SMALL AREA ESTIMATION USING NATURAL EXPONENTIAL FAMILIES WITH QUADRATIC VARIANCE FUNCTION (NEF-QVF) FOR BINARY DATA 1

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1 SMALL AREA ESIMAION USING NAURAL EXPONENIAL FAMILIES WIH QUADRAIC VARIANCE FUNCION NEF-QVF FOR BINARY DAA Ksmat Departmet of Mathematcs Educato,,Yogakarta State Uverst Karagmalag, Yogakarta 558, Idoesa e-mal : ksm_u@ahoo.com Abstract. Small area estmato s a techque for hadlg a sample sze ot large eough to eld drect estmator of adequate precso. Small area tpcall refers to a small geographc area or a demographc group, such as cout, mucpalt ad age-sex-race group. Emprcal Best Lear Ubased Predcto LUP, Emprcal Baes ad Herarchcal Baes HB are commol used for small area estmato. ad HB methods are applcable more geerall to bar ad cout data. But s more dffcult for bar data th covarates. Most of the exstg small area estmato methods do ot make use of surve eghts. Hoever, drect estmates are ofte surve eghted. hs paper descrbes a applcato of atural expoetal famles th quadratc varace fucto NEF-QVF for bar data th covarates ad eght to get LUP ad estmators. Keords: small area estmato, NEF-QVF, LUP ad estmators.. Itroducto Whe a aalss s targeted toards specfc subpopulato such as cout, mucpalt, sub mucpalt ad other small domas here the sample sze s ot hgh, e ru to the usual small area problems. here are ma methods to get a estmator for small area estmato SAE.e. Emprcal Best Lear Ubased Predcto LUP, Emprcal Baes ad Herarchcal Baes HB Rao, 003. ad HB methods are applcable more geerall the sese of hadlg models for bar ad cout data. he stud of Fa ad Herrot 979 as oe of the frst to appl lear emprcal Baes models to small area estmato. Others have studed the estmato of small area rates ad bomal parameters usg emprcal ad herarchcal Baes approach. MacGbbo & omberl 989 proposed small area estmates of proporto va emprcal Baes techque for estmatg cesus udercout for local areas hch as based o a logstc regresso model cotag fxed ad radom effects. Malec, Sedrask, Morart ad Le Clere 997 used full Baes approaches to estmate proportos usg data from the Natoal Health Iterve Surve. Farrel 000 used a herarchcal Baes methodolog for estmatg small area proportos, the dea cossts of corporatg to a logstc regresso model cotag predctor varables, radom effects hch reflect the structure of the sample desg. Ufortuatel emprcal Baes techque for the logstc regresso model cotag fxed ad radom effects s qute cumbersome computatoall Rao, 003. So HB method s more regular used to estmate proporto. Most of the exstg small area estmato methods do ot make use of surve eghts. Hoever, drect estmates are ofte surve eghted. Ghosh & Mat 004 as develop a geeral methodolog for fdg emprcal best lear ubased predctors of small area meas based o drect surve eghted estmators ad atural expoetal faml quadratc varace fucto NEF-QVF. he estmate also the proporto of poor chldre the 5-7 ears age-group for the dfferet coutes oe of the states the Uted States. Sha 004 as proposed estmators of the small area meas usg geeral NEF-QVF to estmate the proporto of usured persos a mort subpopulato. hs paper descrbe a applcato atural expoetal famles th quadratc varace fucto NEF-QVF for LUP ad estmators to estmate the proporto the status of the oershp health card 4 sub mucpaltes at Yogakarta ct. hs artcle s preseted at 3 rd Iteratoal Coferece o Mathematcs ad Statstcs, August 6 th 008, Bogor Agrcultural Uverst, Idoesa

2 .. Natural Expoetal Famles th Quadratc Varace Fucto NEF-QVF Let s the respose of the jth ut the th small area j, K, ;, K, k, ad let assumed that has probablt fucto or probablt dest fucto belogg to the atural expoetal faml,.e. f exp + c here the θ [ ξ { θ ψ θ }, ξ ] ξ are assumed to be, V θ υ + υ µ + υ µ Q µ υ, υ υ are ot 0, ad 0, smultaeousl zero. he varace s at most a quadratc fucto of the mea, ad the ths faml of dstrbuto s usuall referred to as atural expoetal faml quadratc varace fucto NEF-QVF Morrs, 98 & 983. here are sx basc NEF-QVF dstrbutos: ormal, Posso, gamma, bomal, egatve bomal, ad geeralzed hperbolc secat GHS dstrbutos Morrs, 983. For the bomal dstrbuto, υ 0 0, υ ad υ. For the Posso dstrbuto, υ0 υ 0 ad υ. For the ormal dstrbuto th ko varaceσ, ξ σ, υ 0 ad υ υ Emprcal Best Lear Ubased Predctor Estmator for Bar Data Let s the respose of the jth ut the th small area j, K, ;, K, k, ad let s varable of terest, here be the eghted attached to. It s assumed that the eghts are depedet of the. Suppose that the are depedet ad have probablt fucto belogg to the atural expoetal faml at equato, th the quadratc varace fucto structure, V µ µ µ, because follos bomal dstrbuto. he, cosder the cojugate pror th probablt dest fucto for θ s π θ exp λ m θ ψ θ + c λ, m [ { } ] here g b m x,, K, k. Here x s the desg vector for the th small area, b s the regresso coeffcet ad g s the lk fucto. he Morrs 983, E m µ V m λ λ > max 0, υ 3 µ ; V υ, ad the a emprcal best lear ubased predctor of µ based o Ghosh & Mat, 004 s ˆ µ rˆ + rˆ mˆ 4 here rˆ + ˆ λ δ, λ s usuall 0.5 ad, x bˆ m ˆ exp [ + exp x bˆ ] Morrs 983 proposed that δ., 4. Emprcal Baes Estmator for Bar Data Let s the respose of the jth ut the th small area j, K, ;, K, k, have probablt fucto belogg to the atural expoetal faml at equato, th the quadratc varace fucto structure, V θ µ µ Q µ, because follos bomal dstrbuto. Assumed that the surve eght are depedet of the, ad varable of terest s.

