What and why ”small area”

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Small Area Estimation

Research Activity

in Bogor Agricultural University

1)

Khairil A. Notodiputro Anang Kurnia

Department of Statistics Bogor Agricultural University Jl. Meranti, Wing 22 Level 4 Kampus IPB Darmaga, Bogor

1) Paper has presented in Seminar on Use, Analysis and Application of Small Area Statistics, March 7, 2007. BPS-JICA

Definition :

A sub-population is

small

_{if the domain specific sample size}

is not large enough to support direct estimates of adequate

precision.

Small area, small domain,

local area

Small geographical area

Domain : age-sex-race,

poverty status

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Direct estimates:

Use area-specific sample data only.

Indirect estimates:

Borrow strength from sample

observations of related areas

through auxiliary data (recent

census and current

administrative records) to

increase

effective

_{sample size.}

What and why ”small area”

Can we minimize or even eliminate the use of indirect estimates ?

The attention of SAE has increased along with increasing of

government or private sector demand to provide accurate

information quickly, not only for national (large domain) but also

for small domain such as sub-district.

In Indonesia, it’s important to develop SAE because nowadays

there is moving away from centralization to decentralization in

making decision of public policy that a local government can

manage their districts, allocate their funds and make regional

planning well. Certainly, the decision maker in local government

will require some statistics for their districts.

Statistics Indonesia (BPS) regularly conducts surveys such as

“SUSENAS, etc” but it’s based on national designed.

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There are essentially two-types of SAE models:

Basic area level model

_{that relate small area direct}

estimator to area-specific auxiliary data

Basic unit level model

_{that the information is available at}

the sampling unit level and modeling is done based on

individual data

Consider the following Fay-Herriot (1979) model for area level

y

_i

= x

_i

’

β

+

υ

_i

+ e

_i

where

υ

_i

and e

_i

are independent with

υ

_i

~ N(0, A),

e

_i

~ N(0, D

_i

) for i = 1, 2, ..., k.

We assume that

β

and A unknown but D

_i

are known.

Small Area Model

Model description :

1. x_i = (x_i1, x_i2, ..., x_ip)Æ auxiliary data

2. θ_i= x_i’β + υ_i Æ the parameter that is a function of auxiliary

data and random effect υ_i

3. y_i = θ_i + e_i Æ direct estimate with sampling error 4. y_i = x_i’β + υ_i + e_i Æ the special case of GLMM

Small Area Model

The best predictor (BP) of θ_i = x_i’β + υ_i if β and A = σ2

υ known is

given by:

where B_i = σ2

ei /(σ2υ + σ2ei).

The best predictor is equivalent with empirical bayes approach for

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1. Estimates of small area characteristics based on fixed effect models are reffered to as synthetic estimator (Levy and French, 1977), composite estimator (Schaibel et al, 1977), and prediction estimator (Holt et al, 1979, Sarndal 1984, Marker 1999)

2. Mixed models have been used to improve estimation of small area characteristics of small area based on survey sampling or cencus data by Fay and Herriot (1979), Ghosh and Rao (1994), Rao (1999) and Pfeffermann (1999)

3. In addition to EBLUP, empirical Bayes (EB) and hierarchical Bayes (HB) estimation and inference methods have been also applied to small area estimation.

4. Ghosh and Rao (1994) review the application of these estimation. Maiti (1998) has used non-informative priors for hyperparameters in HB methods and You and Rao (2000) have used HB methods to estimate small area means under random effect models.

A Brief Review of SAE Techniques

6. A general approach for SAE based on GLM is describe in Ghosh et al (1998), Malec et al (1999). Farrel et al (1997) extended the mixed logistic model and Moura and Migon (2001) further extend with introducing a component to account for spatially correlated structure in the biner respon data.

7. A measure of uncertainty of EBLUP or EB has been developed in recent years. Rao (2003) described the result of simulation study of Jiang, Lahiri and Wan (2002). They reported the simulation results on the relative performance of estimator of MSE under the simple model.

