MEAN SQUARED ERROR ESTIMATION FOR SMALL AREAS WHEN THE SMALL
AREA VARIANCES ARE ESTIMATED
Louis-Paul Rivest and Nathalie Vandal, Université Laval
L.-P. Rivest, Département de mathématiques et de statistique
Université Laval, Ste-Foy Québec, G1K 7P4
[email protected]
Abstract
This paper suggests a generalization to Prasad and Rao’s estimator for the mean squared errors of
small area estimators. This new approach uses the conditional mean squared error estimator of Rivest
and Belmonte (2000) as an intermediate step in the derivation. It is used in this paper to incorporate,
in the mean squared error estimator for a small area, uncertainty concerning the estimation of the small
area variances σi2 . The impact of adding a term to Prasad-Rao mean squared error estimator for the
estimation of σi2 is investigated in a Monte Carlo experiment. An example concerned with the estimation
of the under coverage of the Canadian census in sub-provincial ages-sex category is also presented.
Key words and phrases: Area level model Empirical Bayes estimation; Estimated variances; Small
areas; Stein’s Lemma.
1. INTRODUCTION
Jon Rao has made important contributions to the development of a statistical theory for the estimation of
small areas characteristics in survey sampling. Prasad and Rao (1990) is a pioneering paper that showed
how to modify the posterior variance of a small area estimator to take into account the estimation of the
parameters of the smoothing model used as a prior distribution. This paper was followed by several original
contributions; a non exhaustive list is Ghosh and Rao (1994), Lahiri and Rao (1995), Prasad and Rao (1999)
and Rao (1999). Additional relevant references for small are estimation are Purcell and Kish (1979), Singh,
Gambino and Mantel (1994), Booth and Hobert (1998), and Singh, Stukel and Pfeffermann (1998).
This paper suggests a generalization of Prasad and Rao (1990) estimator to situations where the variances
of both, the small area direct estimators and the smoothing model, are estimated. Prasad and Rao (1990)
correction to the posterior variance only accounts for the estimation of the latter variance; an additional
correction term for the estimation of the variance of the direct estimator is derived in this work. Such a
correction applies to situations where the regression modeling is done at the area level. This is the Fay and
Herriot (1979) model, as opposed to Battese, Harter and Fuller (1988) nested regression where the modeling
occurs at the unit level.
The technical manipulations underlying the derivations in Prasad and Rao (1990) rely on Kackar and Harville
(1984) method for approximating mean squared errors; Datta and Lahiri (2000) provide some recent extensions of this technique. Accounting for the estimation of the small area variances brings in an additional
complexity. One is now facing three models, for the estimation of the small area characteristics, for the
estimation of their sampling variances and for the smoothing of the direct estimators using explanatory
variables. In an attempt to simplify technical manipulations, this paper derives the correction term for the
estimation of σi2 when all the parameters of the smoothing model are known. The correction factor for the
estimation of σi2 also applies to the general case where all the parameters are estimated as proved in Wang
and Fuller (2003).
2. EMPIRICAL BAYES ESTIMATORS FOR SMALL AREAS
Let yi denote for i = 1, . . . , m the direct estimators for m small areas obtained in a survey. The sampling
distributions of the yi ’s are assumed to be independent N (µi , σi2 ) where µi denote the true population
characteristic for the ith small area and σi2 is the sampling variance of the estimates. The n × 1 vector of
the yi ’s is denoted y. The subscript S is used to represent moments taken with respect to the sampling
distribution of y.
In a Bayesian setting, the parameters µi ’s are random variables whose distributions depend on a p-variate
auxiliary variable xi (Maritz and Lwin, 1989, chapter 3), known for the m small areas
µi = xti β + vi ,
(2.1)
where β is a p×1 vector of regression parameters and the vi ’s are independent random variables normally
distributed with mean 0 and variance σv2 . Model (2.1) is the smoothing model for the direct estimates yi .
The marginal distribution of yi , with respect to both the sampling design S and the smoothing model M , is
then N (xti β, σi2 + σv2 ), for i = 1, . . . , m.
