EB-EBLUP MSE ESTIMATOR ON SMALL AREA ESTIMATION WITH APPLICATION TO BPS DATA 1,2

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1 EB-EBLUP MSE ESTIMATOR ON SMALL AREA ESTIMATION WITH APPLICATION TO BPS DAT, Anang Kurna and Kharl A. Notodputro Department of Statstcs, Bogor Agrcultural Unversty and Center for Statstcs and Publc Opnon (CESPO) Abstract Small Area Estmaton (SAE) s a statstcal technque to estmate parameters of subpopulaton contanng small sze of samples. Ths technque s very mportant to be developed n Indonesa due to the hgher needs of subpopulaton data n the reform era n whch decentralzaton has become a central ssue. Indonesan Bureau of Statstcs (BPS) regularly conducts surveys such as SUSENAS, SAKERNAS, SUSI, etc apart from census whch s carred out every decade and PODES whch s carred out every three years. Some SAE technques have been developed n Canada, USA, and UE based on real data. Snce our data n Indonesa posses specfc characterstcs t s necessary to develop SAE methods sutable for the data. Smulaton data s used to evaluate varous standard SAE technques and real data from Susenas 003 and Podes 003 for West Java Provnce s utlzed to llustrate the method. Keywords: small area estmaton, emprcal bayes, emprcal best lnear unbased predcton 1. Introducton Small Area Estmaton s the most mportant concept n survey samplng especally for ndrect parameter estmaton of relatvely small samples. Ths method can be used to estmate characterstcs of sub populaton (a doman whch s smaller than populaton). Drect estmaton for sub populaton fals to provde enough precson because the sample sze to yeld the estmator s small. Another method whch can be used to obtan hgher precson n small area estmaton may be developed by lnkng some nformaton n partcular area wth some other areas through approprate model. Ths procedure s called ndrect estmaton. The procedure nvolves data from other domans. In the other words, small area estmaton model s borrowng strength from sample observaton of related areas through auxlary data (recent cencus and current admnstratve records) to ncrease effectve sample sze (Rao, 003). In ths paper we wll dscuss small area estmaton through ndrect method or estmaton based models. One of the problems found n usng ths procedure s low precson of MSE estmate for parameter θ. Smulaton s carred out to evaluate some estmaton methods whch s developed recently. Ths paper also presents applcaton of ndrect method on small area estmaton usng poverty data from Susenas 003 and Podes 003 at West Java Provnce. 1 Ths paper has supported by research grant of graduate team : Development of Small Area Estmaton and Its Applcaton for BPS Data, Batch IV (006) Paper presented n ICoMS-1, Bandung 19-1 June 006.

2 . Small Area Models and EB-EBLUP Methods There are essentally two-types of models n small area estmaton. The frst s basc area level model that relate small area drect estmator to area-specfc auxalary data x = (x 1, x,, x p ). We assume the parameter of nterest θ = x β + υ where υ ~ N(0, A) and drect estmator θˆ = θ + e where e θ ~ N(0, D ) and D known. The model combnes the parameter of nterest and the ndrect estmates forms θˆ = x β + υ + e where s a case of generalzed lnear mxed model. The second s basc unt level model. In ths model the nformaton s avalable at the samplng unt level and modelng s done based on ndvdual data x j = (x 1j, x j,, x pj ) and we have model y j = x T j β + υ + e that s more complex model. We consder the followng Fay-Herrot model (see Fay and Herrot, 1979) for the basc area level model y = x β + υ + e (1) where υ and e are ndependent wth υ ~ N(0, A) and e ~ N(0, D ) for = 1,,..., k. We assume that β and A unknown but D ( = 1,,..., k) are known. The best predctor (BP) of θ = x β + υ f β and A known s gven by θˆ BP = θˆ (y β, A) = x β + (1 B )(y - x β) () where B = D /(A + D ) for = 1,,..., k. Let θˆ BP = θˆ (y β, A) s also Bayes estmator of θ under followng Bayesan models: () y θ ~ N(θ, D ) () θ ~ N(x β, A) s pror dstrbuton for θ, = 1,,..., k. The Bayesan model s gven by: 1 f(y θ ) = exp - ( y - ) θ πd k ( ) f(y, θ β, A) = exp - ( y -θ ) =1 πd for y = (y 1, y,..., y k ), θ = (θ 1, θ,, θ k ), 1 1 ( ) dan π(θ ) = exp - ( θ - x ' β) πa ( ) 1 exp - ( θ - x ' β) πa ( ) Look the two exponental functon wthout (-1/) factor from f(y, θ β, A), ( ) ( ) y -θ + θ - x ' β = (y - y + θ θ ) + ( θ - x ' β + (x ' β) ) y x ' β + θ - + θ + a, and = ( ) ( ) D A = ( ) y x ' y x ' y x ' + { - ( + ) ( + ) + [( + ) ( + ) x ( + ) ( + )] } β β D A D A β θ θ D A + a y x' β = ( + ){ θ - ( + ) ( + )} + a D A where a s constant and ndependent from θ. So, y x' β -1 (θ y, β, A) ~ N ( ) ) D + A,( + ) AD = N ( x ' β + ) A+D A (y - x ' β), A + D

