Model-based small area estimation under informative sampling
1. IntroductionModel-based small area estimation under informative sampling
1. Introduction
Estimates of population totals and means are
often required for small subpopulations (or areas). Traditional area-specific
direct estimators are not reliable if the area sample size is small. As a
result, it becomes necessary to “borrow strength” across areas through indirect
estimation based on models that provide a link to related areas. Linking models
make use of auxiliary population information either at the area level or at the
unit level. Rao (2003, Chapter 7) gives a detailed account of area level and
unit level models that are widely used for small area estimation.
Suppose that the population of interest,
U
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbGaai
ilaaaa@39EF@
consists of
M
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbaaaa@3937@
non-overlapping areas with
N
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobWaaS
baaSqaaiaadMgaaeqaaaaa@3A52@
elements in the
i
th
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGPbWaaW
baaSqabeaacaqG0bGaaeiAaaaaaaa@3B62@
area
(
i
=
1
,
…
,
M
)
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai
aadMgacqGH9aqpcaaIXaGaaiilaiablAciljaacYcacaWGnbaacaGL
OaGaayzkaaGaaiOlaaaa@40A3@
A sample,
s
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGZbGaai
ilaaaa@3A0D@
of
m
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGTbaaaa@3957@
areas is first selected using a
specified sampling scheme with inclusion probabilities
π
i
=
m
p
i
(
i
=
1
,
…
,
M
)
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHapaCda
WgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGTbGaamiCamaaBaaaleaa
caWGPbaabeaakmaabmaabaGaamyAaiabg2da9iaaigdacaGGSaGaeS
OjGSKaaiilaiaad2eaaiaawIcacaGLPaaacaGGSaaaaa@4793@
where
p
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaaS
baaSqaaiaadMgaaeqaaaaa@3A74@
denotes the selection probability
of area
i
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGPbGaai
Olaaaa@3A05@
Subsamples
s
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaS
baaSqaaiaadMgaaeqaaaaa@3A77@
of specified sizes
n
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbWaaS
baaSqaaiaadMgaaeqaaaaa@3A72@
are then independently selected
from the sampled areas
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGPbaaaa@3953@
according to specified sampling
schemes with selection probabilities
p
j
|
i
(
∑
j
=
1
N
i
p
j
|
i
=
1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaaS
baaSqaaiaadQgadaabbaqaaiaadMgaaiaawEa7aaqabaGcdaqadaqa
amaaqadabaGaamiCamaaBaaaleaacaWGQbWaaqqaaeaacaWGPbaaca
GLhWoaaeqaaaqaaiaadQgacqGH9aqpcaaIXaaabaGaamOtamaaBaaa
meaacaWGPbaabeaaa0GaeyyeIuoakiabg2da9iaaigdaaiaawIcaca
GLPaaaaaa@4B7C@
such that the second-stage inclusion
probabilities are
π
j
|
i
=
n
i
p
j
|
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHapaCda
WgaaWcbaGaamOAamaaeeaabaGaamyAaaGaay5bSdaabeaakiabg2da
