POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
Poverty mapping at local level with suitablemodelling of income
Isabel MolinaDept. of Statistics, Univ. Carlos III de Madrid
(Coauthors: M. Graf and J.M. Marın)
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
POPULATION AT RISK OF POVERTY
2
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
POVERTY INDICATORS
DIRECT ESTIMATORS
EB METHOD
APPLICATION
SIMULATIONS WITH OUTLIERS
PROPOSAL
RESULTS
3
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
NOTATION
• U finite population of size N.
• U partitioned into D subsets U1, . . . ,UD of sizes N1, . . . ,ND ,called domains or areas.
• s sample of size n drawn from the population U.
• sd = s ∩ Ud sub-sample from domain d of size nd .
• rd = Ud − sd out-of-sample elements from domain d .
• Small area problem: nd too small for some domains.
4
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
POVERTY INDICATORS
• Edi welfare measure for indiv. i in domain d .
• z = poverty line.
• Poverty indicator of order α ≥ 0 for domain d :
Fαd =1
Nd
Nd∑i=1
(z − Edi
z
)αI (Edi < z).
• When α = 0⇒ Poverty incidence
• When α = 1⇒ Poverty gap
• Other: Quintile share ratio, Gini coef., Sen index, Theil index,Generalized entropy, Fuzzy monetary/supplementary index.
X Foster, Greer & Thornbecke (1984), Econom.X Neri, Ballini & Betti (2005), Stat. in Transition 5
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
DIRECT ESTIMATORS
• Direct estimator for a domain d : Estimator that uses onlythe sample data from that domain.
• Example: For linear parameters such as means or totals, theHorwitz-Thompson estimator.
• FGT poverty indicators as means:
Fαd =1
Nd
Nd∑i=1
Fαdi , Fαdi =
(z − Edi
z
)αI (Edi < z).
• πdi inclusion probability of unit i in sd .• wdi = 1/πdi sampling weight.• Horwitz-Thompson estimator (random sampling without
replacement):
FDIRαd =
1
Nd
∑i∈sd
wdiFαdi .
6
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
EB METHOD
• Assumption: For a given one-to-one transformation of thewelfare variables, Ydi = T (Edi ), it holds
Ydi = x′diβ + ud + edi , udiid∼ N(0, σ2u), edi
iid∼ N(0, σ2e ).
• FGT poverty indicator in terms of transformed variables Ydi :
Fαd =1
Nd
Nd∑i=1
{z − T−1(Ydi )
z
}αI{T−1(Ydi ) < z
}= hα(Yd),
where Yd = (Yd1, . . . ,YdNd)′.
X Molina and Rao (2010), CJS 7
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
EB METHOD
• Partition Yd into sample and out-of-sample: Yd = (Y′ds ,Y′dr )′
• General parameter: δd = h(Yd), h(·) real measurable.
• Best estimator: δd with minimum MSE (δd) = E (δd − δd)2
• The best estimator is given by
δBd = EYdr[δd |yds ]
• For δd = hα(Yd) = Fαd , the best estimator is
FBαd = EYdr
[Fαd |yds ] = FBαd(β, σ2u, σ
2e )
• Empirical best (EB) estimator: FEBαd = FB
αd(β, σ2u, σ2e ).
X Molina and Rao (2010), CJS 8
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
POVERTY MAPPING IN SPAIN
• Data source: Spanish Survey on Income and LivingConditions (EU-SILC) of 2006.
• Areas: D = 52 provinces for each gender. We fit a separatemodel for each gender.
• Transformation: We consider the nested-error model for thelog-equivalized disposable income: Ydi = T (Edi ) = log Edi .
• Explanatory variables: indicators of 5 age groups, of havingSpanish nationality, of 3 education levels and of labor forcestatus (unemployed, employed or inactive).
9
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
MODEL DIAGNOSTICS
Residuals
−10 −8 −6 −4 −2 0 2
0.0
0.2
0.4
0.6
0.8
−4 −2 0 2 4−1
0−8
−6−4
−20
2
Theoretical Quantiles
Sam
ple
Qua
ntile
s
10
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
MODEL DIAGNOSTICS
0 5000 10000 15000
−10
−8−6
−4−2
02
Index
Res
idua
ls
9.2 9.4 9.6 9.8 10.0 10.2−1
0−8
−6−4
−20
2
Fitted values
Res
idua
ls
11
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
SIMULATION: NO OUTLIERS
y
De
nsity
−2 0 2 4 6
0.0
00
.05
0.1
00
.15
0.2
00
.25
0.3
00
.35
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−1 0 1 2
−2
02
4
x1s
ys
12
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
POV. INCIDENCE: NO OUTLIERS
0 20 40 60 80
0.1
0.2
0.3
0.4
0.5
0.6
Area
Dire
ct, T
rue
●●●●
●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●
●
● True Direct
0 20 40 60 80
0.1
0.2
0.3
0.4
0.5
0.6
Area
EB
, Tru
e
●●●●
●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●
●
● True EB
0 20 40 60 80
0.1
0.2
0.3
0.4
0.5
0.6
Area
FH
, Tru
e
●●●●
●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●
●
● True FH
0 20 40 60 80
0.1
0.2
0.3
0.4
0.5
0.6
Area
EL
L, Tru
e
●●●●
●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●
●
● True ELL
13
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
SIMULATION: 2 % OUTLIERS
y
De
nsity
−10 −5 0 5
0.0
00
.05
0.1
00
.15
0.2
00
.25
0.3
0
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−1 0 1 2
−1
0−
50
5
x1s
ys
14
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
POV. INCIDENCE: 2 % OUTLIERS
0 20 40 60 80
0.1
0.2
0.3
0.4
0.5
0.6
Area
Dire
ct, T
rue
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●●●●●●●●●●●●●●●●●●●
●●●
●
●
● True Direct
0 20 40 60 80
0.1
0.2
0.3
0.4
0.5
0.6
Area
EB
, Tru
e
●●●●
●●●●●
●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
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●●●
●
●
● True EB
0 20 40 60 80
0.1
0.2
0.3
0.4
0.5
0.6
Area
FH
, Tru
e
●●●●
●●●●●
●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●
●●●
●
●
● True FH
0 20 40 60 80
0.1
0.2
0.3
0.4
0.5
0.6
Area
EL
L, Tru
e
●●●●
●●●●●
●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●
●●●
●
●
● True ELL
15
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
GENERALIZED GAMMA
• Gamma distribution: Y ∼ G (k , θ), k shape, θ scale
f (y ; k , θ) =(y/θ)k−1
Γ(k)
e−y/θ
θ, y > 0, k, θ > 0.