The best linear unbiased predictor for µi given yi is the posterior mean of µi given yi ,
yi
σv2
σi2
0
+
x
β
i
σv2 + σi2
σv2 + σi2
(2.2)
Its precision is measured by the posterior variance of µi given yi , σi2 σv2 /(σv2 + σi2 ).
This paper considers the situation where the sampling variances σi2 are unknown and are estimated by s2i ,
a statistic with a N (σi2 , γi ) distribution. We assume that yi and s2i are independent and we let V denote
moments taken with respect to the s2i ’s. Suppose for instance that yi is the mean of the ni measurements in
2
small area i, and that the measurements are independently distributed according to a N (µ i , σ0i
) distribution.
2
2
2
2
2
Then σi = σ0i /ni and si = s0i /ni , where s0i is the sampling variance within the ith small area. The
normality assumption implies that
χ2
d
(2.3)
s2i = σi2 ni −1 ,
ni − 1
where χ2ni −1 is a chi-square distribution with ni − 1 degrees of freedom. In this setting the normality
assumption holds, at least approximately, with γi = 2σi4 /(ni − 1).
When the parameters β and σv2 of the smoothing model are known, the empirical best linear unbiased
estimator for µi is obtained by replacing the unknown quantity σi2 by its estimator in the posterior mean of
µi given yi ,
σ2
s2
s2
µ̂i = yi 2 v 2 + x0i β 2 i 2 = yi − 2 i 2 (yi − x0i β) = yi + gi (y),
(2.4)
σv + s i
σv + s i
σv + s i
where gi (y) = −(yi − x0i β)s2i /(σv2 + s2i ).
The mean squared error of µ̂i as a predictor for µi is M SE(µ̂i ) = E{(µ̂i − µi )2 } = EV EM ES {(µ̂i − µi )2 }.
An estimator for this quantity is derived in the next section.
3. MEAN SQUARED ERROR ESTIMATION
An estimator for the mean squared error of µ̂i is derived assuming that the parameters of the smoothing
model are known. The only unknown is σi2 ; it is estimated by s2i which is distributed according to a N (σi2 , γi )
distribution. The derivation uses a technique that relies on Stein’s (1981) lemma which is given next,
Proposition 1: Stein’s Lemma. Let Y be a N (µ, σ 2 ) random variable and g(y) be a differentiable function
of y then
d
E{(Y − µ)g(Y )} = σ 2 E
g(Y ) ,
dY
provided that these expectations exist.
Suppose for the time being that µi is fixed, one has
ES {(µ̂i − µi )2 } = ES {(yi − µi )2 } + 2ES {gi (y)(yi − µi )} + ES {gi (y)2 }.
Using Stein’s lemma to evaluate the second term of the right hand side, the expectation of the random
variable
d gi (y)
σ 2 s2
s4 (yi − x0i β)2
s2i + 2σi2
+ gi (y)2 = s2i − 2 2 i i 2 + i 2
dyi
si + σ v
(si + σv2 )2
is equal to ES {(µ̂i −µi )2 }. To turn this random variable into an estimator, one needs to replace the unknown
parameter σi2 by its estimator s2i . This produces a bias that can be corrected using Stein’s lemma. Since
2
γi σv2
(si − σi2 )s2i
=
E
,
EV
V
s2i + σv2
(s2i + σv2 )2
an unbiased estimator of EV ES {(µ̂i − µi )2 } is
s2i − 2
s4i (yi − x0i β)2
γi σv2
s4i
+
+
2
s2i + σv2
(s2i + σv2 )2
(s2i + σv2 )2
(2.5)
This is an unbiased mean squared error estimator. Note however that this estimator can be unstable since
it depends on a squared residual, (yi − x0i β)2 . In order to improve the stability of this estimator, note that
4
s4i (σi2 + σv2 )
si (yi − x0i β)2
=
EM ES
(s2i + σv2 )2
(s2i + σv2 )2
A bias corrected estimator of this term is given by
s2 σ 2
s4
s2 σ 2
s4i (s2i + σv2 )
− 2γi 2 i v 2 3 = 2 i 2 − 2γi 2 i v2 3 .