9iaad6gadaWgaaWcbaGaamyAaaqabaGccaWGWbWaaSbaaSqaamaaei
aabaGaamOAaaGaayjcSdGaamyAaaqabaaaaa@457A@
for unit
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGQbaaaa@3954@
in area
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGPbaaaa@3953@
(
j
=
1
,
…
,
N
i
)
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai
aadQgacqGH9aqpcaaIXaGaaiilaiablAciljaacYcacaWGobWaaSba
aSqaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaaiOlaaaa@41C9@
Typically, the selection
probability
p
j
|
i
=
b
i
j
/
∑
k
=
1
N
i
b
i
k
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaaS
baaSqaaiaadQgadaabbaqaaiaadMgaaiaawEa7aaqabaGccqGH9aqp
daWcgaqaaiaadkgadaWgaaWcbaGaamyAaiaadQgaaeqaaaGcbaWaaa
bmaeaacaWGIbWaaSbaaSqaaiaadMgacaWGRbaabeaaaeaacaWGRbGa
eyypa0JaaGymaaqaaiaad6eadaWgaaadbaGaamyAaaqabaaaniabgg
HiLdaaaOGaaiilaaaa@4B58@
where
b
i
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGIbWaaS
baaSqaaiaadMgacaWGQbaabeaaaaa@3B55@
is a size measure related to the
response variable
y
i
j
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS
baaSqaaiaadMgacaWGQbaabeaakiaac6caaaa@3C28@
In this paper, we focus on the
special case where all the areas are sampled,
m
=
M
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGTbGaey
ypa0Jaamytaiaac6caaaa@3BE1@
We assume a nested error
linear regression model for the population, based on covariates
x
i
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bWaaS
baaSqaaiaadMgacaWGQbaabeaaaaa@3B6F@
related to the response variable
y
i
j
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS
baaSqaaiaadMgacaWGQbaabeaakiaac6caaaa@3C28@
The population model is assumed
to be given by
y
i j
=
x
i j
T
β +
v
i
+
e
i j
; j = 1 , … ,
N
i
; i = 1 , … , M , ( 1.1 )
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS
baaSqaaiaadMgacaWGQbaabeaakiabg2da9iaahIhadaqhaaWcbaGa
amyAaiaadQgaaeaacaWGubaaaOGaaCOSdiabgUcaRiaadAhadaWgaa
WcbaGaamyAaaqabaGccqGHRaWkcaWGLbWaaSbaaSqaaiaadMgacaWG
QbaabeaakiaacUdacaqGGaGaamOAaiabg2da9iaaigdacaGGSaGaeS
OjGSKaaiilaiaad6eadaWgaaWcbaGaamyAaaqabaGccaGG7aGaaeii
aiaadMgacqGH9aqpcaaIXaGaaiilaiablAciljaacYcacaWGnbGaai
ilaiaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGymaiaac6cacaaI
XaGaaiykaaaa@62E0@
where
v
i
∼
iid
N
(
0
,
σ
v
2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG2bWaaS
baaSqaaiaadMgaaeqaaOWaaCbiaeaacqWI8iIoaSqabeaacaqGPbGa
aeyAaiaabsgaaaGccaWGobWaaeWaaeaacaaIWaGaaiilaiabeo8aZn
aaDaaaleaacaWG2baabaGaaGOmaaaaaOGaayjkaiaawMcaaaaa@4636@
are random small area effects
that are independent of the unit-level errors
e
i
j
∼
iid
N
(
0
,
σ
e
2
)
,
x
i
j
=
(
1
,
x
i
j
1
,
x
i
j
2
,
…
,
x
i
j
p
)
T
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGLbWaaS
baaSqaaiaadMgacaWGQbaabeaakmaaxacabaGaeSipIOdaleqabaGa
aeyAaiaabMgacaqGKbaaaOGaamOtamaabmaabaGaaGimaiaacYcacq
aHdpWCdaqhaaWcbaGaamyzaaqaaiaaikdaaaaakiaawIcacaGLPaaa
caGGSaGaaCiEamaaBaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpda
qadaqaaiaaigdacaGGSaGaamiEamaaBaaaleaacaWGPbGaamOAaiaa
igdaaeqaaOGaaiilaiaadIhadaWgaaWcbaGaamyAaiaadQgacaaIYa
