A note on regression estimation with unknown population size
2. Regression estimatorsA note on regression estimation with unknown population size
2. Regression estimators
Under
general regularity conditions (Isaki and Fuller 1982; Montanari 1987), an
approximation to the regression estimator (1.1) is
Y
˜
REG
=
Y
^
π
+
(
X
−
X
^
π
)
Τ
B
,
(
2.1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaG
aadaWgaaWcbaGaaeOuaiaabweacaqGhbaabeaakiaai2daceWGzbGb
aKaadaWgaaWcbaGaeqiWdahabeaakiabgUcaRmaabmaabaGaaCiwai
abgkHiTiqahIfagaqcamaaBaaaleaacqaHapaCaeqaaaGccaGLOaGa
ayzkaaWaaWbaaSqabeaacqGHKoavaaGccaWHcbGaaGilaiaaywW7ca
aMf8UaaGzbVlaaywW7caaMf8UaaiikaiaaikdacaGGUaGaaGymaiaa
cMcaaaa@5535@
where
B
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHcbaaaa@392B@
is the limit in probability of
B
^
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHcbGbaK
aaaaa@393B@
when both the sample and the population sizes
tend to infinity. For large samples, the variance of regression estimator (1.1)
can be studied via (2.1). Note that
Y
˜
REG
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaG
aadaWgaaWcbaGaaeOuaiaabweacaqGhbaabeaaaaa@3BE0@
is unbiased under the sampling plan
p
(
s
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbWaae
WaaeaacaWGZbaacaGLOaGaayzkaaaaaa@3BD6@
and can be re-expressed as:
Y
˜
REG
=
X
Τ
B
+
∑
i
∈
s
d
i
E
i
,
(
2.2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaG
aadaWgaaWcbaGaaeOuaiaabweacaqGhbaabeaakiaai2dacaWHybWa
aWbaaSqabeaacqGHKoavaaGccaWHcbGaey4kaSYaaabuaeqaleaaca
WGPbGaeyicI4Saam4Caaqab0GaeyyeIuoakiaaykW7caWGKbWaaSba
aSqaaiaadMgaaeqaaOGaamyramaaBaaaleaacaWGPbaabeaakiaaiY
cacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIYaGaaiOl
aiaaikdacaGGPaaaaa@5819@
where
E
i
=
y
i
−
x
i
Τ
B
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaS
baaSqaaiaadMgaaeqaaOGaaGypaiaadMhadaWgaaWcbaGaamyAaaqa
baGccqGHsislcaWH4bWaa0baaSqaaiaadMgaaeaacqGHKoavaaGcca
WHcbGaaiOlaaaa@434E@
The
design variance for
Y
^
REG
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK
aadaWgaaWcbaGaaeOuaiaabweacaqGhbaabeaaaaa@3BE1@
can be approximated by
AV
p
(
Y
^
REG
)
=
∑
i
∈
U
∑
j
∈
U
Δ
i
j
E
i
π
i
E
j
π
j
,
(
2.3
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaqGbbGaae
OvamaaBaaaleaacaWGWbaabeaakmaabmaabaGabmywayaajaWaaSba
aSqaaiaabkfacaqGfbGaae4raaqabaaakiaawIcacaGLPaaacaaI9a
WaaabuaeqaleaacaWGPbGaeyicI4Saamyvaaqab0GaeyyeIuoakmaa
qafabeWcbaGaamOAaiabgIGiolaadwfaaeqaniabggHiLdGccaaMc8
UaeuiLdq0aaSbaaSqaaiaadMgacaWGQbaabeaakmaalaaabaGaamyr
amaaBaaaleaacaWGPbaabeaaaOqaaiabec8aWnaaBaaaleaacaWGPb
aabeaaaaGcdaWcaaqaaiaadweadaWgaaWcbaGaamOAaaqabaaakeaa
cqaHapaCdaWgaaWcbaGaamOAaaqabaaaaOGaaGilaiaaywW7caaMf8
UaaGzbVlaaywW7caaMf8UaaiikaiaaikdacaGGUaGaaG4maiaacMca
aaa@66BB@
where
Δ
i
j
=
π
i
j
−
π
i
π
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqqHuoarda
WgaaWcbaGaamyAaiaadQgaaeqaaOGaaGypaiabec8aWnaaBaaaleaa
caWGPbGaamOAaaqabaGccqGHsislcqaHapaCdaWgaaWcbaGaamyAaa
qabaGccqaHapaCdaWgaaWcbaGaamOAaaqabaaaaa@4716@
and
π
i
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHapaCda
WgaaWcbaGaamyAaiaadQgaaeqaaaaa@3C26@
is the second order inclusion probability for
units
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGPbaaaa@394E@
and
j
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGQbGaai
Olaaaa@3A01@
Both the model-assisted (Särndal, Swensson and
Wretman 1992) and the optimal-variance (Montanari 1987) approaches can be used
to estimate
B
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHcbGaai
Olaaaa@39DD@
They both yield approximately unbiased
estimators. In the case of the model-assisted approach, the basic properties
(bias and variance terms) are valid even when the model is not correctly
specified. Under the optimal-variance approach no assumption is made on the
variable of interest.