2
2
2
(si + σv )
(si + σv )
si + σ v
(si + σv )
Replacing the third term of (2.5) by this quantity leads to
mse(µ̂i ) = s2i − 2
s2i
s4i
s4
s2 σ 2
γi σ 2
s2 σ 2
γi σ 4
+ 2 i 2 − 2γi 2 i v2 3 + 2 2 v 2 2 = 2 i v 2 + 2 2 v 2 2
2
+ σv
si + σ v
(si + σv )
(si + σv )
si + σ v
(si + σv )
(2.6)
as a mean squared error estimator. This mean squared error estimator is the estimated posterior variance
plus a term for the estimation of σi2 . Thus, 2γi σv4 /(s2i + σv2 )2 represents the variance attributable to the
estimation of σi2 . When γi is unknown, an unbiased mean squared error estimator is obtained by replacing
γi by an unbiased estimator γ̂i in (2.6).
In a general setting, where the parameters β, σv2 , and σi2 are estimated, the derivation presented above
suggests to estimate the mean squared error of small area characteristics by adding a term, 2γ i σ̂v4 /(s2i + σ̂v2 )2
to Prasad and Rao’s (1990) mean squared error estimator. This yields the generalized Prasad and Rao
estimator,
mseP Rg (µ̂i ) =
ˆ v2 ) + σ̂v4 γ̂i
s4 xt Â−1 xi
s4 Var(σ̂
s2i σv2
+ i2i 2 2 + 2 i
,
2
+ σ̂v
(si + σ̂v )
(s2i + σ̂v2 )3
s2i
(2.7)
scenario
(a)
σv2
σi2
scenario
(b)
ni
σv2
σi2
ni
1-5
.2
.5
4
1
1
4
m = 20
i
6-10 11-15
.2
.2
.33
.25
6
8
1
1
.67
.5
6
8
16-20
.2
.2
10
1
.4
10
1-6
.2
.5
4
1
1
4
7-12
.2
.33
6
1
.67
6
m = 30
i
13-18 18-24
.2
.2
.25
.2
8
10
1
1
.5
.4
8
10
25-30
.2
.17
12
1
.33
12
Table 1. Variances σv2 and σi2 used in the simulations.
P
ˆ v2 ) is an estimator of the variance of σv2 . When the variances of σ̂v2
where  =
xi x0i /(s2i + σ̂v2 ) and Var(σ̂
2
and si are of the same order, say O(1/m), it is conjectured that the bias of (2.7) is o(1/m) as an estimator
of EV EM ES {(µ̂i − µi )2 }. A proof of this result is provided in Wang and Fuller (2003).
The remainder of this paper presents Monte Carlo simulations and data analyses to assess the impact of
the additional term for the estimation of σi2 on the accuracy of the mean squared error estimator. As an
estimator of σv2 , we use the moment estimator suggested by Prasad and Rao
σ̂v2 =
m
X
1
max{0,
(yi − x0i β̂)2 − (1 − hii )s2i },
m−p
i=1
where hii is an element of the hat matrix for the matrix of explanatory variables X and β̂ is the ordinary
least squares estimator of β. Once σ̂v2 is estimated, the final estimator of β is calculated by weighted least
squares. Furthermore, the estimator for the variance of σ̂v2 is
2 X 2
2
ˆ
(si + σ̂v2 )2 .
Var(σ̂
v) =
m2
4. MONTE CARLO SIMULATIONS
4.1. A general presentation of the simulation programs
The parameter values in the simulations are similar to those used in Lahiri and Rao (1995). Situations where
the small area variances are either all equal or differ between regions are considered. The fixed part of (2.1)
is now set to xti β = µ for all small areas, thus p = 1 and xi = 1 for each i; the value µ = 0 is used in the
simulations. Simulations involving an X matrix with p = 5 are presented at the end of section 4.2. The
number of small areas is set to either m = 20 or m = 30. Table 1 gives sets of parameters values used in
the simulation. For a given value of σi2 , s2i was simulated according to (2.3). The values of µi and yi where
generated with normal random variables.
The calculations were carried out using S-plus; all the results are based on N = 10000 Monte Carlo replicates.
