Register-based sampling for household panels
4. Sample size determinationRegister-based sampling for household panels
4. Sample size determination
The purpose of the RIS is to publish
income distributions for households and persons at different geographical
levels. Income distributions for households for region or area
r
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@3823@
are defined as
P
l
r
=
M
l
r
M
+
r
,
l
=
1
,
…
,
L
,
(
4.1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa
aaleaacaWGSbGaamOCaaqabaGccqGH9aqpdaWcaaqaaiaad2eadaWg
aaWcbaGaamiBaiaadkhaaeqaaaGcbaGaamytamaaBaaaleaacqGHRa
WkcaWGYbaabeaaaaGccaGGSaGaaGzbVlaaywW7caWGSbGaeyypa0Ja
aGymaiaacYcacqWIMaYscaGGSaGaamitaiaacYcacaaMf8UaaGzbVl
aaywW7caaMf8UaaGzbVlaacIcacaaI0aGaaiOlaiaaigdacaGGPaaa
aa@56D0@
where
M
l
r
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamytamaaBa
aaleaacaWGSbGaamOCaaqabaaaaa@3A12@
denotes
the number of households from region
r
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOCaiaacY
caaaa@38D3@
belonging to the
l
th
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiBamaaCa
aaleqabaGaaeiDaiaabIgaaaaaaa@3A2C@
income category, and
M
+
r
=
∑
l
M
l
r
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamytamaaBa
aaleaacqGHRaWkcaWGYbaabeaakiabg2da9maaqababaGaamytamaa
BaaaleaacaWGSbGaamOCaaqabaaabaGaamiBaaqab0GaeyyeIuoaki
aacYcaaaa@417C@
the
total number of households in area
r
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOCaiaac6
caaaa@38D5@
This
income distribution is estimated as
P
^
l
r
=
M
^
l
r
M
+
r
,
l
=
1
,
…
,
L
,
(
4.2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiuayaaja
WaaSbaaSqaaiaadYgacaWGYbaabeaakiabg2da9maalaaabaGabmyt
ayaajaWaaSbaaSqaaiaadYgacaWGYbaabeaaaOqaaiaad2eadaWgaa
WcbaGaey4kaSIaamOCaaqabaaaaOGaaiilaiaaywW7caaMf8UaamiB
aiabg2da9iaaigdacaGGSaGaeSOjGSKaaiilaiaadYeacaGGSaGaaG
zbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGinaiaac6cacaaI
YaGaaiykaaaa@56F1@
where
M
^
l
r
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmytayaaja
WaaSbaaSqaaiaadYgacaWGYbaabeaaaaa@3A22@
denotes
an appropriate direct estimator for the total number of households from area
r
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOCaiaacY
caaaa@38D3@
classified to the
l
th
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiBamaaCa
aaleqabaGaaeiDaiaabIgaaaaaaa@3A2C@
income category. For the moment the HT
estimator is assumed as an appropriate estimator for
M
l
r
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamytamaaBa
aaleaacaWGSbGaamOCaaqabaGccaGGSaaaaa@3ACC@
i.e. ,
M
^
l
r
=
∑
h
∈
r
∑
k
=
1
m
h
y
k
h
l
π
k
,
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmytayaaja
WaaSbaaSqaaiaadYgacaWGYbaabeaakiabg2da9maaqafabaWaaabC
aeaadaWcaaqaaiaadMhadaWgaaWcbaGaam4AaiaadIgacaWGSbaabe
aaaOqaaiabec8aWnaaBaaaleaacaWGRbaabeaaaaaabaGaam4Aaiab
g2da9iaaigdaaeaacaWGTbWaaSbaaWqaaiaadIgaaeqaaaqdcqGHri
s5aaWcbaGaamiAaiabgIGiolaadkhaaeqaniabggHiLdGccaGGSaaa
aa@4F55@
where
y
k
h
l
=
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa
aaleaacaWGRbGaamiAaiaadYgaaeqaaOGaeyypa0JaaGymaaaa@3CEF@
if
household
k
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@
from
stratum
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiAaaaa@3819@
is
classified to the
l
th
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiBamaaCa
aaleqabaGaaeiDaiaabIgaaaaaaa@3A2C@
income
class and
y
k
h
l
=
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa
aaleaacaWGRbGaamiAaiaadYgaaeqaaOGaeyypa0JaaGimaaaa@3CEE@
otherwise and
m
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa
aaleaacaWGObaabeaaaaa@3937@
the
total number of households selected in stratum
h
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiAaiaac6
caaaa@38CB@
In the
RIS
L
=
10.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamitaiabg2