The model-assisted estimator
of Särndal et al. (1992) assumes a working model between the variable of
interest
(
y
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai
aadMhaaiaawIcacaGLPaaaaaa@3AE7@
and the auxiliary variables
(
x
)
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai
aahIhaaiaawIcacaGLPaaacaGGUaaaaa@3B9C@
The working model is denoted by
m
:
y
i
=
x
i
Τ
β
+
ε
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGTbGaaG
OoaiaadMhadaWgaaWcbaGaamyAaaqabaGccaaI9aGaaCiEamaaDaaa
leaacaWGPbaabaGaeyiPdqfaaOGaaCOSdiabgUcaRiabew7aLnaaBa
aaleaacaWGPbaabeaaaaa@458D@
where
β
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHYoaaaa@399E@
is a vector of
p
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGWbaaaa@3955@
unknown parameters,
E
m
(
ε
i
|
x
i
)
=
0
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaS
baaSqaaiaad2gaaeqaaOWaaeWaaeaadaabcaqaaiabew7aLnaaBaaa
leaacaWGPbaabeaakiaaykW7aiaawIa7aiaaykW7caWH4bWaaSbaaS
qaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaaGypaiaaicdacaGGSaaa
aa@47A8@
V
m
(
ε
i
|
x
i
)
=
σ
i
2
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGwbWaaS
baaSqaaiaad2gaaeqaaOWaaeWaaeaadaabcaqaaiabew7aLnaaBaaa
leaacaWGPbaabeaakiaaykW7aiaawIa7aiaaykW7caWH4bWaaSbaaS
qaaiaadMgaaeqaaaGccaGLOaGaayzkaaGaaGypaiabeo8aZnaaDaaa
leaacaWGPbaabaGaaGOmaaaakiaacYcaaaa@4AA3@
and
Cov
m
(
ε
i
,
ε
j
|
x
i
,
x
j
)
=
0,
i
≠
j
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaqGdbGaae
4BaiaabAhadaWgaaWcbaGaamyBaaqabaGcdaqadaqaamaaeiaabaGa
eqyTdu2aaSbaaSqaaiaadMgaaeqaaOGaaGilaiabew7aLnaaBaaale
aacaWGQbaabeaakiaaykW7aiaawIa7aiaaykW7caWH4bWaaSbaaSqa
aiaadMgaaeqaaOGaaGilaiaahIhadaWgaaWcbaGaamOAaaqabaaaki
aawIcacaGLPaaacaaI9aGaaGimaiaaiYcacaWGPbGaeyiyIKRaamOA
aiaac6caaaa@5449@
Under this approach,
B
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHcbaaaa@392B@
in equation (2.1) is the ordinary least
squares estimator of
β
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHYoaaaa@399E@
in the population and it is given by
B
GREG
=
(
∑
i
∈
U
c
i
x
i
x
i
Τ
)
−
1
(
∑
i
∈
U
c
i
x
i
y
i
)
,
(
2.4
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHcbWaaS
baaSqaaiaabEeacaqGsbGaaeyraiaabEeaaeqaaOGaaGypamaabmaa
baWaaabuaeqaleaacaWGPbGaeyicI4Saamyvaaqab0GaeyyeIuoaki
aaykW7caWGJbWaaSbaaSqaaiaadMgaaeqaaOGaaCiEamaaBaaaleaa
caWGPbaabeaakiaahIhadaqhaaWcbaGaamyAaaqaaiabgs6aubaaaO
GaayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaabmaa
baWaaabuaeqaleaacaWGPbGaeyicI4Saamyvaaqab0GaeyyeIuoaki
aaykW7caWGJbWaaSbaaSqaaiaadMgaaeqaaOGaaCiEamaaBaaaleaa
caWGPbaabeaakiaadMhadaWgaaWcbaGaamyAaaqabaaakiaawIcaca
GLPaaacaaISaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGa
aGOmaiaac6cacaaI0aGaaiykaaaa@6A85@
where
c
i
=
σ
i
−
2
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGJbWaaS
baaSqaaiaadMgaaeqaaOGaaGypaiabeo8aZnaaDaaaleaacaWGPbaa
baGaeyOeI0IaaGOmaaaakiaac6caaaa@4076@
This yields the following estimator for the
total
Y
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGzbaaaa@393E@
Y
^
GREG
=
Y
^
π
+
(
X
−
X
^
π
)
Τ
B
^
GREG
,
(
2.5
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK
aadaWgaaWcbaGaae4raiaabkfacaqGfbGaae4raaqabaGccaaI9aGa
bmywayaajaWaaSbaaSqaaiabec8aWbqabaGccqGHRaWkdaqadaqaai
aahIfacqGHsislceWHybGbaKaadaWgaaWcbaGaeqiWdahabeaaaOGa
ayjkaiaawMcaamaaCaaaleqabaGaeyiPdqfaaOGabCOqayaajaWaaS
baaSqaaiaabEeacaqGsbGaaeyraiaabEeaaeqaaOGaaGilaiaaywW7
caaMf8UaaGzbVlaaywW7caaMf8UaaiikaiaaikdacaGGUaGaaGynai
aacMcaaaa@597B@
where
B
^
GREG
=
(
∑
i
∈
s
c
i
d
i
x
i
x
i
Τ
)
−
1
(
∑
i
∈
s
c
i
d
i
x
i
y
i
)
.