3 the eghted small area meas. µ µ. he drect ubased estmator of µ s gve b he frst step to get estmators of the small area meas s bega th the geeral NEF-QVF faml of dstrbutos alog th a cojugate pror for the caocal parameter of the expoetal model,.e. π θ exp λ m θ ψ θ + c λ, m 5 [ { } ] here m g b, j, K, ;, K, k x, ad x s the desg vector for the jth ut the th small area, ad g s the lk fucto. he Morrs 983, E m µ Q m λ max 0, µ ; V υ, υ Further, th logstc represetato, m exp x b [ + exp x b ] a estmator of pˆ λ + p Sha, 004 s gve b λ + m λ + bˆ ad the a estmator of µ p s B λ > 6, after resultg estmator b bˆ, ˆµ pˆ 8 here λ s usuall 0.5 ad. I ths paper eght s assumed be. 5. Data Aalss Bar data s used to llustrate the LUP ad methods to estmate the proporto of the status of the oershp health card. he data s obtaed from Surve Sosal Ekoom Nasoal SUSENAS 003 th formato based o household ad PODES 003 as sources of covarates data. he data s take from 4 sub mucpaltes at Yogakarta ct. A varable of terest s proporto of the status of the oershp health card, respose varable s sum of household hch has the status of the oershp health card the th sub mucpalt, s sum of household the th sub mucpalt. As covarates are proporto of pre-elfare ad elfare household, proporto of PLN electrcs costumer household, ad proporto of telephoe customer household. Aalss of the data usg SAS 9..e. PROC GENMOD to get bˆ, ad PROC IML to get estmator of the proporto of the status of the oershp health card. able Estmato of the proporto of the status of the oershp health card No. Sub Sample Drect LUP Mucpaltes Sze Estmator Estmator Estmator Matrero Krato Mergagsa Umbulharjo Kotagede Godokusuma Daureja Pakualama Godomaa

4 0 Ngampla Wrobraja Gedog tege Jets egalrejo able sho that each sub mucpalt has a small umber of proportos. It meas most household does ot have the health card. I Pakualama ad Ngampla sub mucpaltes have zero data for sum of household hch has the status of the oershp health card, so drect estmators s zero, but LUP ad methods ca perform the estmator values. Proporto of the oershp of health card Scatterplot of Drect, LUP, Estmators vs Sub Mucpaltes Sub Mucpaltes 3 4 Varable Drect Estmator LUP Estmator Estmator Fgure. Proporto of the oershp of health card based o drect, LUP ad estmators Fgure shos proporto estmates form a varet method, LUP estmators s more closed to the drect estmators tha estmators. 6. Coclusos here are a umber of problems th NEF-QVF for estmates the proporto usg LUP ad descrbe here. he eght determato s ecessar to fd out for estmators. A comparso for drect, LUP ad estmators ca be doe b calculated mea square error from each method. 7. Ackoledgemets I ould lke to thak small area estmato group at IPB for scece recogto ad dscusso. 8. Refereces Farrel PJ Baesa ferece for small area proportos. Sakhā: he Ida Joural of Statstcs 6: Fa RE, Herrot RA Estmates of come for small places: a applcato of James-Ste procedures to cesus data. Joural of the Amerca Statstcal Assocato 74: Ghosh M, Mat Small-area estmato based o atural expoetal faml quadratc varace fucto models ad surve eghts. Bometrka 9:

5 Malec D, Sedrask J, Morart CL & LeClere FB Small area ferece for bar varables the Natoal Health Iterve surve. Joural of the Amerca Statstcal Assocato 9: MacGbbo, B. ad omberl,.j Small area estmates of proportos va emprcal Baes techques. Surve Methodolog 5: Morrs CN. 98. Natural expoetal famles th quadratc varace fuctos. he Aals of Statstcs 0: Morrs CN Natural expoetal famles th quadratc varace fuctos: statstcal theor. he Aals of Statstcs : Rao JNK Small Area Estmato. Ne Jerse: Joh Wle & Sos. Sha K Some cotrbutos to small area estmato. Upublshed Ph.D. Dssertato, Uverst of Florda, Florda. 5