da9iaaigdacaaIWaGaaiOlaaaa@3B2A@
Income distributions for persons are defined
and estimated similarly to (4.1), (4.2), with
M
l
r
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamytamaaBa
aaleaacaWGSbGaamOCaaqabaaaaa@3A12@
the
number of persons from area
r
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOCaiaacY
caaaa@38D3@
belonging to the
l
th
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiBamaaCa
aaleqabaGaaeiDaiaabIgaaaaaaa@3A2C@
income
category. The HT estimator for
M
l
r
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamytamaaBa
aaleaacaWGSbGaamOCaaqabaaaaa@3A12@
is now
defined as
M
^
l
r
=
∑
h
∈
r
∑
k
=
1
m
h
1
π
k
∑
j
=
1
N
k
y
k
j
h
l
,
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmytayaaja
WaaSbaaSqaaiaadYgacaWGYbaabeaakiabg2da9maaqafabaWaaabC
aeaadaWcaaqaaiaaigdaaeaacqaHapaCdaWgaaWcbaGaam4Aaaqaba
aaaOWaaabCaeaacaWG5bWaaSbaaSqaaiaadUgacaWGQbGaamiAaiaa
dYgaaeqaaaqaaiaadQgacqGH9aqpcaaIXaaabaGaamOtamaaBaaame
aacaWGRbaabeaaa0GaeyyeIuoaaSqaaiaadUgacqGH9aqpcaaIXaaa
baGaamyBamaaBaaameaacaWGObaabeaaa0GaeyyeIuoaaSqaaiaadI
gacqGHiiIZcaWGYbaabeqdcqGHris5aOGaaiilaaaa@57E1@
where
y
k
j
h
l
=
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa
aaleaacaWGRbGaamOAaiaadIgacaWGSbaabeaakiabg2da9iaaigda
aaa@3DDE@
if
person
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@381B@
from household
k
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@
and
stratum
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiAaaaa@3819@
is
classified to the
l
th
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiBamaaCa
aaleqabaGaaeiDaiaabIgaaaaaaa@3A2C@
income
class and
y
k
j
h
l
=
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa
aaleaacaWGRbGaamOAaiaadIgacaWGSbaabeaakiabg2da9iaaicda
aaa@3DDD@
otherwise.
For sample size determination,
precision specifications for the estimated income distributions are required.
For stratified sampling designs, Neyman allocations are often considered to
determine minimum sample sizes and optimal allocations to meet precision
requirements at aggregated levels (Cochran 1977). Power allocations are useful
to find the right balance between precision requirements for aggregates and
strata (Bankier 1988). In this application the minimum sample size is based on
precision requirements for the individual strata, i.e. , neighbourhoods, which
is the most detailed publication level.
If precision requirements are
specified for the separate classes of the income distributions, then the income
class with the largest population variance determines the minimum required
sample size, resulting in unnecessarily large sample sizes. As an alternative
the square root of the mean over the variances of the estimated income classes
of an income distribution is proposed as a precision measure for the estimated
income distributions. With this measure the influence of the most imprecise
income class on the minimum sample size will be reduced. The square root of the
mean over the variances of the estimated income classes of an income
distribution is called the average standard error measure and is defined as
s
=
1
L
∑
l
=
1
L
V
(
P
^
l
r
)
.
(
4.3
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Caiabg2
da9maakaaabaWaaSaaaeaacaaIXaaabaGaamitaaaadaaeWbqaaiaa
dAfadaqadaqaaiqadcfagaqcamaaBaaaleaacaWGSbGaamOCaaqaba
aakiaawIcacaGLPaaaaSqaaiaadYgacqGH9aqpcaaIXaaabaGaamit
aaqdcqGHris5aaWcbeaakiaac6cacaaMf8UaaGzbVlaaywW7caaMf8
UaaGzbVlaacIcacaaI0aGaaiOlaiaaiodacaGGPaaaaa@5214@
In this section an exact expression
for
s
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@3824@
will be derived as well as an approximation that can be used to estimate
the minimum required sample size which does not require information about income
distributions or variances.