(
2.6
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHcbGbaK
aadaWgaaWcbaGaae4raiaabkfacaqGfbGaae4raaqabaGccaaI9aWa
aeWaaeaadaaeqbqabSqaaiaadMgacqGHiiIZcaWGZbaabeqdcqGHri
s5aOGaaGPaVlaadogadaWgaaWcbaGaamyAaaqabaGccaWGKbWaaSba
aSqaaiaadMgaaeqaaOGaaCiEamaaBaaaleaacaWGPbaabeaakiaahI
hadaqhaaWcbaGaamyAaaqaaiabgs6aubaaaOGaayjkaiaawMcaamaa
CaaaleqabaGaeyOeI0IaaGymaaaakmaabmaabaWaaabuaeqaleaaca
WGPbGaeyicI4Saam4Caaqab0GaeyyeIuoakiaaykW7caWGJbWaaSba
aSqaaiaadMgaaeqaaOGaamizamaaBaaaleaacaWGPbaabeaakiaahI
hadaWgaaWcbaGaamyAaaqabaGccaWG5bWaaSbaaSqaaiaadMgaaeqa
aaGccaGLOaGaayzkaaGaaGOlaiaaywW7caaMf8UaaGzbVlaaywW7ca
aMf8UaaiikaiaaikdacaGGUaGaaGOnaiaacMcaaaa@6EEF@
The
optimal estimator of Montanari (1987), obtained by minimizing the design
variance of
Y
˜
REG
=
Y
^
π
+
(
X
−
X
^
π
)
Τ
B
,
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaG
aadaWgaaWcbaGaaeOuaiaabweacaqGhbaabeaakiaai2daceWGzbGb
aKaadaWgaaWcbaGaeqiWdahabeaakiabgUcaRmaabmaabaGaaCiwai
abgkHiTiqahIfagaqcamaaBaaaleaacqaHapaCaeqaaaGccaGLOaGa
ayzkaaWaaWbaaSqabeaacqGHKoavaaGccaWHcbGaaGilaaaa@49ED@
is
Y
˜
OPT
=
Y
^
π
+
(
X
−
X
^
π
)
Τ
B
OPT
,
(
2.7
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaG
aadaWgaaWcbaGaae4taiaabcfacaqGubaabeaakiaai2daceWGzbGb
aKaadaWgaaWcbaGaeqiWdahabeaakiabgUcaRmaabmaabaGaaCiwai
abgkHiTiqahIfagaqcamaaBaaaleaacqaHapaCaeqaaaGccaGLOaGa
ayzkaaWaaWbaaSqabeaacqGHKoavaaGccaWHcbWaaSbaaSqaaiaab+
eacaqGqbGaaeivaaqabaGccaaISaGaaGzbVlaaywW7caaMf8UaaGzb
VlaaywW7caGGOaGaaGOmaiaac6cacaaI3aGaaiykaaaa@5802@
where
B
OPT
=
{
V
(
X
^
π
)
}
−
1
Cov
(
X
^
π
,
Y
^
π
)
=
(
∑
i
∈
U
∑
j
∈
U
Δ
i
j
x
i
π
i
x
j
Τ
π
j
)
−
1
(
∑
i
∈
U
∑
j
∈
U
Δ
i
j
x
i
π
i
y
j
π
j
)
.
(
2.8
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaafaqaaeGaca
aabaGaaCOqamaaBaaaleaacaqGpbGaaeiuaiaabsfaaeqaaaGcbaGa
aGypamaacmaabaGaamOvamaabmaabaGabCiwayaajaWaaSbaaSqaai
abec8aWbqabaaakiaawIcacaGLPaaaaiaawUhacaGL9baadaahaaWc
beqaaiabgkHiTiaaigdaaaGccaqGdbGaae4BaiaabAhadaqadaqaai
qahIfagaqcamaaBaaaleaacqaHapaCaeqaaOGaaGilaiqadMfagaqc
amaaBaaaleaacqaHapaCaeqaaaGccaGLOaGaayzkaaaabaaabaGaaG
ypamaabmaabaWaaabuaeqaleaacaWGPbGaeyicI4Saamyvaaqab0Ga
eyyeIuoakmaaqafabeWcbaGaamOAaiabgIGiolaadwfaaeqaniabgg
HiLdGccaaMc8UaeuiLdq0aaSbaaSqaaiaadMgacaWGQbaabeaakmaa
laaabaGaaCiEamaaBaaaleaacaWGPbaabeaaaOqaaiabec8aWnaaBa
aaleaacaWGPbaabeaaaaGcdaWcaaqaaiaahIhadaqhaaWcbaGaamOA
aaqaaiabgs6aubaaaOqaaiabec8aWnaaBaaaleaacaWGQbaabeaaaa
aakiaawIcacaGLPaaadaahaaWcbeqaaiabgkHiTiaaigdaaaGcdaqa
daqaamaaqafabeWcbaGaamyAaiabgIGiolaadwfaaeqaniabggHiLd
GcdaaeqbqabSqaaiaadQgacqGHiiIZcaWGvbaabeqdcqGHris5aOGa
aGPaVlabfs5aenaaBaaaleaacaWGPbGaamOAaaqabaGcdaWcaaqaai
aahIhadaWgaaWcbaGaamyAaaqabaaakeaacqaHapaCdaWgaaWcbaGa
amyAaaqabaaaaOWaaSaaaeaacaWG5bWaaSbaaSqaaiaadQgaaeqaaa
GcbaGaeqiWda3aaSbaaSqaaiaadQgaaeqaaaaaaOGaayjkaiaawMca
aiaai6caaaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaG
Omaiaac6cacaaI4aGaaiykaaaa@9851@
The optimal estimator for the total
Y
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGzbaaaa@393E@
is estimated by
Y
^
OPT
=
Y
^
π
+
(
X
−
X
^
π
)
Τ
B
^
OPT
,
(
2.9
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK
aadaWgaaWcbaGaae4taiaabcfacaqGubaabeaakiaai2daceWGzbGb
aKaadaWgaaWcbaGaeqiWdahabeaakiabgUcaRmaabmaabaGaaCiwai
abgkHiTiqahIfagaqcamaaBaaaleaacqaHapaCaeqaaaGccaGLOaGa
ayzkaaWaaWbaaSqabeaacqGHKoavaaGcceWHcbGbaKaadaWgaaWcba
Gaae4taiaabcfacaqGubaabeaakiaaiYcacaaMf8UaaGzbVlaaywW7
caaMf8UaaGzbVlaacIcacaaIYaGaaiOlaiaaiMdacaGGPaaaaa@5815@
where
B
^
OPT
=
(
∑
i
∈
s
∑
j
∈
s
Δ
i
j
π
i
j
x
i
π
i
x
j
Τ
π
j
)
−
1
(
∑
i
∈
s
∑
j
∈
s
Δ
i
j
π
i
j
x
i
π
i
y
j
π
j
)
.