3 Based on the formulaton, we could vew that where MSE( θˆ EB ) = Var(θ y, β, A) = θˆ EB are dentcs for normal cases. θˆ EB = E(θ y, β, A) = x β + (1 B )(y - x β) (3) AD A + D = (1 B )D = g 1 (A). The estmator θˆ BP and When A s known, β could be estmate by followng maxmum lkelhood method log L(β, V) = - 1 log V - 1 (Y - Xβ) V -1 (Y - Xβ) wth V = Dag(A +, A + D,..., A + D k ). The dfferental of log L(β, V) wrt β s d log L(β, V) = X V -1 (Y - Xβ) dβ = X V -1 Y - (X V -1 X)β ( = 0 ) (X V -1 X) β = X V -1 Y β = (X V -1 X) -1 X V -1 Y Let β* = βˆ (A) = (X V -1 X) -1 X V -1 Y and the replacng β by β* n the θˆ BP, we get the best lnear unbased predctor (BLUP) of θ I gven by θˆ BLUP = θˆ (y A) = x β* + (1 B )(y - x β*) (4) Ghosh and Rao (1994) descrbe the MSE( θˆ BLUP ) = g 1 (A) + g (A), where AD g 1 (A) = = (1 B )D, and A + D g (A) = D /(A + D ) [x (X V -1 X) -1 x ] = D (1 B ) [x (X V -1 X) -1 x ] untuk = 1,,, k. However, n practce both β and A are unknown. To estmated A, we can use maxmum lkelhood (ML), restrcted/resdual maxmum lkelhood (REML) or method of moment (MM). Jang (1996) reported that the estmator of A usng REML was consstence although there are volated normal assumton. If we replacng β by βˆ and A by  n the BLUP ( θˆ BLUP ) estmator, we get the emprcal best lnear unbased predctor (EBLUP) θˆ EBLUP = θˆ (y  ) = x βˆ + (1 Bˆ )(y - x βˆ ) (5) If defned MSE of θˆ EBLUP s MSE( θˆ EBLUP ) = E( θˆ EBLUP - θ ) Kacker and Harvlle (1984) reformulated by MSE( θˆ EBLUP ) = MSE( θˆ BLUP ) + E( θˆ EBLUP - θˆ BLUP ) = H 1 (A) + H (A) Where H 1 (A) = MSE( θˆ BLUP ) = g 1 (A) + g (A) and H (A) = E( θˆ EBLUP term g 1 (A) lead to large reducton n MSE relatve to the MSE of the drect estmator, g (A) s due to estmatng of β and H (A) s due to estmatng A. = Var( θˆ EBLUP ) + (Bas θˆ EBLUP ), - θˆ BLUP ). Leadng Prasad and Rao (1990) used the Taylor seres method to estmate g 1 (A), g (A) and H (A). The MSE estmator of θˆ approxmaton by MSE( θˆ EBLUP ) PR = g 1 (  ) + g (  ) + g 3 (  ) (6) D where g 3 (  ) = k (A + D 3 j) k (A + D) j= 1 defned Butar and Lahr (003).. The MSE( θˆ ) PR s dentcal to the Bayes rsk as 3

4 3. The Smulaton Study and Applcaton to BPS Data A measure of uncertanty θˆ EBLUP or θˆ EB has been developed n recent years. Rao (003) descrbed the result of smulaton study of Jang, Lahr and Wan (00). They reported the smulaton results on the relatve performance of estmator of MSE( θˆ EB ) under the smple model θˆ θ ~ N(θ I, ψ) and θ ~ N(µ, σ υ). Samples were smulated from the model by lettng µ = 0, σ υ = ψ = 1 and k = 30, 60 and 90. The result showed that the mse PR (Prasad and Rao, 1990), mse BL (Butar and Lahr, 003), Var MR (Morrs, 1983) and mse J (Jang, Lahr and Wan, 00) are nearly unbased. Tabel 1. Average MSE Estmator Area E(mse J ) E(mse N ) E(mse PR ) Area E(mse J ) E(mse N ) E(mse PR ) SA SA SA SA SA SA SA SA SA SA SA SA SA SA SA SA SA SA SA SA SA SA SA SA SA SA SA SA SA SA Note : E(mse ) = average(mse ), ARB = k -1 Σ RB, RB =[E(mse ) MSE ]/MSE Table. The lustrated usng real data from SUSENAS 003 and PODES 003 Kabupaten/Kota Desgn Based Model Based Theta_hat MSE Theta_hat MSE_J MSE_N MSE_P Kab. Bogor Kab. Sukabum Kab. Canjur Kab. Bandung Kab. Garut Kab. Taskmalaya Kab. Cams Kab. Kunngan Kab. Crebon Kab. Majalengka Kab. Sumedang Kab. Indramayu Kab. Subang Kab. Purwakarta Kab. Karawang Kab. Bekas Kota Bogor Kota Sukabum Kota Bandung Kota Crebon Kota Bekas Kota Depok