aabeaakiaacYcacqWIMaYscaGGSaGaamiEamaaBaaaleaacaWGPbGa
amOAaiaadchaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaWGub
aaaaaa@5E95@
and
β
=
(
β
0
,
β
1
,
…
,
β
p
)
T
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHYoGaey
ypa0ZaaeWaaeaacqaHYoGydaWgaaWcbaGaaGimaaqabaGccaGGSaGa
eqOSdi2aaSbaaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacq
aHYoGydaWgaaWcbaGaamiCaaqabaaakiaawIcacaGLPaaadaahaaWc
beqaaiaadsfaaaGccaGGUaaaaa@4915@
Parameters of interest are the
small area means
Y
¯
i
=
N
i
−
1
∑
j
=
1
N
i
y
i
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbae
badaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGobWaa0baaSqaaiaa
dMgaaeaacqGHsislcaaIXaaaaOWaaabmaeaacaWG5bWaaSbaaSqaai
aadMgacaWGQbaabeaaaeaacaWGQbGaeyypa0JaaGymaaqaaiaad6ea
daWgaaadbaGaamyAaaqabaaaniabggHiLdaaaa@48C1@
which may be approximated by
μ
i
=
X
¯
i
T
β
+
v
i
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda
WgaaWcbaGaamyAaaqabaGccqGH9aqpceWHybGbaebadaqhaaWcbaGa
amyAaaqaaiaadsfaaaGccqaHYoGycqGHRaWkcaWG2bWaaSbaaSqaai
aadMgaaeqaaOGaaiilaaaa@448E@
if the area sizes
N
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobWaaS
baaSqaaiaadMgaaeqaaaaa@3A52@
are large, where
X
¯
i
=
N
i
−
1
∑
j
=
1
N
i
x
i
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHybGbae
badaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGobWaa0baaSqaaiaa
dMgaaeaacqGHsislcaaIXaaaaOWaaabmaeaacaWH4bWaaSbaaSqaai
aadMgacaWGQbaabeaaaeaacaWGQbGaeyypa0JaaGymaaqaaiaad6ea
daWgaaadbaGaamyAaaqabaaaniabggHiLdaaaa@48C7@
is
the known population mean of
x
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4baaaa@3966@
for area
i
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGPbGaai
Olaaaa@3A05@
Efficient model-based estimators of the area
means
μ
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda
WgaaWcbaGaamyAaaqabaaaaa@3B35@
may be obtained if the sampling
design is non-informative for the model, which implies that the sample and the
population models coincide. In particular, empirical best linear unbiased
prediction (EBLUP) estimators (Henderson 1975), based on the assumed sample
model under non-informative sampling, may be used to estimate small area means
Y
¯
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbae
badaWgaaWcbaGaamyAaaqabaaaaa@3A75@
or
μ
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda
WgaaWcbaGaamyAaaqabaaaaa@3B35@
(see Section 2 and Rao 2003,
Chapter 7). However, in many practical situations the selection probabilities
p
j
|
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaaS
baaSqaaiaadQgadaabbaqaaiaadMgaaiaawEa7aaqabaaaaa@3CF7@
may be related to the associated
y
i
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS
baaSqaaiaadMgacaWGQbaabeaaaaa@3B6C@
even after conditioning on the
covariates
x
i
j
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bWaaS
baaSqaaiaadMgacaWGQbaabeaakiaac6caaaa@3C2B@
In such cases, we have
“informative sampling” in the sense that the population model (1.1) no longer
