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Comparingthreemethodsofstandarderrorestimationforpovertymeasures
Accordingtothissamplestructureproperweightsareconstructed.Every
statisticanditsvariabilityareestimatedusingtheseweights.
Complexsamplingdesignandcomplex(non-linear)statisticsoften
makesexactexpressionforvarianceestimatenotpossibletofind[Särndal,
Swensson,Wretman,1995].Inthiscaseapproximatetechniquescanbe
applied.Wecomparetheeffectsestimatedunderthethreesuchapproaches.
3.Thelinearizationmethods
Basicideaoflinearizationmethodconsistsofderivingfromacomplex
non-linearstatisticalinearstatisticwhichhasthesameasymptoticvariance
[Osier,2006].
V
(
ˆ
0
)
≈
V
(
T
ˆ
)
=
V
(
i
∑
=1
n
w
i
T
i
)
,
(9)
where:
θ
ˆ-complexnon-linearstatistic,
n–thesamplesize,
wi–sampleweightofitemi,
Ti-linearisedvariable(variablewhoseexpressiondependson
θ
ˆ).
Linearizingisthewaytomakeanalyticalvariancecalculationsincase
ofthecomplexstructureoftheLaekenindicators.Theasymptoticvarianceof
theestimatoristhevarianceofitslinearization[Niemiro,Wieczorkowski,
2005]:
V
(
T
ˆ
)
=
∑
h
L
=
1
a
h
a
h
−
1
∑∑
c
a
=
h
1
⎛
⎜
⎜
⎜
⎜
n
i
=
hc
1
w
hci
T
hci
−
∑∑
c
a
=
h
1
n
i
=
hc
1
a
w
h
hci
T
hci
⎞
⎟
⎟
⎟
⎟
2
,
⎝
⎠
(10)
where:
whci–thesurveyweightattachedtoi
thsampleunit(individual)inthecthcluster
(PSU)oftheh
thstratum,
Thci–correspondinglinearisedvariable,
T
ˆ
=
∑
w
hciT
hci
–theestimatoroftargetparameter
θ
,
h–numberofstrata(h=1,2,…,L),
ah–numberofclusters(PSUs)instratumh,
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