5 In ths study, we evaluate several measure of uncertanty of θˆ EBLUP usng mse PR, mse J and mse N (Ghosh and Rao, 1994). The samples were smulated from the model θˆ = x β + υ + e where x β fxed, υ ~ N(0, 1), e ~ N(0, ψ ) where ψ were generated from unform dstrbuton (0.75, 1.5) and k = 30. As presented n Table 1, the ARB for each mse J, mse N, and mse PR are %, -4.01% and %. Regardng the applcaton on BPS Data, we used per capta expendture of household n the small area. The data s obtaned from SUSENAS 003 wth tems of nformaton based on household, and PODES 003 as source of auxalary data that assumed depct expendture of household at one partcular regon. The result of applcaton reported n Table and for detaled see Kurna and Notodputro (006). In the context of the smulaton study conducted here, both the jackknfe and Prasad- Rao MSE estmator better than the Naïve MSE estmator although the accurate of predcton less than when homogenety samplng varance. We ndcated there are two man problems n our research (real Indonesan Statstcs data) are complex samplng desgn and heterogenety samplng varance. Our future research goal s enhancng the standard methods to solve these problems. 4. Bblography. Butar, F. and Lahr, P. (003). On Measures of Uncertanty of Emprcal Bayes Small Area Estmators, Journal of Statstcal Plannng and Inference, Vol 11, ssue 1-, pp Fay, R.E. and Herrot, R.A., (1979), Estmates of ncome for small places: an applcaton of James-Sten procedures to Census data. Journal of the Amercan Statstcal Assocaton, Vol. 74, p:69-77 Ghosh, M. and Rao, J.N.K Small Area Estmaton: An Apprasal. Statstcal Scence, 9, No.1 p: Jang, J REML estmaton: Asymptotc behavor and related topcs, Annals of Statstcs, 4, : Jang, J., Lahr, P. and Wan, S.M. 00. A Unfed Jackknfe Theory, Annals of Statstcs, 30. Kackar, R.N. and Harvlle, D.A Approxmaton for Standard Errors of Estmator of Fxed and Random Effects n Mxed Lnear Model. Journal of Amercan Statstcal Assocaton, 79, p: Kurna, A. dan Notodputro, K.A. 005a. Generalzed Lnear Mxed Model pada Small Area Estmaton. Forum Statstka dan Komputas ISSN Vol.10 No.. Kurna, A dan Notodputro, K.A. 005b. Aplkas Metode Bayes pada Small Area Estmaton. Makalah dsampakan pada Semnar Nasonal Statstka VII. ITS Surabaya, 6 November 005. Kurna, A. dan Notodputro, K.A Penerapan Metode Jackknfe dalam Pendugaan Area Kecl. Forum Statstka dan Komputas ISSN Vol.11 No.1. Lahr, P A Rvew of Emprcal Best Lnear Unbased Predcton for The Fay- Herrot Small-Area Model. The Phlppne Statstcan, Vol.5 p:1-15. Morrs, C.N Parametrc Emprcal Bayes Inference: Theory and Applcaton. Journal of Amercan Statstcal Assocaton, 78, p:

6 Prasad, N.G.N. and Rao, J.N.K The Estmaton of Mean Squared Errors of Small Area Estmators. Journal of Amercan Statstcal Assocaton, 85, p: Rao, J.N.K Some Recent Advances n Model-Based Small Area Estmaton, Survey Methodology, Vol.5 No., p: Rao, J.N.K Small Area Estmaton, New York : John Wley and Sons. Rao, J.N.K Inferental Issues In Small Area Estmaton: Some New Developments. Statstcs In Transton, December 005 Vol. 7, No. 3, Pp Sae, A. dan R. Chambers, (003), Small Area Estmaton: A Revew of Methods Based on the Applcaton of Mxed Models, S 3 RI Methodolog Workng Paper M03/16, Unversty of Southampton, UK. 6