(
2.10
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHcbGbaK
aadaWgaaWcbaGaae4taiaabcfacaqGubaabeaakiaai2dadaqadaqa
amaaqafabeWcbaGaamyAaiabgIGiolaadohaaeqaniabggHiLdGcda
aeqbqabSqaaiaadQgacqGHiiIZcaWGZbaabeqdcqGHris5aOWaaSaa
aeaacqqHuoardaWgaaWcbaGaamyAaiaadQgaaeqaaaGcbaGaeqiWda
3aaSbaaSqaaiaadMgacaWGQbaabeaaaaGcdaWcaaqaaiaahIhadaWg
aaWcbaGaamyAaaqabaaakeaacqaHapaCdaWgaaWcbaGaamyAaaqaba
aaaOWaaSaaaeaacaWH4bWaa0baaSqaaiaadQgaaeaacqGHKoavaaaa
keaacqaHapaCdaWgaaWcbaGaamOAaaqabaaaaaGccaGLOaGaayzkaa
WaaWbaaSqabeaacqGHsislcaaIXaaaaOWaaeWaaeaadaaeqbqabSqa
aiaadMgacqGHiiIZcaWGZbaabeqdcqGHris5aOWaaabuaeqaleaaca
WGQbGaeyicI4Saam4Caaqab0GaeyyeIuoakmaalaaabaGaeuiLdq0a
aSbaaSqaaiaadMgacaWGQbaabeaaaOqaaiabec8aWnaaBaaaleaaca
WGPbGaamOAaaqabaaaaOWaaSaaaeaacaWH4bWaaSbaaSqaaiaadMga
aeqaaaGcbaGaeqiWda3aaSbaaSqaaiaadMgaaeqaaaaakmaalaaaba
GaamyEamaaBaaaleaacaWGQbaabeaaaOqaaiabec8aWnaaBaaaleaa
caWGQbaabeaaaaaakiaawIcacaGLPaaacaaIUaGaaGzbVlaaywW7ca
aMf8UaaGzbVlaaywW7caGGOaGaaGOmaiaac6cacaaIXaGaaGimaiaa
cMcaaaa@8953@
Note that the computation of
the regression vectors requires that the first component that defines them is
invertible. We can ensure this by reducing the number of auxiliary variables
that are input into the regression if not much loss in efficiency of the
resulting regression estimator is incurred. If, on the other hand, there is a
significant loss in efficiency, then we can invert these singular matrices
using generalised inverses.
As mentioned in the
introduction, not all population totals may be known for each component of the
auxiliary vector
x
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bGaai
Olaaaa@3A13@
The regression normally uses the auxiliary
variables for which a corresponding population total is known. Decomposing
x
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bWaaS
baaSqaaiaadMgaaeqaaaaa@3A7B@
as
(
1,
x
i
*
Τ
)
Τ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai
aaigdacaaISaGaaCiEamaaDaaaleaacaWGPbaabaGaaGOkaiabgs6a
ubaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeyiPdqfaaaaa@416F@
where
x
i
*
=
(
x
2
i
,
…
,
x
p
i
)
Τ
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bWaa0
baaSqaaiaadMgaaeaacaaIQaaaaOGaaGypamaabmaabaGaamiEamaa
BaaaleaacaaIYaGaamyAaaqabaGccaaISaGaeSOjGSKaaGilaiaadI
hadaWgaaWcbaGaamiCaiaadMgaaeqaaaGccaGLOaGaayzkaaWaaWba
aSqabeaacqGHKoavaaGccaGGSaaaaa@4879@
Singh and Raghunath (2011) proposed a
GREG -like estimator that assumes that the regression is based on an intercept
and the variable
x
*
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bWaaW
baaSqabeaacaaIQaaaaOGaaiilaaaa@3AFC@
even though only the population total of the
x
*
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bWaaW
baaSqabeaacaaIQaaaaaaa@3A42@
is known.