8. Some author who concern in a measure of uncertainty are Butar and Lahiri (2001, 2003) on Bootstraping methods; Wang and Fuller (2003), Rivest and Vandal (2003) on aspect of unknown sampling variance; Rao (2003), Jiang, Lahiri and Wan (2003) on jackknife

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1. Empirical Best Linear Unbiased Predictor (EBLUP)

Î Estimation of variance component

2. Empirical Bayes (EB)

Î Mean of posterior distribution, the parameter was estimated from empirical data

3. Hierachical Bayes (HB)

Î Mean of posterior distribution, prior distribution

Inference of Area Level Model

One of recent problem on SAE is uncertainty and MSE estimator

Ghosh and Rao (1994), Prasad and Rao (1990), Butar and Lahiri (2001, 2003), Jiang, Lahiri and Wan (2002), Chen and Lahiri (2001, 2005), Hall and Maiti (2005) give a contribution for this problem.

The approximation that proposed by some authors could be eliminated the problem of underestimate especially for case of A = D_i = 1 and Xβ = 0. However Kurnia and Notodiputro (2006) showed that for heterogenity of D_i (sampling error) and all of parameter model must be estimate, the underestimate of MSE large enough about 13% - 19%.

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The Chronology

1. A discussion in ”Forum Masyarakat Statistik”

held in Solo on December, 3, 2004, identified

the need of quality research in small area

estimation models for Indonesia Case

2. In 2003 Smeru Research Institute developed

small area statistics map for several provinces

3. Kurnia and Notodiputro (2005) carried out a

study of generalized linear mixed model

approach and hierarchical Bayes for SAE

applied to BPS data.

The Development of SAE Research in IPB

Research on

Small Area Estimation

_{at IPB has been}

carried out through support from DGHE

Developing Small Area Estimation Models for BPS Data

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Development Stages of SAE Research at IPB

Tinjauan

M etodologi Pemilihan Peubah

Survey untuk M etode (Unggul)

Desain Software Uji Coba

Kajian untuk Pembandingan

M etode

Evaluasi M etode Terhadap Data

Simulasi yang M engikuti Sampling BPS

Up dating data BPS

Tahun I Tahun II Tahun III

Roadmap

Papers in conference proceeding or journals:

Kurnia, A. dan Notodiputro, K.A. 2006. The Jacknife Method in

Small Area Estimation. Forum Statistika dan Komputasi, Vol. 11 No.1, p:12-15.

Sadik, K. dan Notodiputro, K.A. 2006. Small Area Estimation

based on Random Walk Models. Forum Statistika dan Komputasi, Vol. 11 No.1.

Sadik, K. dan Notodiputro, K.A. 2006. Small Area Estimation

with Time and Area Effects Using Two Stage Estimation, ICoMS-1 : Bandung.

Kurnia, A. dan Notodiputro, K.A. 2006. EB-EBLUP MSE Estimator

on Small Area Estimation with Application to BPS Data, ICoMS-1 : Bandung.

Kismiantini, Kurnia, A. and Notodiputro, K.A. 2006. Risk of

Dengue Haemorrhagic Fever In Bekasi Municipality With Small

Area Approach, ICoMS-1 : Bandung.

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The Development of SAE Research in IPB

Papers in conference proceedings or journals (continued):

Indahwati and Notodiputro, K.A. 2006. Effect of Inappropiate

Sampling Design on Reliability of Small Area Estimates, ICoMS-1 : Bandung.

Sadik, K. dan Notodiputro, K.A. P-Spline M-Quantile Approach in

Small Area Estimation. Jurnal Matematika Aplikasi dan Pembelajaran, Vol. 5 No.2 Jilid 1, p:142-147.

Kurnia, A. dan Notodiputro, K.A. Effects of Sampling Variance

Estimation in Small Area Estimation. Jurnal Matematika Aplikasi dan Pembelajaran, Vol. 5 No.2 Jilid 2.