Poverty Estimation in Small Areas
Agne Bikauskaite
European Conference on Quality in Official Statistics (Q2014)Vienna, 3-5 June 2014
Data:
• The population size N = 3000
• The sample size n = 300
• Number of mutually exclusive strata H = 7
• The income of individuals (yh1, ..., yhN)
• The auxiliary information (x1i, ..., xji, ..., xJi)
• 1000 simple random samples
Income distribution
Data:
• The population size N = 3000
• The sample size n = 300
• Number of mutually exclusive strata H = 7
• The income of individuals (yh1, ..., yhN)
• The auxiliary information (x1i, ..., xji, ..., xJi)
• 1000 simple random samples
Strata size
hN hnNumber of strata Population size Strata size
1 496 50
2 333 33
3 177 18
4 119 12
5 92 9
6 794 79
7 989 99
Total 3000 300
Stratified Sampling
• The sample design probability when element i belongs to stratum h is
• The sampling weight for selected person i from the h stratum is
;h
hih N
n
h
h
ihih n
Nw
1
Estimated parameters
• The Average Income
• The Poverty Line
• The Headcount Index
• The Poverty Gap Index
The Average Income
• The average income in strata h is
• The average income estimate is
n
iii
h
h ywN 1ˆ1̂
hN
ii
hh yN 1
1
The Poverty Line
• The Poverty Line is defined as 60 per cent of the median equivalent disposable income
• The Poverty Line estimate is
Mz %60
Mz ˆ%60ˆ
The Headcount Index
• The headcount index is defined as the number of persons below the poverty line divided by the population number
• The Headcount index estimate is
N
izyi
IN
P1
)(0
1
n
iziyi IwN
P1
)ˆ(0 ˆ1ˆ
The Poverty Gap
• The poverty gap G is defined as an amount of difference between poverty line and income value y of ith person living in poverty or social exclusion
• The poverty gap estimate
zyii iIyzG
zyiii iIwyzG
The Poverty Gap Index
• The poverty gap index is a proportion of the poverty gap and the poverty line
• The poverty gap index estimate is
zy
N
i
iq
i
i
iI
z
yz
Nz
G
NP
11
1
11
.ˆ
ˆˆ1ˆ
ˆ1
1 zy
n
ii
ii
Iwz
yz
NP
Population
What is Small Area?
Sampling in Small Areas
Direct and Indirect Estimates
• Direct Estimates:– Not using auxiliary information– Using auxiliary information from the same
area
• Indirect Estimates:– Using auxiliary information from adjacent
areas
Simulated Estimation Methods
• The Horvitz-Thompson (HT)
• The Generalised Regression (GREG)
• The Synthetic (S)
The Absolute Relative Bias
• The Absolute Relative Bias (ARB) assessed the accuracy of the estimates
K
k h
hh
KARB
1
ˆ1
The Horvitz-Thompson estimator
• The sum estimate is
i
n
ii
n
i i
i ywy
t
11
ˆ
The ARB of the average income estimates
StratumHorvitz-Thompson
estimate’s ARB (%)Generalised Regression
estimate’s ARB (%)Synthetic estimate’s
ARB (%)
Population -0.06447544 0.098310539 -0.08398375
1 -0.3211974 -0.31518106 -0.34310121
2 -0.02643092 -0.014056 -0.06902109
3 0.465571393 0.551799055 0.403882282
4 -0.81562095 -0.88208503 -0.65375062
5 0.485715332 0.510841272 0.492216146
6 -0.1417938 -0.13401672 -0.14913289
7 0.079252793 0.090945055 0.188597999
The ARB of the headcount index estimates
StratumHorvitz-Thompson
estimate’s ARB (%)Generalised regression
estimate’s ARB (%)Synthetic estimate’s
ARB (%)
Population 0.36396329 0.147665664 0.152247869
1 -3.51959494 -3.7958481 -3.8266288
2 1.468493151 1.192029888 1.003491015
3 4.644761905 4.757144543 5.058255185
4 2.859782609 2.601086957 2.877924901
5 -2.80634921 -2.90370419 -1.68042706
6 -0.63675717 -0.78252971 -0.8622043
7 1.344097079 1.068357786 1.298860988
ARB of the poverty gap index estimate
StratumHorvitz-Thompson
estimate’s ARB (%)Generalised regression
estimate’s ARB (%)Synthetic estimate’s
ARB (%)
Population -0.1594528 -0.35543944 -0.41065525
1 -1.37126072 -1.59705543 -1.57592157
2 -0.9619282 -1.23793553 -1.64985282
3 -0.49766038 -0.6453229 -0.69178013
4 -1.21749012 -1.2989069 -1.45047831
5 -0.73358855 -0.95628486 0.02299011
6 0.702989553 0.477610719 0.357964671
7 0.19625962 0.02379632 0.278143276
The GREG estimator
• The sum estimate