For the case that
N
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobaaaa@3933@
is not known and that the population total of
x
*
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bWaaW
baaSqabeaacaaIQaaaaaaa@3A42@
is known, their estimator is
Y
^
SREG
=
Y
^
π
+
(
X
*
−
X
^
π
*
)
Τ
B
^
2
,
GREG
,
(
2.11
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK
aadaWgaaWcbaGaae4uaiaabkfacaqGfbGaae4raaqabaGccaaI9aGa
bmywayaajaWaaSbaaSqaaiabec8aWbqabaGccqGHRaWkdaqadaqaai
aahIfadaahaaWcbeqaaiaaiQcaaaGccqGHsislceWHybGbaKaadaqh
aaWcbaGaeqiWdahabaGaaGOkaaaaaOGaayjkaiaawMcaamaaCaaale
qabaGaeyiPdqfaaOGabCOqayaajaWaaSbaaSqaaiaaikdacaGGSaGa
ae4raiaabkfacaqGfbGaae4raaqabaGccaaISaGaaGzbVlaaywW7ca
aMf8UaaGzbVlaaywW7caGGOaGaaGOmaiaac6cacaaIXaGaaGymaiaa
cMcaaaa@5D4A@
where
X
*
=
∑
i
∈
U
x
i
*
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHybWaaW
baaSqabeaacaaIQaaaaOGaaGypamaaqababeWcbaGaamyAaiabgIGi
olaadwfaaeqaniabggHiLdGccaaMc8UaaCiEamaaDaaaleaacaWGPb
aabaGaaGOkaaaaaaa@4488@
and
X
^
π
*
=
∑
i
∈
s
d
i
x
i
*
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHybGbaK
aadaqhaaWcbaGaeqiWdahabaGaaGOkaaaakiaai2dadaaeqaqabSqa
aiaadMgacqGHiiIZcaWGZbaabeqdcqGHris5aOGaaGPaVlaadsgada
WgaaWcbaGaamyAaaqabaGccaWH4bWaa0baaSqaaiaadMgaaeaacaaI
QaaaaOGaaiOlaaaa@493C@
The regression vector of estimated
coefficients
B
^
2
,
GREG
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHcbGbaK
aadaWgaaWcbaGaaGOmaiaacYcacaqGhbGaaeOuaiaabweacaqGhbaa
beaaaaa@3E04@
is obtained from
B
^
GREG
=
(
B
^
1
,
G
R
E
G
,
B
^
2
,
GREG
Τ
)
Τ
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHcbGbaK
aadaWgaaWcbaGaae4raiaabkfacaqGfbGaae4raaqabaGccaaI9aWa
aeWaaeaaceWGcbGbaKaadaWgaaWcbaGaaGymaiaacYcacaGGhbGaai
OuaiaacweacaGGhbaabeaakiaaiYcaceWHcbGbaKaadaqhaaWcbaGa
aGOmaiaacYcacaqGhbGaaeOuaiaabweacaqGhbaabaGaeyiPdqfaaa
GccaGLOaGaayzkaaWaaWbaaSqabeaacqGHKoavaaaaaa@4E3E@
given by (2.6). The approximate design
variance for
Y
^
SREG
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK
aadaWgaaWcbaGaae4uaiaabkfacaqGfbGaae4raaqabaaaaa@3CB7@
takes the same form as equation (2.3), with
E
i
=
y
i
−
x
i
*
Τ
B
2
,
GREG
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaS
baaSqaaiaadMgaaeqaaOGaaGypaiaadMhadaWgaaWcbaGaamyAaaqa
baGccqGHsislcaWH4bWaa0baaSqaaiaadMgaaeaacaaIQaGaeyiPdq
faaOGaaCOqamaaBaaaleaacaaIYaGaaiilaiaabEeacaqGsbGaaeyr
aiaabEeaaeqaaOGaaiilaaaa@48D3@
where
B
2,
GREG
=
{
∑
i
∈
U
c
i
(
x
i
*
−
X
¯
N
*
)
(
x
i
*
−
X
¯
N
*
)
Τ
}
−
1
∑
i
∈
U
c
i
(
x
i
*
−
X
¯
N
*
)
y
i
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHcbWaaS
baaSqaaiaaikdacaaISaGaae4raiaabkfacaqGfbGaae4raaqabaGc
caaI9aWaaiWaaeaadaaeqbqabSqaaiaadMgacqGHiiIZcaWGvbaabe
qdcqGHris5aOGaaGPaVlaadogadaWgaaWcbaGaamyAaaqabaGcdaqa
daqaaiaahIhadaqhaaWcbaGaamyAaaqaaiaaiQcaaaGccqGHsislce
WHybGbaebadaqhaaWcbaGaamOtaaqaaiaaiQcaaaaakiaawIcacaGL
PaaadaqadaqaaiaahIhadaqhaaWcbaGaamyAaaqaaiaaiQcaaaGccq
GHsislceWHybGbaebadaqhaaWcbaGaamOtaaqaaiaaiQcaaaaakiaa
wIcacaGLPaaadaahaaWcbeqaaiabgs6aubaaaOGaay5Eaiaaw2haam
aaCaaaleqabaGaeyOeI0IaaGymaaaakmaaqafabeWcbaGaamyAaiab
gIGiolaadwfaaeqaniabggHiLdGccaaMc8Uaam4yamaaBaaaleaaca
WGPbaabeaakmaabmaabaGaaCiEamaaDaaaleaacaWGPbaabaGaaGOk
aaaakiabgkHiTiqahIfagaqeamaaDaaaleaacaWGobaabaGaaGOkaa
aaaOGaayjkaiaawMcaaiaadMhadaWgaaWcbaGaamyAaaqabaaaaa@70E7@
and
X
¯
N
*
=
∑
i
∈
U
x
i
*
/
N
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHybGbae
badaqhaaWcbaGaamOtaaqaaiaaiQcaaaGccaaI9aWaaSGbaeaadaae
qaqabSqaaiaadMgacqGHiiIZcaWGvbaabeqdcqGHris5aOGaaGPaVl
aahIhadaqhaaWcbaGaamyAaaqaaiaaiQcaaaaakeaacaWGobaaaiaa
c6caaaa@4718@
The
properties of (2.11) can be obtained by noting that
Y
^
SREG
−
Y
=
Y
^
π
−
Y
+
(
X
*
−
X
^
π
*
)
Τ
B
^
2
,
GREG
=
Y
^
π
−
Y
+
(
X
*
−
X
^
π
*
)
Τ
B
2
,
GREG
+
(
X
*
−
X
^
π
*
)
Τ
(
B
^
2
,
GREG
−
B
2
,
GREG
)
.