Since neighbourhoods are the most
detailed areas for which income distributions are published, precision
requirements for sample size determination are specified at this level. Since
neighbourhoods are used as the stratification variable in the sample design,
expressions for
s
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@3824@
can be derived under simple random sampling
without replacement of core persons within each neighbourhood. It is proved in
the appendix that an expression for the average standard error measure
s
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa
aaleaacaWGObaabeaaaaa@393D@
in (4.3) for an income distribution is given
by
s
h
=
1
L
N
h
−
n
h
n
h
1
N
h
−
1
(
N
h
M
h
2
∑
l
=
1
L
∑
k
=
1
M
h
y
k
h
l
g
k
h
−
∑
l
=
1
L
(
M
l
h
M
h
)
2
)
,
(
4.4
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9
vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr
0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa
aaleaacaWGObaabeaakiabg2da9maakaaabaWaaSaaaeaacaaIXaaa
baGaamitaaaadaWcaaqaaiaad6eadaWgaaWcbaGaamiAaaqabaGccq
GHsislcaWGUbWaaSbaaSqaaiaadIgaaeqaaaGcbaGaamOBamaaBaaa
leaacaWGObaabeaaaaGcdaWcaaqaaiaaigdaaeaacaWGobWaaSbaaS
qaaiaadIgaaeqaaOGaeyOeI0IaaGymaaaadaqadaqaamaalaaabaGa
amOtamaaBaaaleaacaWGObaabeaaaOqaaiaad2eadaqhaaWcbaGaam
iAaaqaaiaaikdaaaaaaOWaaabCaeaadaaeWbqaamaalaaabaGaamyE
amaaBaaaleaacaWGRbGaamiAaiaadYgaaeqaaaGcbaGaam4zamaaBa
aaleaacaWGRbGaamiAaaqabaaaaaqaaiaadUgacqGH9aqpcaaIXaaa
baGaamytamaaBaaameaacaWGObaabeaaa0GaeyyeIuoaaSqaaiaadY
gacqGH9aqpcaaIXaaabaGaamitaaqdcqGHris5aOGaeyOeI0YaaabC
aeaadaqadaqaamaalaaabaGaamytamaaBaaaleaacaWGSbGaamiAaa
qabaaakeaacaWGnbWaaSbaaSqaaiaadIgaaeqaaaaaaOGaayjkaiaa
wMcaaaWcbaGaamiBaiabg2da9iaaigdaaeaacaWGmbaaniabggHiLd
GcdaahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaaaSqabaGccaGG
SaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGinaiaac6
cacaaI0aGaaiykaaaa@7BA0@
with
M
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS
baaSqaaiaadIgaaeqaaaaa@3A4B@
the
number of households in stratum
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGObaaaa@394D@
and
M
l
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS
baaSqaaiaadYgacaWGObaabeaaaaa@3B3C@
the
number of households in stratum
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGObaaaa@394D@
belonging to the
l
th
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGSbWaaW
baaSqabeaacaqG0bGaaeiAaaaaaaa@3B60@
income class. Note that if
g
k
h
=
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGNbWaaS
baaSqaaiaadUgacaWGObaabeaakiabg2da9iaaigdaaaa@3D20@
for all
households in the population of stratum
h
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGObGaai
ilaaaa@39FD@
then it
follows that
M
h
=
N
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS
baaSqaaiaadIgaaeqaaOGaeyypa0JaamOtamaaBaaaleaacaWGObaa
beaaaaa@3D47@
and that
formula (4.1) simplifies to
V
(
P
^
l
h
)
=
N
h
−
n
h
n
h
1
N
h
−
1
(
P
l
h
(
1
−
P
l
h
)
)
,
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGwbWaae
WaaeaaceWGqbGbaKaadaWgaaWcbaGaamiBaiaadIgaaeqaaaGccaGL
OaGaayzkaaGaeyypa0ZaaSaaaeaacaWGobWaaSbaaSqaaiaadIgaae
qaaOGaeyOeI0IaamOBamaaBaaaleaacaWGObaabeaaaOqaaiaad6ga
daWgaaWcbaGaamiAaaqabaaaaOWaaSaaaeaacaaIXaaabaGaamOtam
aaBaaaleaacaWGObaabeaakiabgkHiTiaaigdaaaWaaeWaaeaacaWG
qbWaaSbaaSqaaiaadYgacaWGObaabeaakmaabmaabaGaaGymaiabgk
HiTiaadcfadaWgaaWcbaGaamiBaiaadIgaaeqaaaGccaGLOaGaayzk
aaaacaGLOaGaayzkaaGaaiilaaaa@5586@
which can be recognized as the
variance of an estimated fraction under simple random sampling without
replacement (Cochran 1977, Chapter 3).
