct
estimators (the restriction) and the weighted sum of the EBLUPs is relatively small.
Equation (36) gives the increase in MSE caused by augmentation, and the results in Table
1 and document this increase. For the set where the weight increases with the sample
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size, benchmarking has a small effect on MSE; the amount added to the MSE of the
EBLUP is less than 6% of the original MSE.
Table 2: Simulation MSE for predictions
Simulation set 1:  i increases as ni increases
g ( xi , β)
ni
i
MSE( ˆi )
Increase in MSE
Raking
Aug BHF
Aug 
1
16
16
0.20
0.50
0.01
0.01
11
21
0.0
0.2
0.0
0.0
0.0
0.1
204
204
0.20
0.50
0.25
0.25
5
12
0.2
0.2
0.2
0.5
0.3
0.1
□□
The results for the second set of parameters in the simulation, where the weights  i
decrease as the sample size increases, are given in Table 3. In this set the amount added
to the MSE due to benchmarking is large. For raking, the increase in MSE is roughly
proportional to the true proportions, an increase about equal to twice the original MSE for
ni=16 and nearly four times the original MSE for ni=204. For the augmented model with
the weights proposed by BHF, the increase in MSE is associated with the weight, but the
method gives the smallest average increase in MSE. The amount added to the MSE using
the third augmented model is roughly constant, and approximately equal to the variance
of the estimated coefficient β̂aug .The results in Table 3 illustrate that one can determine
the nature of the added variance by the choice of  i .
Table 3: Simulation MSE for predictions
Simulation set 2:  i decreases as ni increases
ni
g ( xi , β)
i
MSE( ˆi )
Increase in MSE
Raking
Aug BHF
Aug 
16
16
0.20
0.50
0.25
0.25
11
21
20
46
26
68
28
30
204
204
0.20
0.50
0.01
0.01
5
12
19
45
0
0
28
28
1
Table 4 contains the empirical coverages of nominal 95% prediction intervals for the
predictions constructed using the weights proposed by BHF. The results are averaged
across areas with the same sample size, for each set of parameters. The prediction
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.5
intervals are constructed as ˆi  t ni MSˆEi0,aug
, where the MSE is computed using Taylor
approximations similar to (36). The MSE estimator is approximately unbiased even if the
working model used for the variance of
differs from the true variance. (See Appendix
2 for details of the MSE estimator.) The empirical coverages for set 1 are between 93%
and 95%, slightly smaller than the empirical coverages for set 2, that range between 93%
and 96%. We consider the empirical coverages to be satisfactory.
Table 4: Empirical Coverages (BHF)
Simulation set 1:  i increases as ni increases
Simulation set 2:  i decreases as ni increases
ni  16
ni  30 ni  60 ni  204
Set 1
0.94
0.93
0.94
0.95
Set 2
0.96
0.93
0.93
0.95
5. Conclusions
We presented an augmented approach for benchmarking nonlinear small area models,
considering multiple benchmarking restrictions. In the simulation example we used a
nonlinear small area model for proportions and analyzed the benchmarking effect on
MSE. When the weights were inversely related to the variance, there was a small increase
in MSE. When the weights were positively related to the variance, there was a large
increase in MSE. We considered different weights in the objective function and the
weights proposed by Battese, Harter and Fuller (1988) gave the smallest average amount
added to MSE. The augmented model with i1  i1 gave a nearly constant increase in
MSE.
Appendix 1: Augmented Model Benchmarking for the Simulation
The benchmarked predictor for area i based on the augmented model with weight  i is of
the form,
̂
(
̂ ̂
̂ ) g{(xi , zi ), (βˆ , βˆ aug )}
where
{ g (x i , βˆ ) (
g (x i , βˆ ) )}
̂ ) i and x i  (1, xi )'. The β̂aug is
(
( 0)
 0, let
obtained through iteration as follows. Starting with βˆ aug
1
m
 m
( j 1)
( j) 
βˆ aug
 βˆ aug
   zi2 i1 [ g (x i , βˆ )(1  g (x i , βˆ ))]2   zi  i1 [ g (x i , βˆ )(1  g (x i , βˆ ))] ai( j ) ,
 i 1
 i 1
( j)
( j)
ˆ
ˆ
g{(xi , zi ), (β, βaug )} , and terminate the iteration when
where ai  ̂
| βˆ ( j )  βˆ ( j 1) | is less than 10-5. The benchmarking restriction is satisfied because
aug
aug
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Section on Survey Research Methods – JSM 2012
m
[ g (x i , βˆ )(1  g (x i , βˆ ))] i1ai( j )   i (1  ˆi )ai( j )  0
∑
i 1
in the last iteration.
Appendix 2: MSE Estimator for Simulation
In the approximation for the MSE in (36), β̂ and β̂aug are uncorrelated with
)
(
. We define an MSE estimator to account for a correlation between ( β̂ , β̂aug ) and
(
)
that may result from misspecification of the working model for the
variance of . A Taylor approximation for the error in the vector of benchmarked
predictors is
̂
(
)
(
)
(
)
where and ̂ are the vectors of benchmarked predictors and true values, respectively,
and are the vectors of area effects and sampling errors,
is a diagonal matrix with
on the diagonal, is an identity matrix of dimension m, and
, defined below, is
based on a linear approximation for β̂ and β̂aug . To define
, let
and Ψ be
diagonal matrices with g (xi , βˆ )(1  g (xi , βˆ )) and  i , respectively, on the diagonal, let
be the matrix with ith column x i , and let
Then,
be the column vector with ith element zi .
(
)(
),
where
(
)
(
(
Ψ
)
1
)
Ψ 1
is a diagonal matrix with ˆg (x i , βˆ )(1  g (x i , βˆ ))  ni1ˆ ei2 ,w on the diagonal. An
approximation for the MSE of ̂ , the vector of benchmarked predictors constructed with
the true , is
and
(
) { }(
)
. The MSE estimator is the sum of an estimator of
(
)
and a second term to account for the variance of the estimator of
See Berg
and Fuller (2012) for the estimator of the variance of the estimator of
where
Acknowledgements
This research was funded in part by the USDA Natural Resources Conservation Service
via cooperative agreement 68748211534.
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