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaafaqaaeGaca
aabaGabmywayaajaWaaSbaaSqaaiaabofacaqGsbGaaeyraiaabEea
aOqabaGaeyOeI0Iaamywaaqaaiaai2daceWGzbGbaKaadaWgaaWcba
GaeqiWdahabeaakiabgkHiTiaadMfacqGHRaWkdaqadaqaaiaahIfa
daahaaWcbeqaaiaaiQcaaaGccqGHsislceWHybGbaKaadaqhaaWcba
GaeqiWdahabaGaaGOkaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGa
eyiPdqfaaOGabCOqayaajaWaaSbaaSqaaiaaikdacaGGSaGaae4rai
aabkfacaqGfbGaae4raaGcbeaaaeaaaeaacaaI9aGabmywayaajaWa
aSbaaSqaaiabec8aWbqabaGccqGHsislcaWGzbGaey4kaSYaaeWaae
aacaWHybWaaWbaaSqabeaacaaIQaaaaOGaeyOeI0IabCiwayaajaWa
a0baaSqaaiabec8aWbqaaiaaiQcaaaaakiaawIcacaGLPaaadaahaa
Wcbeqaaiabgs6aubaakiaahkeadaWgaaWcbaGaaGOmaiaacYcacaqG
hbGaaeOuaiaabweacaqGhbaabeaakiabgUcaRmaabmaabaGaaCiwam
aaCaaaleqabaGaaGOkaaaakiabgkHiTiqahIfagaqcamaaDaaaleaa
cqaHapaCaeaacaaIQaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacq
GHKoavaaGcdaqadaqaaiqahkeagaqcamaaBaaaleaacaaIYaGaaiil
aiaabEeacaqGsbGaaeyraiaabEeaaOqabaGaeyOeI0IaaCOqamaaBa
aaleaacaaIYaGaaiilaiaabEeacaqGsbGaaeyraiaabEeaaeqaaaGc
caGLOaGaayzkaaGaaGOlaaaaaaa@82B7@
Since
B
^
2,
GREG
−
B
2,
GREG
=
O
p
(
n
−
1
/
2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHcbGbaK
aadaWgaaWcbaGaaGOmaiaaiYcacaqGhbGaaeOuaiaabweacaqGhbaa
beaakiabgkHiTiaahkeadaWgaaWcbaGaaGOmaiaaiYcacaqGhbGaae
OuaiaabweacaqGhbaabeaakiaai2dacaWGpbWaaSbaaSqaaiaadcha
aeqaaOWaaeWaaeaacaWGUbWaaWbaaSqabeaacqGHsisldaWcgaqaai
aaigdaaeaacaaIYaaaaaaaaOGaayjkaiaawMcaaaaa@4C98@
under some regularity conditions discussed in
Fuller (2009, Chapter 2), the last term is of smaller order. Thus,
ignoring the smaller order terms, we get the following approximation
Y
^
SREG
−
Y
≅
∑
i
∈
s
d
i
E
i
−
∑
i
∈
U
E
i
,
(
2.12
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK
aadaWgaaWcbaGaae4uaiaabkfacaqGfbGaae4raaqabaGccqGHsisl
caWGzbGaeyyrIa0aaabuaeqaleaacaWGPbGaeyicI4Saam4Caaqab0
GaeyyeIuoakiaaykW7caWGKbWaaSbaaSqaaiaadMgaaeqaaOGaamyr
amaaBaaaleaacaWGPbaabeaakiabgkHiTmaaqafabeWcbaGaamyAai
abgIGiolaadwfaaeqaniabggHiLdGccaaMc8UaamyramaaBaaaleaa
caWGPbaabeaakiaaiYcacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVl
aacIcacaaIYaGaaiOlaiaaigdacaaIYaGaaiykaaaa@6176@
where
E
i
=
y
i
−
x
i
*
Τ
B
2,
GREG
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGfbWaaS
baaSqaaiaadMgaaeqaaOGaaGypaiaadMhadaWgaaWcbaGaamyAaaqa
baGccqGHsislcaWH4bWaa0baaSqaaiaadMgaaeaacaaIQaGaeyiPdq
faaOGaaCOqamaaBaaaleaacaaIYaGaaGilaiaabEeacaqGsbGaaeyr
aiaabEeaaeqaaOGaaiOlaaaa@48DB@
Thus,
Y
^
SREG
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK
aadaWgaaWcbaGaae4uaiaabkfacaqGfbGaae4raaqabaaaaa@3CB7@
is approximately design-unbiased. The
asymptotic variance can be computed using
V
{
∑
i
∈
s
d
i
E
i
−
∑
i
∈
U
E
i
}
=
E
{
(
∑
i
∈
s
d
i
E
i
−
∑
i
∈
U
E
i
)
2
}
.