If vij represents the error term in (2.1) for the ith small area in the j replicates, the mean squared error for
µ̂i is evaluated as
N
X
(µ̂ij − vij )2
M SE(µ̂i ) =
N
j=1
where µ̂ij is the estimate for the ith small area in the jth replicate calculated using (2.2) with the parameter
values replaced by their estimates for the jth Monte Carlo replicate. The relative bias, in percentage, of
(2.7) as a mean squared error estimator for the ith small area is
biais(mse(µ̂i )) =
meanE[mse(µ̂i )] − meanM SE(µ̂i )
× 100
meanM SE(µ̂i )
ni
σi2
M SE(µ̂i )
Relative bias mseP R (µ̂i ) (%)
Relative bias mseP Rg (µ̂i ) (%)
Coverage (PR)
Coverage (PRg)
∞
.5
.18
20
.95
-
σv2 = .2
20 10
.5
.5
.19 .20
16 15
19 20
.95 .95
.95 .95
5
.5
.22
17
27
.92
.93
∞
1
.58
2
.94
-
σv2 = 1
20 10
1
1
.59 .61
-3
-7
1
0
.93 .92
.94 .93
5
1
.64
-14
0
.90
.91
Table 2: Comparisons between the Prasad-Rao estimator (PR) and its generalization (PRg) when all the
small area variances are the same and m = 20, p = 1.
ni
σi2
M SE(µ̂i )
Relative bias mseP R (µ̂i ) (%)
Relative bias mseP Rg (µ̂i ) (%)
Coverage (PR)
Coverage (PRg)
∞
.5
.17
10
.94
-
σv2 = .2
20 10
.5
.5
.18 .19
7
3
10
9
.94 .93
.94 .94
5
.5
.21
-2
10
.91
.92
∞
1
.55
1
.94
-
σv2 = 1
20 10
1
1
.56 .58
-4
-8
0
-1
.93 .92
.94 .93
5
1
.61
-18
-3
.89
.90
Table 3: Comparisons between the Prasad-Rao estimator (PR) and its generalization (PRg) when all the
small area variances are the same and m = 30, p = 1.
where E[mse(µ̂i )] is the average on the N replicates of the mean squared error estimates for the ith small
area, and the mean is taken on all the small areas j such that σi2 = σj2 . A 95% prediction interval for µi is
given by
p
µ̂i ± 1.96 mse(µ̂i ).
The real coverage of this interval is also calculated in the simulations. The results are presented by combining
all the small areas with the same characteristics.
4.2. Comparisons between Prasad-Rao estimator and its generalization
Tables 2 and 3 present findings of the Monte Carlo analysis when all the small areas have the same variances.
The Prasad-Rao estimator mseP R (µ̂i ) refers to (2.7) with γ̂i set equal to 0. Tables 2 and 3 show that, as
the size of the sample within each small area increases, M SE(µ̂i ) decreases. When σv2 = 1, the bias of
mseP R (µ̂i ) can be severe especially when relatively few degrees of freedom are available to estimate σ i2 . The
generalized estimator provides protection against this bias. The real coverage of the 95% prediction interval
is below the nominal level, even when calculated with the generalized estimator. This could possibly be
improved by replacing the normal critical, 1.96, value by a t critical value. When σ v2 = .2, the two mean
squared error estimators have a positive bias. This is similar to the findings of Lahiri and Rao (1995); the
relatively small value for σv2 makes the occurrence of σ̂v2 = 0 more likely. Then mseP R (µ̂i ) and mseP Rg (µ̂i )
are positively biased.
Tables 4 and 5 present simulation results when the small area variances vary according to the scenarios of
Table 1. The findings are similar to those of Tables 2 and 3. When the sample size ni within a small area is
small, mseP R (µ̂i ) underestimates the mean squared error severely, especially when σv2 = 1.0. The correction
derived in Section 3 decreases this bias.