holds for the sample. For example, Pfeffermann and Sverchkov (2007) assumed
that the sampled unit design weight
w
j
|
i
=
π
j
|
i
−
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG3bWaaS
baaSqaamaaeiaabaGaamOAaaGaayjcSdGaamyAaaqabaGccqGH9aqp
cqaHapaCdaqhaaWcbaWaaqGaaeaacaWGQbaacaGLiWoacaWGPbaaba
GaeyOeI0IaaGymaaaaaaa@4515@
is random with conditional
expectation
E
s
i
(
w
j
|
i
|
x
i
j
,
y
i
j
,
v
i
)
=
E
s
i
(
w
j
|
i
|
x
i
j
,
y
i
j
)
=
k
i
exp
(
x
i
j
T
a
+
b
y
i
j
)
,
(
1.2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaafaqaaeOada
aabaGaamyramaaBaaaleaacaWGZbWaaSbaaWqaaiaadMgaaeqaaaWc
beaakmaabmaabaWaaqGaaeaacaWG3bWaaSbaaSqaamaaeiaabaGaam
OAaaGaayjcSdGaamyAaaqabaaakiaawIa7aiaahIhadaWgaaWcbaGa
amyAaiaadQgaaeqaaOGaaiilaiaadMhadaWgaaWcbaGaamyAaiaadQ
gaaeqaaOGaaiilaiaadAhadaWgaaWcbaGaamyAaaqabaaakiaawIca
caGLPaaaaeaacqGH9aqpaeaacaWGfbWaaSbaaSqaaiaadohadaWgaa
adbaGaamyAaaqabaaaleqaaOWaaeWaaeaadaabcaqaaiaadEhadaWg
aaWcbaWaaqGaaeaacaWGQbaacaGLiWoacaWGPbaabeaaaOGaayjcSd
GaaCiEamaaBaaaleaacaWGPbGaamOAaaqabaGccaGGSaGaamyEamaa
BaaaleaacaWGPbGaamOAaaqabaaakiaawIcacaGLPaaaaeaaaeaacq
GH9aqpaeaacaWGRbWaaSbaaSqaaiaadMgaaeqaaOGaciyzaiaacIha
caGGWbWaaeWaaeaacaWH4bWaa0baaSqaaiaadMgacaWGQbaabaGaam
ivaaaakiaahggacqGHRaWkcaWGIbGaamyEamaaBaaaleaacaWGPbGa
amOAaaqabaaakiaawIcacaGLPaaacaGGSaaaaiaaywW7caaMf8UaaG
zbVlaaywW7caaMf8UaaiikaiaaigdacaGGUaGaaGOmaiaacMcaaaa@7D61@
where
a
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHHbaaaa@394F@
and
b
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGIbaaaa@394C@
are fixed unknown constants and
k
i
=
N
i
n
i
−
1
{
∑
j
=
1
N
i
exp
(
−
x
i
j
T
a
−
b
y
i
j
)
/
N
i
}
.
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGRbWaaS
baaSqaaiaadMgaaeqaaOGaeyypa0JaamOtamaaBaaaleaacaWGPbaa
beaakiaad6gadaqhaaWcbaGaamyAaaqaaiabgkHiTiaaigdaaaGcda
GadaqaamaalyaabaWaaabCaeaaciGGLbGaaiiEaiaacchadaqadaqa
aiabgkHiTiaahIhadaqhaaWcbaGaamyAaiaadQgaaeaacaWGubaaaO
GaaCyyaiabgkHiTiaadkgacaWG5bWaaSbaaSqaaiaadMgacaWGQbaa
beaaaOGaayjkaiaawMcaaaWcbaGaamOAaiabg2da9iaaigdaaeaaca
WGobWaaSbaaWqaaiaadMgaaeqaaaqdcqGHris5aaGcbaGaamOtamaa
BaaaleaacaWGPbaabeaaaaaakiaawUhacaGL9baacaGGUaaaaa@5C1D@
Under informative sampling within areas, the
EBLUP estimator of
Y
¯
i
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbae
badaWgaaWcbaGaamyAaaqabaGccaGGSaaaaa@3B2F@
assuming that model (1.1) holds
for the sample, may be heavily biased. It is, therefore, necessary to develop
estimators that can account for sample selection bias and thus reduce
estimation bias. Pfeffermann and Sverchkov (2007) developed a bias-adjusted
estimator of the mean
Y
¯
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbae
badaWgaaWcbaGaamyAaaqabaaaaa@3A75@
under the assumption (1.2) on the
design weights
w
j
|
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG3bWaaS
baaSqaamaaeiaabaGaamOAaaGaayjcSdGaamyAaaqabaaaaa@3D00@
and assuming that the sample
model is a nested error model
y
i j
=
x
i j
T
α +
u
i
+
h
i j
; j = 1 , … ,
n
i
; i = 1 , … , M , ( 1.3 )