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGwbWaai
WaaeaadaaeqbqabSqaaiaadMgacqGHiiIZcaWGZbaabeqdcqGHris5
aOGaaGPaVlaadsgadaWgaaWcbaGaamyAaaqabaGccaWGfbWaaSbaaS
qaaiaadMgaaeqaaOGaeyOeI0YaaabuaeqaleaacaWGPbGaeyicI4Sa
amyvaaqab0GaeyyeIuoakiaaykW7caWGfbWaaSbaaSqaaiaadMgaae
qaaaGccaGL7bGaayzFaaGaaGypaiaadweadaGadaqaamaabmaabaWa
aabuaeqaleaacaWGPbGaeyicI4Saam4Caaqab0GaeyyeIuoakiaayk
W7caWGKbWaaSbaaSqaaiaadMgaaeqaaOGaamyramaaBaaaleaacaWG
PbaabeaakiabgkHiTmaaqafabeWcbaGaamyAaiabgIGiolaadwfaae
qaniabggHiLdGccaaMc8UaamyramaaBaaaleaacaWGPbaabeaaaOGa
ayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOGaay5Eaiaaw2haai
aai6caaaa@6C5D@
As we can see, the asymptotic variance can be quite large unless
∑
i
∈
U
E
i
=
0.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaaeqaqabS
qaaiaadMgacqGHiiIZcaWGvbaabeqdcqGHris5aOGaaGPaVlaadwea
daWgaaWcbaGaamyAaaqabaGccaaI9aGaaGimaiaac6caaaa@4346@
Remark 2.1 If
y
i
=
a
+
b
x
i
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS
baaSqaaiaadMgaaeqaaOGaaGypaiaadggacqGHRaWkcaWGIbGaamiE
amaaBaaaleaacaWGPbaabeaakiaacYcaaaa@40C9@
we have
Y
^
SREG
−
Y
=
(
N
^
π
−
N
)
a
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK
aadaWgaaWcbaGaae4uaiaabkfacaqGfbGaae4raaqabaGccqGHsisl
caWGzbGaaGypamaabmaabaGabmOtayaajaWaaSbaaSqaaiabec8aWb
qabaGccqGHsislcaWGobaacaGLOaGaayzkaaGaamyyaaaa@4658@
and this implies that
V
(
Y
^
SREG
)
=
a
2
V
(
N
^
π
)
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGwbWaae
WaaeaaceWGzbGbaKaadaWgaaWcbaGaae4uaiaabkfacaqGfbGaae4r
aaqabaaakiaawIcacaGLPaaacaaI9aGaamyyamaaCaaaleqabaGaaG
OmaaaakiaadAfadaqadaqaaiqad6eagaqcamaaBaaaleaacqaHapaC
aeqaaaGccaGLOaGaayzkaaGaaiOlaaaa@47B1@
This means that if
V
(
N
^
π
)
>
0
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGwbWaae
WaaeaaceWGobGbaKaadaWgaaWcbaGaeqiWdahabeaaaOGaayjkaiaa
wMcaaiaai6dacaaIWaGaaiilaaaa@3FCC@
we can artificially
increases
a
2
V
(
N
^
π
)
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGHbWaaW
baaSqabeaacaaIYaaaaOGaamOvamaabmaabaGabmOtayaajaWaaSba
aSqaaiabec8aWbqabaaakiaawIcacaGLPaaacaGGSaaaaa@4023@
the variance of
Y
^
SREG
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK
aadaWgaaWcbaGaae4uaiaabkfacaqGfbGaae4raaqabaGccaGGSaaa
aa@3D71@
by choosing large
values of
a
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGHbGaai
Olaaaa@39F8@
Note that the optimal
regression estimator using
x
*
=
(
x
2
,
…
,
x
p
)
Τ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bWaaW
baaSqabeaacaaIQaaaaOGaaGypamaabmaabaGaamiEamaaBaaaleaa
caaIYaaabeaakiaaiYcacqWIMaYscaaISaGaamiEamaaBaaaleaaca
WGWbaabeaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeyiPdqfaaaaa
@44F5@
is also approximately design unbiased because
Y
^
OPT
*
−
Y
=
Y
^
π
−
Y
+
(
X
*
−
X
^
π
*
)
Τ
B
^
OPT
*
=
Y
^
π
−
Y
+
(
X
*
−
X
^
π
*
)
Τ
B
OPT
*
+
(
X
*
−
X
^
π
*
)
Τ
(
B
^
OPT
*
−
B
OPT
*
)
,
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaafaqaaeGaca
aabaGabmywayaajaWaa0baaSqaaiaab+eacaqGqbGaaeivaaqaaiaa
cQcaaaGccqGHsislcaWGzbaabaGaaGypaiqadMfagaqcamaaBaaale
aacqaHapaCaeqaaOGaeyOeI0IaamywaiabgUcaRmaabmaabaGaaCiw
amaaCaaaleqabaGaaGOkaaaakiabgkHiTiqahIfagaqcamaaDaaale
aacqaHapaCaeaacaaIQaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaa
cqGHKoavaaGcceWHcbGbaKaadaqhaaWcbaGaae4taiaabcfacaqGub