J
jXXjyGREGy jjttBtt
1,
ˆˆˆˆ
The ARB of the average income estimates
StratumHorvitz-Thompson
estimate’s ARB (%)Generalised Regression
estimate’s ARB (%)Synthetic estimate’s
ARB (%)
Population -0.06447544 0.098310539 -0.08398375
1 -0.3211974 -0.31518106 -0.34310121
2 -0.02643092 -0.014056 -0.06902109
3 0.465571393 0.551799055 0.403882282
4 -0.81562095 -0.88208503 -0.65375062
5 0.485715332 0.510841272 0.492216146
6 -0.1417938 -0.13401672 -0.14913289
7 0.079252793 0.090945055 0.188597999
The ARB of the headcount index estimates
StratumHorvitz-Thompson
estimate’s ARB (%)Generalised regression
estimate’s ARB (%)Synthetic estimate’s
ARB (%)
Population 0.36396329 0.147665664 0.152247869
1 -3.51959494 -3.7958481 -3.8266288
2 1.468493151 1.192029888 1.003491015
3 4.644761905 4.757144543 5.058255185
4 2.859782609 2.601086957 2.877924901
5 -2.80634921 -2.90370419 -1.68042706
6 -0.63675717 -0.78252971 -0.8622043
7 1.344097079 1.068357786 1.298860988
ARB of the poverty gap index estimate
StratumHorvitz-Thompson
estimate’s ARB (%)Generalised regression
estimate’s ARB (%)Synthetic estimate’s
ARB (%)
Population -0.1594528 -0.35543944 -0.41065525
1 -1.37126072 -1.59705543 -1.57592157
2 -0.9619282 -1.23793553 -1.64985282
3 -0.49766038 -0.6453229 -0.69178013
4 -1.21749012 -1.2989069 -1.45047831
5 -0.73358855 -0.95628486 0.02299011
6 0.702989553 0.477610719 0.357964671
7 0.19625962 0.02379632 0.278143276
The Synethetic estimator
• The sum estimate is
K
khkyhk
tyh Nt
0
sin ˆˆ
The ARB of the average income estimates
StratumHorvitz-Thompson
estimate’s ARB (%)Generalised Regression
estimate’s ARB (%)Synthetic estimate’s
ARB (%)
Population -0.06447544 0.098310539 -0.08398375
1 -0.3211974 -0.31518106 -0.34310121
2 -0.02643092 -0.014056 -0.06902109
3 0.465571393 0.551799055 0.403882282
4 -0.81562095 -0.88208503 -0.65375062
5 0.485715332 0.510841272 0.492216146
6 -0.1417938 -0.13401672 -0.14913289
7 0.079252793 0.090945055 0.188597999
The ARB of the headcount index estimates
StratumHorvitz-Thompson
estimate’s ARB (%)Generalised regression
estimate’s ARB (%)Synthetic estimate’s
ARB (%)
Population 0.36396329 0.147665664 0.152247869
1 -3.51959494 -3.7958481 -3.8266288
2 1.468493151 1.192029888 1.003491015
3 4.644761905 4.757144543 5.058255185
4 2.859782609 2.601086957 2.877924901
5 -2.80634921 -2.90370419 -1.68042706
6 -0.63675717 -0.78252971 -0.8622043
7 1.344097079 1.068357786 1.298860988
ARB of the poverty gap index estimate
StratumHorvitz-Thompson
estimate’s ARB (%)Generalised regression
estimate’s ARB (%)Synthetic estimate’s
ARB (%)
Population -0.1594528 -0.35543944 -0.41065525
1 -1.37126072 -1.59705543 -1.57592157
2 -0.9619282 -1.23793553 -1.64985282
3 -0.49766038 -0.6453229 -0.69178013
4 -1.21749012 -1.2989069 -1.45047831
5 -0.73358855 -0.95628486 0.02299011
6 0.702989553 0.477610719 0.357964671
7 0.19625962 0.02379632 0.278143276
The mean estimate’s variance
,)ˆˆ(1000
1ˆ
1000
1
2
i
iD
The Jack-Knife method
• The Jack-Knife method’s idea is to divide stratified sample into mutually exclusive subgroups.
• The modified sampling weights
stratum. tobelongselement when ,1
subgroup, and stratum tobelongselement when,0
stratum, tobelongnot doeselement hen w ,
thth
thth
thth
hiwn
nkhi
hiw
w
ii
i
thi
hki
The Jack-Knife variance estimator
• Then the Jack-Knife variance estimator of estimated parameter is
2
11
ˆˆ1ˆˆ
hK
khkhk
H
h h
hhkJACK K
KD
hK
khk
hhk K 1
ˆ1ˆ
Conclusions:Poverty parameters estimation
• Different estimation methods for large and for small areas
• The Synthetic method for poverty estimation in small areas
• If auxiliary information from adjacent areas is not available then the most appropriate estimation method is Horvitz-Thompson
Conclusions:Variances estimation of the
estimated parameters
• Large ARBs
• The best results of estimation are given by the Horvitz-Thompson method
• Applying Jack-Knife method precision of the estimates increases when the group size is extremely small