• Generalized Gamma distribution: Y ∼ GG (a, θ, k),
f (y ; a, θ, k) =a(y/θ)k−1
Γ(k/a)
e−(y/θ)a
θ, y > 0, a, θ, k > 0.
• Reparameterization: Y ∼ GG (a, θ, p), p = k/a⇔ k = ap
f (y ; a, θ, p) =a(y/θ)ap−1
Γ(p)
e−(y/θ)a
θ, y > 0, a, θ, p > 0.
• For p large, GG (a, θ, p) ≈ logN(a, θ).
16
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
GENERALIZED BETA OF THE SECOND KIND
• It can be obtained as a scale mixture:
Y |u ∼ GG (a, bu1/a, p)U ∼ InvG (1, q)
}⇒ Y ∼ GB2(a, b, p, q).
• Density of Y ∼ GB2(a, b, p, q):
f (y ; a, b, p, q) =a
bB(p, q)
(y/b)ap−1
{1 + (y/b)a}p+q , y > 0, a, b, p, q > 0.
• a shape, b scale, p length of left tail, q length of right tail.
• logY has location parameter log b.
17
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
MULTIVARIATE GB2 MODEL
• MGB2 mixed regression model:
Ydi |udind∼ GG (a, bdiu
1/ad , p),
ud ∼ InvG (1, q),log bdi = x′diβ, i = 1, . . . ,Nd , d = 1, . . . ,D.
• Equivalent model:
Yd = (Yd1, . . . ,YdNd)′
ind∼ MGB2(a, {bdi}Ndi=1, p, q),
log bdi = x′diβ, i = 1, . . . ,Nd , d = 1, . . . ,D.
• Marginals are univariate GB2:
Ydi ∼ GB2(a, bdi , p, q), i = 1, . . . ,Nd , d = 1, . . . ,D.
X Graf, Marın and Molina (2014), Unpublished 18
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
MULTIVARIATE GB2 MODEL
Moments of log variables:
• E [log(Ydi )|ud ] = x′diβ +1
a[log ud + ψ(p)]︸ ︷︷ ︸
vd
,
• E [log(Ydi )] = x′diβ +1
a[ψ(p)− ψ(q)]︸ ︷︷ ︸
c
,
• V [log(Ydi )] =1
a2
[ψ(1)(p) + ψ(1)(q)
],
• Cov [log(Ydi ), log(Ydj)] =1
a2ψ(1)(q), j 6= i ,
where ψ(x) =∂ log Γ(x)
∂xand ψ(1)(x) =
∂ψ(x)
∂x.
X Graf, Marın and Molina (2014), Unpublished 19
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
MULTIVARIATE GB2 MODEL
• Decomposition into non-sample and sample:
Yd = (Y′dr ,Y′ds)′
• Conditional distribution of non-sample given sample:
Ydr |ydsind∼ MGB2(a, {b∗di}i∈rd , p, q
∗d),
b∗di = bdi
1 +∑i∈sd
(ydibdi
)a1/a
, i ∈ rd ,
q∗d = q + ndp, d = 1, . . . ,D
X Graf, Marın and Molina (2014), Unpublished 20
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
SIMULATION RESULTS: POV. INCIDENCE• If income simulated from the MGB2 model fitted to realincome, EB estimates based on Log Normal biased upwards!
Mean values, Pov. Inc. ( %)
0 20 40 60 80
1820
2224
2628
3032
Area
Pov
. Inc
iden
ce
●
●
●
●
●
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●
●
●
●
●
●
●
●
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●
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●
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●
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●
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●
●
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●
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●
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●
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●
●
●
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●●
● TrueEB GB2
EB Log NormalDirect
MSEs, Pov. Inc. ( %)
0 20 40 60 80
1015
2025
3035
40
Area
MS
E P
ov. I
ncid
ence
EB GB2 EB Log Normal Direct
21
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
SIMULATION RESULTS: POV. INCIDENCE• If income simulated from the Log Normal model fitted to realincome, EB estimates based on MGB2 are good!
Mean values, Pov. Inc. ( %)
0 20 40 60 80
1015
20
Area
Pov
. Inc
iden
ce
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●
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●
●
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●
●
●●●●●
● TrueEB GB2
EB Log NormalDirect
MSEs, Pov. Inc. ( %)
0 20 40 60 80
510
1520
2530
35
Area
MS
E P
ov. I
ncid
ence
EB GB2 EB Log Normal Direct
22
POV. INDICATORS DIRECT ESTIMATORS EB METHOD APPLICATION OUTLIERS PROPOSAL RESULTS
DOMO ARIGATO!!EZKERRIK ASKO!!GRAZIE MILLE!!MERCI BEAUCOUP!!¡¡MOITAS GRAZAS!!¡¡MOLTES GRACIES!!¡¡MUCHAS GRACIAS!!THANK YOU VERY MUCH!!VIELEN DANK!!
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