To end this section, simulations carried out with p = 5 explanatory variables are presented. These variables
have skewed distributions so that the diagonal elements of the hat matrix, hii , vary widely, from 0.05 to
ni
σi2
M SE(µ̂i )
Relative bias mseP R (µ̂i ) (%)
Relative bias mseP Rg (µ̂i ) (%)
Coverage (PR)
Coverage (PRg)
scenario (a), σv2 = 0.2
4
6
8
10
.5
.33 .25
.2
.21 .17 .14
.13
-15
3
18
26
-2
13
26
32
.92 .92 .92
.92
.93 .93 .93
.93
ni
σi2
M SE(µ̂i )
Relative bias mseP R (µ̂i ) (%)
Relative bias mseP Rg (µ̂i ) (%)
Coverage (PR)
Coverage (PRg)
scenario (b), σv2 = 1.0
4
6
8
10
1
.67 .5
.4
.63 .47 .39
.33
-24 -11 -6
-2
-6
1
2
4
.90 .90 .90
.90
.91 .91 .92
.91
Table 4: Comparisons between the Prasad-Rao estimator (PR) and its generalization (PRg)when the small
area variances vary according to the scenarios of Table 1 and m = 20, p = 1.
ni
σi2
M SE(µ̂i )
Relative bias mseP R (µ̂i ) (%)
Relative bias mseP Rg (µ̂i ) (%)
Coverage (PR)
Coverage (PRg)
scenario
4
6
.5
.33
.20 .16
-24 -10
-10
1
.92 .91
.93 .92
(a), σv2 = 0.2
8
10 12
.25 .2 .17
.14 .12 .11
-3
3
8
5
9
14
.92 .92 .91
.93 .93 .92
ni
σi2
M SE(µ̂i )
Relative bias mseP R (µ̂i ) (%)
Relative bias mseP Rg (µ̂i ) (%)
Coverage (PR)
Coverage (PRg)
scenario
4
6
1
.67
.61 .46
-25 -15
-6
-2
.90 .91
.92 .92
(b), σv2 = 1.0
8
10 12
.5
.4 .33
.37 .31 .27
-10 -7
-5
0
1
1
.90 .90 .90
.92 .92 .92
Table 5: Comparisons between the Prasad-Rao estimator (PR) and its generalization (PRg)when the small
area variances vary according to the scenarios of Table 1 and m = 30, p = 1.
ni
σi2
M SE(µ̂i )
Relative bias mseP R (µ̂i ) (%)
Relative bias mseP Rg (µ̂i ) (%)
Coverage (PR)
Coverage (PRg)
σv2 = .2
20 10
5
.5
.5
.5
.23 .24 .26
4
-1
-8
6
4
0
.94 .93 .90
.94 .94 .91
20
1
.63
-4
0
.93
.94
σv2 = 1
10
1
.65
-9
-2
.92
.93
5
1
.69
-18
-4
.88
.90
Table 6: Comparisons between the Prasad-Rao estimator (PR) and its generalization (PRg) when all the
small area variances are the same and m = 30, p = 5.
ni
σi2
M SE(µ̂i )
Relative bias mseP R (µ̂i ) (%)
Relative bias mseP Rg (µ̂i ) (%)
Coverage (PR)
Coverage (PRg)
scenario
4
6
.5
.33
.24 .18
-27 -12
-15 -2
.91 .91
.92 .92
(a), σv2 = 0.2
8
10 12
.25 .2 .17
.15 .14 .12
-4
4
9
4
10 14
.91 .91 .92
.92 .92 .93
ni
σi2
M SE(µ̂i )
Relative bias mseP R (µ̂i ) (%)
Relative bias mseP Rg (µ̂i ) (%)
Coverage (PR)
Coverage (PRg)
scenario
4
6
1
.67
.67 .49
-25 -15
-8
-2
.90 .90
.92 .92
(b), σv2 = 1.0
8
10 12
.5
.4 .33
.39 .33 .29
-10 -5
-5
0
2
1
.90 .90 .90
.92 .92 .92
Table 7: Comparisons between the Prasad-Rao estimator (PR) and its generalization (PRg)when the small
area variances vary according to the scenarios of Table 1 and m = 30, p = 5.