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS
baaSqaaiaadMgacaWGQbaabeaakiabg2da9iaahIhadaqhaaWcbaGa
amyAaiaadQgaaeaacaWGubaaaOGaaCySdiabgUcaRiaadwhadaWgaa
WcbaGaamyAaaqabaGccqGHRaWkcaWGObWaaSbaaSqaaiaadMgacaWG
QbaabeaakiaacUdacaqGGaGaamOAaiabg2da9iaaigdacaGGSaGaeS
OjGSKaaiilaiaad6gadaWgaaWcbaGaamyAaaqabaGccaGG7aGaaeii
aiaadMgacqGH9aqpcaaIXaGaaiilaiablAciljaacYcacaWGnbGaai
ilaiaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGymaiaac6cacaaI
ZaGaaiykaaaa@6303@
where
u
i
∼
iid
N
(
0
,
σ
u
2
)
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG1bWaaS
baaSqaaiaadMgaaeqaaOWaaCbiaeaacqWI8iIoaSqabeaacaqGPbGa
aeyAaiaabsgaaaGccaWGobWaaeWaaeaacaaIWaGaaiilaiabeo8aZn
aaDaaaleaacaWG1baabaGaaGOmaaaaaOGaayjkaiaawMcaaiaabYca
aaa@46E3@
and
h
i
j
|
j
∈
s
i
∼
iid
N
(
0
,
σ
h
2
)
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaabcaqaai
aadIgadaWgaaWcbaGaamyAaiaadQgaaeqaaaGccaGLiWoacaWGQbGa
eyicI4Saam4CamaaBaaaleaacaWGPbaabeaakmaaxacabaGaeSipIO
daleqabaGaaeyAaiaabMgacaqGKbaaaOGaamOtamaabmaabaGaaGim
aiaacYcacqaHdpWCdaqhaaWcbaGaamiAaaqaaiaaikdaaaaakiaawI
cacaGLPaaacaGGUaaaaa@4DE0@
Pfeffermann and Sverchkov (2007)
noted that under a sampling scheme satisfying (1.2) the population model is
also a nested error model but with different parameters. However, they do not
use the form of the population model. The sample model (1.3) is identified
after fitting the model to the sample data and then doing some model
diagnostics. Similarly, model (1.2) on the weights is identified from the
sample data
{
w
j
|
i
,
y
i
j
,
x
i
j
,
j
∈
s
i
,
i
∈
s
}
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaGadaqaai
aadEhadaWgaaWcbaWaaqGaaeaacaWGQbaacaGLiWoacaWGPbaabeaa
kiaacYcacaWG5bWaaSbaaSqaaiaadMgacaWGQbaabeaakiaacYcaca
WH4bWaaSbaaSqaaiaadMgacaWGQbaabeaakiaacYcacaWGQbGaeyic
I4Saam4CamaaBaaaleaacaWGPbaabeaakiaabYcacaWGPbGaeyicI4
Saam4CaaGaay5Eaiaaw2haaiaac6caaaa@50CA@
Their estimators are noted (PS)
in the following.
Prasad and Rao (1999) and You and Rao (2002)
developed pseudo-EBLUP estimators of small area means
μ
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda
WgaaWcbaGaamyAaaqabaaaaa@3B35@
that depend on the sampling
weights
w
j
|
i
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG3bWaaS
baaSqaamaaeiaabaGaamOAaaGaayjcSdGaamyAaaqabaGccaGGSaaa
aa@3DBA@
assuming non-informative sampling
for the model (1.1). Their motivation for pseudo-EBLUP is to ensure design
consistency as the area sample size,
n
i
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbWaaS
baaSqaaiaadMgaaeqaaOGaaiilaaaa@3B2C@
increases. The estimators of You
and Rao (note (YR) in the following) also satisfy a benchmarking property in
the sense that the associated estimators of area totals add up to a reliable
direct estimator of the total, unlike the EBLUP estimators. Stefan (2005)
studied the empirical performance of pseudo-EBLUP estimators under informative
sampling for model (1.1) and showed that the pseudo-EBLUP leads to smaller bias
compared to the EBLUP .