Minimum sample size requirements
based on (4.4) require information about the income distribution and its
variance from preceding periods. Since this information is generally not
available at the design phase of a panel, it is useful to have an upper bound
for the average standard error measure for the income distribution in (4.4).
This is comparable to taking the variance for a parameter defined as a
proportion, which reaches a maximum when the proportion is 0.5 for calculating
the minimum sample size for a survey. It is shown in the appendix that an upper
bound for the average standard error measure
s
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaS
baaSqaaiaadIgaaeqaaaaa@3A71@
for an income distribution, specified in (4.4)
is given by
s
h
≤
1
L
N
h
−
n
h
n
h
1
N
h
−
1
(
N
h
M
h
2
∑
t
=
1
T
M
t
h
t
−
1
L
)
,
(
4.5
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaS
baaSqaaiaadIgaaeqaaOGaeyizIm6aaOaaaeaadaWcaaqaaiaaigda
aeaacaWGmbaaamaalaaabaGaamOtamaaBaaaleaacaWGObaabeaaki
abgkHiTiaad6gadaWgaaWcbaGaamiAaaqabaaakeaacaWGUbWaaSba
aSqaaiaadIgaaeqaaaaakmaalaaabaGaaGymaaqaaiaad6eadaWgaa
WcbaGaamiAaaqabaGccqGHsislcaaIXaaaamaabmaabaWaaSaaaeaa
caWGobWaaSbaaSqaaiaadIgaaeqaaaGcbaGaamytamaaDaaaleaaca
WGObaabaGaaGOmaaaaaaGcdaaeWbqaamaalaaabaGaamytamaaBaaa
leaacaWG0bGaamiAaaqabaaakeaacaWG0baaaiabgkHiTmaalaaaba
GaaGymaaqaaiaadYeaaaaaleaacaWG0bGaeyypa0JaaGymaaqaaiaa
dsfaa0GaeyyeIuoaaOGaayjkaiaawMcaaaWcbeaakiaacYcacaaMf8
UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaI0aGaaiOlaiaaiwda
caGGPaaaaa@6816@
with
M
t
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS
baaSqaaiaadshacaWGObaabeaaaaa@3B44@
the
number of households of size
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG0baaaa@3959@
in stratum
h
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGObGaai
Olaaaa@39FF@
If
g
k
h
=
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGNbWaaS
baaSqaaiaadUgacaWGObaabeaakiabg2da9iaaigdaaaa@3D20@
for all households in the population of
stratum
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGObaaaa@394D@
and the number of classes of the income
distribution
L
=
2
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbGaey
ypa0JaaGOmaiaacYcaaaa@3BA3@
then it follows that the approximation for the average standard error
measure
s
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaS
baaSqaaiaadIgaaeqaaaaa@3A71@
in (4.5) can be simplified to
s
h
≤
N
h
−
n
h
n
h
1
(
N
h
−
1
)
1
4
,
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaS
baaSqaaiaadIgaaeqaaOGaeyizIm6aaOaaaeaadaWcaaqaaiaad6ea
daWgaaWcbaGaamiAaaqabaGccqGHsislcaWGUbWaaSbaaSqaaiaadI
gaaeqaaaGcbaGaamOBamaaBaaaleaacaWGObaabeaaaaGcdaWcaaqa
aiaaigdaaeaadaqadaqaaiaad6eadaWgaaWcbaGaamiAaaqabaGccq
GHsislcaaIXaaacaGLOaGaayzkaaaaamaalaaabaGaaGymaaqaaiaa
isdaaaaaleqaaOGaaiilaaaa@4B9E@
which equals the square root of
the maximum variance of an estimated fraction at
P
^
=
0.5
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGqbGbaK
aacqGH9aqpcaaIWaGaaiOlaiaaiwdaaaa@3C76@
under
simple random sampling. This illustrates that the approximation for the average
standard error measure in (4.5) can be interpreted as a generalization of the
approximation of the maximum variance of an estimated fraction at
P
^
=
0.5
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGqbGbaK
aacqGH9aqpcaaIWaGaaiOlaiaaiwdacaGGSaaaaa@3D26@
often
used in sample size determination. The average standard error measure has its
maximum value in the case of an equal distribution of the households over the
income categories, i.e. ,
P
^
l
h
=
1
/
L
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGqbGbaK
aadaWgaaWcbaGaamiBaiaadIgaaeqaaOGaeyypa0ZaaSGbaeaacaaI
XaaabaGaamitaaaaaaa@3E01@
for
l
=
1
,
…
,
L
.