0.55. Comparing the findings of Tables 3 and 6, the additional explanatory variables increase the small area
mean squared error by more than 20%. The generalized estimator captures this additional variability well
since the corresponding biases and prediction interval coverages in the two tables are equal for all practical
purposes. Tables 5 and 7 convey a similar message when the sample sizes vary. In Table 7 the average of h ii
varies from 0.11 for small areas 13 to 18 (ni = 8) to 0.23 for small areas 19 to 24 (ni = 10). The changes
in hii do not have any impact on the simulation results. Indeed the biases and the coverage rates obtained
with the generalized estimator are almost identical in Tables 5 and 7.
5. MODELLING CENSUS UNDER-COVERAGE
Every five years, there is census in Canada. Unfortunately the regular census operations fail to count all the
citizens. The under count varies between 2 and 3 %. Statistics Canada has a special survey to estimate the
number of persons missed by the census. The estimates produced by this survey have a large variability and
ways of improving the direct under count estimators have been investigated, see Rivest (1995) for instance.
Following Dick (1995), we consider the 1991 estimates of under coverage in the 96 province × age × sex cells.
There are P = 12 provinces (including the two territories) and A = 8 age-sex groups. The age categories
are less than 19, 20 to 29, 30 to 44, and 45 and over. The 96 cells are considered as small areas. For each
Category
intercept
age*sex interaction
Variable
language*sex*age interaction
province*renters interaction
variance
M * 20-29
M * 30-44
F * 20-29
F * 45+
L * F * 0-19
CB * loc
Ont * loc
Que * loc
NB * loc
Yukon * loc
TNO * loc
σ̂v2
Dick
1.0076
.0563
.0207
.0243
.0802
.0436
.0791
.0253
.1039
.0633
.0687
3.37e-05
Modified Dick
1.0099
.0541
.0185
.0222
-.0102
.0680
.0433
.0789
.0259
.1032
.0634
.0680
2.21e-05
Table 8: Two models for the correction factors for under coverage.
one Ci represent the census count which is less that the true count Ti . The number of persons missed is
Mi = Ti − Ci . The aim of the analysis is to estimate the correction factor µi = Ti /Ci = (Mi + Ci )/Ci .
The direct estimator is yi = (M̂i + Ci )/Ci , where M̂i is an estimate of the number of persons missed by
the census. The standard error of M̂i , v(Mi )1/2 , is also estimated in the coverage study. Two explanatory
variables, besides the province, the age and the sex, are available to model the correction factors. There is
the renter’s rate in a cell (loc) the proportion of people who do not speak one of the two official languages
(L).
Before proceeding with the construction of the small area estimates, Dick (1995) smoothed the small area
variances, using the following model
log(v(Mi )) = α + γ log(Ci ) + ηi
where ηi ∼ N (0, ζ 2 ). The smoothed variance estimate for the ith small area is
v̂(Mi ) = exp(−5.5209 + 1.6775 log(Ci )),
where -5.5209 and 1.6775 are the least squares estimates of the parameters of the above model. The
2
ˆ
smoothed variance estimate for yi = 1 + M̂i /Ci is σ̂i2 = v̂(Mi )/Ci2 . By linearization, Var(σ̂
i ) is exp{2(α̂ +
4
γ̂ log(Ci ))}Var(α̂ + γ̂ log(Ci ))/Ci . This plays the role of γ̂i in the mean squared error calculations. Furthermore the small area variances are estimated by v̂(Mi )/Ci2 in the construction of the small area empirical
Bayes estimators. Rivest and Belmonte (2000) argue that this underestimates the small area variances
however this is not pursued here and the σ̂i2 ’s are considered to be unbiased.
Two regression models have been proposed for this data set, one by Dick (1995) and one by Rivest and
Belmonte (2000) who suggested adding one explanatory variable to those of Dick (1995). The parameter
estimates for the two models are given in Table 8. The empirical best linear unbiased estimator of the small
area characteristics are constructed using (2.4) by replacing parameters by their estimates. The Prasad-Rao
mean squared error estimator and its generalization can be estimated using (2.7).