The main purpose
of our paper is to study augmented sample models of the form
y
i j
=
x
i j
T
β
0
+ g (
p
j | i
)
δ
0
+
v
˜
i
+
e
˜
i j
; j = 1 , … ,
n
i
; i = 1 , … , M ( 1.4 )
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS
baaSqaaiaadMgacaWGQbaabeaakiabg2da9iaahIhadaqhaaWcbaGa
amyAaiaadQgaaeaacaWGubaaaOGaaCOSdmaaBaaaleaacaaIWaaabe
aakiabgUcaRiaadEgadaqadaqaaiaadchadaWgaaWcbaWaaqGaaeaa
caWGQbaacaGLiWoacaWGPbaabeaaaOGaayjkaiaawMcaaiabes7aKn
aaBaaaleaacaaIWaaabeaakiabgUcaRiqadAhagaacamaaBaaaleaa
caWGPbaabeaakiabgUcaRiqadwgagaacamaaBaaaleaacaWGPbGaam
OAaaqabaGccaGG7aGaaeiiaiaadQgacqGH9aqpcaaIXaGaaiilaiab
lAciljaacYcacaWGUbWaaSbaaSqaaiaadMgaaeqaaOGaai4oaiaabc
cacaWGPbGaeyypa0JaaGymaiaacYcacqWIMaYscaGGSaGaamytaiaa
ywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGymaiaac6cacaaI0aGaai
ykaaaa@6DEB@
for a
suitably defined function
g
i
j
=
g
(
p
j
|
i
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGNbWaaS
baaSqaaiaadMgacaWGQbaabeaakiabg2da9iaadEgadaqadaqaaiaa
dchadaWgaaWcbaWaaqGaaeaacaWGQbaacaGLiWoacaWGPbaabeaaaO
GaayjkaiaawMcaaaaa@437D@
of the probability
p
j
|
i
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaaS
baaSqaamaaeiaabaGaamOAaaGaayjcSdGaamyAaaqabaGccaGGSaaa
aa@3DB3@
where
v
˜
i
∼
iid
N
(
0
,
σ
v
0
2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWG2bGbaG
aadaWgaaWcbaGaamyAaaqabaGcdaWfGaqaaiablYJi6aWcbeqaaiaa
bMgacaqGPbGaaeizaaaakiaad6eadaqadaqaaiaaicdacaGGSaGaeq
4Wdm3aa0baaSqaaiaadAhacaaIWaaabaGaaGOmaaaaaOGaayjkaiaa
wMcaaaaa@46FF@
and independent of
e
˜
i
j
∼
iid
N
(
0
,
σ
e
0
2
)
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGLbGbaG
aadaWgaaWcbaGaamyAaiaadQgaaeqaaOWaaCbiaeaacqWI8iIoaSqa
beaacaqGPbGaaeyAaiaabsgaaaGccaWGobWaaeWaaeaacaaIWaGaai
ilaiabeo8aZnaaDaaaleaacaWGLbGaaGimaaqaaiaaikdaaaaakiaa
wIcacaGLPaaacaGGSaaaaa@487C@
and
β
0
=
(
β
00
,
β
01
,
…
,
β
0
p
)
T
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHYoWaaS
baaSqaaiaaicdaaeqaaOGaeyypa0ZaaeWaaeaacqaHYoGydaWgaaWc
baGaaGimaiaaicdaaeqaaOGaaiilaiabek7aInaaBaaaleaacaaIWa
GaaGymaaqabaGccaGGSaGaeSOjGSKaaiilaiabek7aInaaBaaaleaa
caaIWaGaamiCaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiaads
faaaGccaGGUaaaaa@4C33@
The sample model (1.4) is
identified after fitting the model to sample data for different choices of the
function
g
(
⋅
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGNbWaae
WaaeaacqGHflY1aiaawIcacaGLPaaaaaa@3D24@
and checking their adequacy. For
example, residuals
r
i
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGYbWaaS
baaSqaaiaadMgacaWGQbaabeaaaaa@3B65@
from fitting the model (1.4)
without the augmenting variable
g
(
p
j
|
i
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGNbWaae
WaaeaacaWGWbWaaSbaaSqaamaaeiaabaGaamOAaaGaayjcSdGaamyA
aaqabaaakiaawIcacaGLPaaaaaa@3F78@
may be plotted against
g
(
p
j
|
i
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGNbWaae
WaaeaacaWGWbWaaSbaaSqaamaaeiaabaGaamOAaaGaayjcSdGaamyA
aaqabaaakiaawIcacaGLPaaaaaa@3F78@
to select
g
(
⋅
)
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGNbWaae
WaaeaacqGHflY1aiaawIcacaGLPaaacaGGUaaaaa@3DD6@
The identified augmented sample
model will also hold for the population (Skinner 1994, Rao 2003, Section 5.3).