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGSbGaey
ypa0JaaGymaiaacYcacqWIMaYscaGGSaGaamitaiaac6caaaa@3F17@
In this situation the approximation for
s
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaS
baaSqaaiaadIgaaeqaaaaa@3A71@
is
exact, which follows directly from equation (4.3).
Equating the expression for
s
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaS
baaSqaaiaadIgaaeqaaaaa@3A71@
in (4.5) to a pre-specified maximum value, say
Δ
h
,
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqqHuoarda
WgaaWcbaGaamiAaaqabaGccaGGSaaaaa@3B99@
results in the following expression for the
minimum sample size of core persons
n
h
≥
(
N
h
M
h
)
2
∑
t
=
1
T
M
t
h
t
−
N
h
L
(
N
h
−
1
)
L
Δ
h
2
+
N
h
M
h
2
∑
t
=
1
T
M
t
h
t
−
1
L
.
(
4.6
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbWaaS
baaSqaaiaadIgaaeqaaOGaeyyzIm7aaSaaaeaadaqadaqaamaalaaa
baGaamOtamaaBaaaleaacaWGObaabeaaaOqaaiaad2eadaWgaaWcba
GaamiAaaqabaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaa
aOWaaabCaeaadaWcaaqaaiaad2eadaWgaaWcbaGaamiDaiaadIgaae
qaaaGcbaGaamiDaaaacqGHsisldaWcaaqaaiaad6eadaWgaaWcbaGa
amiAaaqabaaakeaacaWGmbaaaaWcbaGaamiDaiabg2da9iaaigdaae
aacaWGubaaniabggHiLdaakeaadaqadaqaaiaad6eadaWgaaWcbaGa
amiAaaqabaGccqGHsislcaaIXaaacaGLOaGaayzkaaGaamitaiabfs
5aenaaDaaaleaacaWGObaabaGaaGOmaaaakiabgUcaRmaalaaabaGa
amOtamaaBaaaleaacaWGObaabeaaaOqaaiaad2eadaqhaaWcbaGaam
iAaaqaaiaaikdaaaaaaOWaaabCaeaadaWcaaqaaiaad2eadaWgaaWc
baGaamiDaiaadIgaaeqaaaGcbaGaamiDaaaacqGHsisldaWcaaqaai
aaigdaaeaacaWGmbaaaaWcbaGaamiDaiabg2da9iaaigdaaeaacaWG
ubaaniabggHiLdaaaOGaaiOlaiaaywW7caaMf8UaaGzbVlaaywW7ca
aMf8UaaiikaiaaisdacaGGUaGaaGOnaiaacMcaaaa@7794@
The information required to
estimate the minimum sample size is the total number of persons and the total
number of equally sized households for neighbourhoods. No information about the
expected income distribution or its variance is required. More precise
estimates for the minimum sample size can be obtained with the expression in (4.4),
but require sample information from, for example, previous periods about the
income distributions.
Expression (4.6) gives the minimum
sample size for core persons. Subsequently all household members of each core
person are included in the sample. As a result, households can be included in
the sample more than once and the sample size in terms of unique households and
unique persons is random. To plan a survey and control survey costs, it is
necessary to know the expected number of unique households and unique persons
if a sample of core persons of size
n
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbWaaS
baaSqaaiaadIgaaeqaaaaa@3A6C@
is drawn. In the appendix it is proved that the
expected number of unique households in a sample of
n
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbWaaS
baaSqaaiaadIgaaeqaaaaa@3A6C@
core persons, drawn by means of simple random
sampling without replacement from a finite population of size
N
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobWaaS
baaSqaaiaadIgaaeqaaaaa@3A4C@
is given by
D
h
=
∑
t
=
1
T
M
t
h
(
1
−
∏
i
=
0
t
−
1
(
N
h
−
n
h
−
i
)
∏
i
=
0
t
−
1
(
N
h
−
i
)
)
.