Adding the term for estimating the variances σi2 to Prasar and Rao mean squared error estimator does
not, for all practical purposes, change the estimator. The component of mseP Rg for the estimation of the
ˆ v2 )/σ̂v4 + Var(s
ˆ 2 )/s4 ). Their relative contribution to this term is
2 variances is proportional to s4i σv4 (Var(σ̂
i
i
proportional to their squared coefficients of variation. For Dick’s model and the modified Dick model, the
CVs of σ̂v2 are 0.73 and 1.05 respectively while for s2i , the CVs are in the range 0.1 to 0.2. Accounting for
the estimation of the small area variances increases mseP R by about 1%, as shown in Figure 1.
0.00014
•
0.00010
••
msePRg
•
•
0.00006
•
••
0.00002
••
• •••
••
•
••••
••••••
•••••
••••
•
•
•
•
•
••
0.00002
0.00004
0.00006
0.00008
0.00010
0.00012
0.00014
msePR
Figure 1: A scatter plot of the generalized Prasad-Rao estimator versus the standard estimator for the
modified Dick model.
6. REFERENCES
Battese, G. E., R. M. Harter, and W. A. Fuller (1988), “An Error-Components Model for Prediction of
Country Crop Areas Using Survey and Satellite Data,” Journal of the American Statistical Association
80, pp. 28–36
Booth, J. G. and J. P. Hobert, (1998), “Standard Errors of Predictions in Generalized Linear Mixed
Models,” Journal of the American Statistical Association 93, pp. 262-272
Datta, G. and P. Lahiri (2000), “A Unified Measure of Uncertainty of Estimated Best Linear Predictors in
Small Area Estimation Problems,” Statistica Sinica, 10, pp. 613–627
Dick, P. (1995), “Modeling Net Under-coverage in the 1991 Canadian Census,” Survey Methodology, 21,
pp. 44–55
Fay, R. E. and, R. A. Herriot (1979), “Estimates of Income for Small Places: An Application of James
Stein Procedure to Census Data,” Journal of the American Statistical Association, 74, pp. 269–277
Ghosh, M. and J. N. K. Rao (1994), “Small Area Estimation: An Appraisal,” Statistical Science, 9,
pp. 55–93
Kackar, R. N. and D. A. Harville (1984), “Approximations for Standard Errors of Estimators for Fixed and
Random Effects in Mixed Models,” Journal of the American Statistical Association, 79, pp. 853–862
Lahiri, P. and , J. N. K. Rao (1995), “Robust Estimation of Mean Squared Error of Small Area Estimators,”
Journal of the American Statistical Association, 90, pp. 758–766
Prasad, N.G.N. and J. N. K. Rao (1990), “The Estimation of Mean Squared Errors of Small Area Estimators,” Journal of the American Statistical Association, 85, pp. 163-171
Prasad, N.G.N. and J. N. K. Rao (1999), “On Robust Small Area Estimation Using a Simple Random
Effect Model,” Survey Methodology, 25, pp. 163–171
Purcell, N. J.and L. Kish (1979), “Estimation for Small Domains,” Biometrics 35, pp. 365–384
Rao, J. N. K. (1999), “ Some Recent Advances in Model Based Small Area Estimation,” Survey Methodology, 25, pp. 175–186
Rivest, L.P. (1995),“ A Composite Estimator for Provincial Under-coverage in the Canadian Census,”
Proceedings of the Survey Methods Section. Statistical Society of Canada. pp. 33–38
Rivest, L.P. and E. Belmonte (2000),“A Conditional Mean Squared Error of Small Area Estimators,”
Survey Methodology, 26, pp. 79–90
Singh, M. P., J. Gambino and H. J. Mantel (1994),“Issues and Strategy for Small Area Data,” Survey
Methodology, 20, pp. 1–22
Singh, A. C. , D. M. Stukel and D. Pfeffermann (1998), “Bayesian Versus Frequentist Measures of Error in
Small Area Estimation,” Journal of the Royal Statistical Society, series B, 60, pp. 377–396
Stein C. (1981),“Estimation of the Mean of a Multivariate Normal Distribution,” The Annals of Statistics,
9, pp. 1135–1151
Wang, J. and W. A. Fuller C. (2003),“The Mean Square Error of Small Area Predictors Constructed with
Estimated Area Variances,” Journal of the American Statistical Association , 98, to appear