Possible choices of
g
(
p
j
|
i
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGNbWaae
WaaeaacaWGWbWaaSbaaSqaamaaeiaabaGaamOAaaGaayjcSdGaamyA
aaqabaaakiaawIcacaGLPaaaaaa@3F78@
are
p
j
|
i
,
log
p
j
|
i
,
w
j
|
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaaS
baaSqaamaaeiaabaGaamOAaaGaayjcSdGaamyAaaqabaGccaGGSaGa
ciiBaiaac+gacaGGNbGaamiCamaaBaaaleaadaabcaqaaiaadQgaai
aawIa7aiaadMgaaeqaaOGaaiilaiaadEhadaWgaaWcbaWaaqGaaeaa
caWGQbaacaGLiWoacaWGPbaabeaaaaa@4A6C@
and
n
i
w
j
|
i
=
p
j
|
i
−
1
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbWaaS
baaSqaaiaadMgaaeqaaOGaam4DamaaBaaaleaadaabcaqaaiaadQga
aiaawIa7aiaadMgaaeqaaOGaeyypa0JaamiCamaaDaaaleaadaabca
qaaiaadQgaaiaawIa7aiaadMgaaeaacqGHsislcaaIXaaaaOGaaiOl
aaaa@4720@
From the augmented sample model (1.4) we obtain
the EBLUP estimators of
Y
¯
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbae
badaWgaaWcbaGaamyAaaqabaaaaa@3A75@
or
μ
i
=
X
¯
i
T
β
0
+
G
¯
i
δ
0
+
v
˜
i
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda
WgaaWcbaGaamyAaaqabaGccqGH9aqpceWHybGbaebadaqhaaWcbaGa
amyAaaqaaiaadsfaaaGccaWHYoWaaSbaaSqaaiaaicdaaeqaaOGaey
4kaSIabm4rayaaraWaaSbaaSqaaiaadMgaaeqaaOGaeqiTdq2aaSba
aSqaaiaaicdaaeqaaOGaey4kaSIabmODayaaiaWaaSbaaSqaaiaadM
gaaeqaaOGaaiilaaaa@4AA9@
the approximate area mean under
the augmented population model, where
G
¯
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGhbGbae
badaWgaaWcbaGaamyAaaqabaaaaa@3A63@
is the area mean of the
population values
g
(
p
j
|
i
)
≡
g
i
j
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGNbWaae
WaaeaacaWGWbWaaSbaaSqaamaaeiaabaGaamOAaaGaayjcSdGaamyA
aaqabaaakiaawIcacaGLPaaacqGHHjIUcaWGNbWaaSbaaSqaaiaadM
gacaWGQbaabeaakiaac6caaaa@44F2@
The EBLUP of
Y
¯
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbae
badaWgaaWcbaGaamyAaaqabaaaaa@3A75@
or
μ
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda
WgaaWcbaGaamyAaaqabaaaaa@3B35@
requires the knowledge of
G
¯
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGhbGbae
badaWgaaWcbaGaamyAaaqabaaaaa@3A63@
which depends on all the
population values
p
j
|
i
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaaS
baaSqaamaaeiaabaGaamOAaaGaayjcSdGaamyAaaqabaGccaGGUaaa
aa@3DB5@
However, the choice
g
(
p
j
|
i
)
=
p
j
|
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGNbWaae
WaaeaacaWGWbWaaSbaaSqaamaaeiaabaGaamOAaaGaayjcSdGaamyA
aaqabaaakiaawIcacaGLPaaacqGH9aqpcaWGWbWaaSbaaSqaamaaei
aabaGaamOAaaGaayjcSdGaamyAaaqabaaaaa@4512@
gives
G
¯
i
=
1
/
N
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGhbGbae
badaWgaaWcbaGaamyAaaqabaGccqGH9aqpdaWcgaqaaiaaigdaaeaa
caWGobWaaSbaaSqaaiaadMgaaeqaaaaaaaa@3E31@
and the choice
g
(
p
j
|
i
)
=
n
i
w
j
|
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGNbWaae
WaaeaacaWGWbWaaSbaaSqaamaaeiaabaGaamOAaaGaayjcSdGaamyA
aaqabaaakiaawIcacaGLPaaacqGH9aqpcaWGUbWaaSbaaSqaaiaadM