aabaGaaiOkaaaaaOqaaaqaaiaai2daceWGzbGbaKaadaWgaaWcbaGa
eqiWdahabeaakiabgkHiTiaadMfacqGHRaWkdaqadaqaaiaahIfada
ahaaWcbeqaaiaaiQcaaaGccqGHsislceWHybGbaKaadaqhaaWcbaGa
eqiWdahabaGaaGOkaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaey
iPdqfaaOGaaCOqamaaDaaaleaacaqGpbGaaeiuaiaabsfaaeaacaGG
QaaaaOGaey4kaSYaaeWaaeaacaWHybWaaWbaaSqabeaacaaIQaaaaO
GaeyOeI0IabCiwayaajaWaa0baaSqaaiabec8aWbqaaiaaiQcaaaaa
kiaawIcacaGLPaaadaahaaWcbeqaaiabgs6aubaakmaabmaabaGabC
OqayaajaWaa0baaSqaaiaab+eacaqGqbGaaeivaaqaaiaacQcaaaGc
cqGHsislcaWHcbWaa0baaSqaaiaab+eacaqGqbGaaeivaaqaaiaacQ
caaaaakiaawIcacaGLPaaacaaISaaaaaaa@7CDB@
where
B
OPT
*
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHcbWaa0
baaSqaaiaab+eacaqGqbGaaeivaaqaaiaacQcaaaaaaa@3C82@
is obtained by replacing
x
i
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bWaaS
baaSqaaiaadMgaaeqaaaaa@3A7B@
by
x
i
*
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bWaa0
baaSqaaiaadMgaaeaacaGGQaaaaaaa@3B2A@
in equation (2.8). Since
B
^
OPT
*
−
B
OPT
*
=
O
p
(
n
− 1 / 2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHcbGbaK
aadaqhaaWcbaGaae4taiaabcfacaqGubaabaGaaiOkaaaakiabgkHi
TiaahkeadaqhaaWcbaGaae4taiaabcfacaqGubaabaGaaiOkaaaaki
aai2dacaWGpbWaaSbaaSqaaiaadchaaeqaaOWaaeWaaeaacaWGUbWa
aWbaaSqabeaacqGHsisldaWcgaqaaiaaigdaaeaacaaIYaaaaaaaaO
GaayjkaiaawMcaaaaa@49A8@
under some regularity conditions discussed
in Fuller (2009, Chapter 2), ignoring the smaller order terms we get
Y
^
OPT
*
− Y ≅
Y
^
π
− Y +
(
X
*
−
X
^
π
*
)
Τ
B
OPT
*
.
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK
aadaqhaaWcbaGaae4taiaabcfacaqGubaabaGaaiOkaaaakiabgkHi
TiaadMfacqGHfjcqceWGzbGbaKaadaWgaaWcbaGaeqiWdahabeaaki
abgkHiTiaadMfacqGHRaWkdaqadaqaaiaahIfadaahaaWcbeqaaiaa
iQcaaaGccqGHsislceWHybGbaKaadaqhaaWcbaGaeqiWdahabaGaaG
OkaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeyiPdqfaaOGaaCOq
amaaDaaaleaacaqGpbGaaeiuaiaabsfaaeaacaGGQaaaaOGaaGOlaa
aa@53B7@
The asymptotic variance of
Y
^
OPT
*
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK
aadaqhaaWcbaGaae4taiaabcfacaqGubaabaGaaiOkaaaaaaa@3CA5@
is smaller than the one associated
with
Y
^
SREG
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK
aadaWgaaWcbaGaae4uaiaabkfacaqGfbGaae4raaqabaGccaGGUaaa
aa@3D73@
The reason for this is that the
optimal estimator minimizes the asymptotic variance among the class of
estimators of the form
Y
^
B
=
Y
^
π
+
(
X
*
−
X
^
π
*
)
Τ
B
^
( 2.13 )
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGzbGbaK
aadaWgaaWcbaGaamOqaaqabaGccaaI9aGabmywayaajaWaaSbaaSqa
aiabec8aWbqabaGccqGHRaWkdaqadaqaaiaahIfadaahaaWcbeqaai
aaiQcaaaGccqGHsislceWHybGbaKaadaqhaaWcbaGaeqiWdahabaGa
aGOkaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeyiPdqfaaOGabC
OqayaajaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGOm
aiaac6cacaaIXaGaaG4maiaacMcaaaa@554D@
indexed by
B
^
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHcbGbaK
aacaGGUaaaaa@39ED@
ISSN : 1492-0921
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Published by authority of the Minister responsible for Statistics Canada.
© Minister of Industry, 2016
Use of this publication is governed by the Statistics Canada Open Licence Agreement .
Catalogue No. 12-001-X
Frequency: semi-annual
Ottawa
Date modified:
2016-06-22