(
4.7
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGebWaaS
baaSqaaiaadIgaaeqaaOGaeyypa0ZaaabCaeaacaWGnbWaaSbaaSqa
aiaadshacaWGObaabeaakmaabmaabaGaaGymaiabgkHiTmaalaaaba
WaaebCaeaadaqadaqaaiaad6eadaWgaaWcbaGaamiAaaqabaGccqGH
sislcaWGUbWaaSbaaSqaaiaadIgaaeqaaOGaeyOeI0IaamyAaaGaay
jkaiaawMcaaaWcbaGaamyAaiabg2da9iaaicdaaeaacaWG0bGaeyOe
I0IaaGymaaqdcqGHpis1aaGcbaWaaebCaeaadaqadaqaaiaad6eada
WgaaWcbaGaamiAaaqabaGccqGHsislcaWGPbaacaGLOaGaayzkaaaa
leaacaWGPbGaeyypa0JaaGimaaqaaiaadshacqGHsislcaaIXaaani
abg+GivdaaaaGccaGLOaGaayzkaaaaleaacaWG0bGaeyypa0JaaGym
aaqaaiaadsfaa0GaeyyeIuoakiaac6cacaaMf8UaaGzbVlaaywW7ca
aMf8UaaGzbVlaacIcacaaI0aGaaiOlaiaaiEdacaGGPaaaaa@702C@
The expected number of unique
persons in a sample of
n
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbWaaS
baaSqaaiaadIgaaeqaaaaa@3A6C@
core persons, drawn by means of simple random
sampling without replacement from a finite population of size
N
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobWaaS
baaSqaaiaadIgaaeqaaaaa@3A4C@
follows directly from (4.7) and is given by
D
h
[
p
]
=
∑
t
=
1
T
t
M
t
h
(
1
−
∏
i
=
0
t
−
1
(
N
h
−
n
h
−
i
)
∏
i
=
0
t
−
1
(
N
h
−
i
)
)
.
(
4.8
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGebWaa0
baaSqaaiaadIgaaeaadaWadaqaaiaadchaaiaawUfacaGLDbaaaaGc
cqGH9aqpdaaeWbqaaiaadshacaWGnbWaaSbaaSqaaiaadshacaWGOb
aabeaakmaabmaabaGaaGymaiabgkHiTmaalaaabaWaaebCaeaadaqa
daqaaiaad6eadaWgaaWcbaGaamiAaaqabaGccqGHsislcaWGUbWaaS
baaSqaaiaadIgaaeqaaOGaeyOeI0IaamyAaaGaayjkaiaawMcaaaWc
baGaamyAaiabg2da9iaaicdaaeaacaWG0bGaeyOeI0IaaGymaaqdcq
GHpis1aaGcbaWaaebCaeaadaqadaqaaiaad6eadaWgaaWcbaGaamiA
aaqabaGccqGHsislcaWGPbaacaGLOaGaayzkaaaaleaacaWGPbGaey
ypa0JaaGimaaqaaiaadshacqGHsislcaaIXaaaniabg+GivdaaaaGc
caGLOaGaayzkaaaaleaacaWG0bGaeyypa0JaaGymaaqaaiaadsfaa0
GaeyyeIuoakiaac6cacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaa
cIcacaaI0aGaaiOlaiaaiIdacaGGPaaaaa@740E@
Since the expected numbers of unique
households and persons are random variables, it would be useful to have an
uncertainty measure for these expected values. Variance expressions for (4.7)
and (4.8) are however not straightforward and therefore left for further
research.
Sample size calculations are conducted
at the level of neighbourhoods. It was finally decided to select core persons
with a sampling fraction of 0.16. With this sample size, the maximum value for
the average standard error measure
s
h
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x
e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9
Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaS
baaSqaaiaadIgaaeqaaaaa@3A71@
at the level of neighbourhoods amounts to about 0.01 for the estimated
household income distributions. With a total population of about 12 million
persons, this resulted in a sample size of about 2.1 million core persons and
an expected sample size of about 4.6 million unique persons. This sample was
drawn in 1994, which was the start of the panel for the Dutch RIS .
ISSN : 1492-0921
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Copyright
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