gaaeqaaOGaam4DamaaBaaaleaadaabcaqaaiaadQgaaiaawIa7aiaa
dMgaaeqaaaaa@4730@
gives
G
¯
i
=
n
i
W
¯
i
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGhbGbae
badaWgaaWcbaGaamyAaaqabaGccqGH9aqpcaWGUbWaaSbaaSqaaiaa
dMgaaeqaaOGabm4vayaaraWaaSbaaSqaaiaadMgaaeqaaOGaaiilaa
aa@4052@
where
W
¯
i
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGxbGbae
badaWgaaWcbaGaamyAaaqabaaaaa@3A72@
is the area population mean of
the weights
w
j
|
i
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG3bWaaS
baaSqaamaaeiaabaGaamOAaaGaayjcSdGaamyAaaqabaGccaGGUaaa
aa@3DBC@
The means
W
¯
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGxbGbae
badaWgaaWcbaGaamyAaaqabaaaaa@3A73@
are often known in practice.
Pseudo-EBLUP estimators under the augmented model are also studied.
We conducted a
simulation study under the design-model (or pm) framework to study the bias and
MSE of the proposed estimators relative to EBLUP and pseudo-EBLUP estimators
based on non-informative sampling, and the bias-adjusted estimators of
Pfeffermann and Sverchkov (2007). We also studied the performance of MSE
estimators in terms of relative bias.
Section 2 summarizes the existing model-based
methods for estimating the small area means
Y
¯
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbae
badaWgaaWcbaGaamyAaaqabaaaaa@3A75@
or
μ
i
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9LqFf0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaH8oqBda
WgaaWcbaGaamyAaaqabaGccaGGUaaaaa@3BF1@
Proposed methods based on the
augmented sample model (1.4) are presented in Section 3 . The results of the
simulation study are reported in Section 4 . Concluding remarks are given in
Section 5 .
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Survey Methodology publishes articles dealing with various aspects of statistical development relevant to a statistical agency, such as design issues in the context of practical constraints, use of different data sources and collection techniques, total survey error, survey evaluation, research in survey methodology, time series analysis, seasonal adjustment, demographic studies, data integration, estimation and data analysis methods, and general survey systems development. The emphasis is placed on the development and evaluation of specific methodologies as applied to data collection or the data themselves. All papers will be refereed. However, the authors retain full responsibility for the contents of their papers and opinions expressed are not necessarily those of the Editorial Board or of Statistics Canada.
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Copyright
Published by authority of the Minister responsible for Statistics Canada.
© Minister of Industry, 2015
Catalogue no. 12-001-X
Frequency: semi-annual
Ottawa
Date modified:
2017-09-20