A COMPARISON OF ADJUSTED BAYES ESTIMATORS OF AN ENSEMBLE OF SMALL AREA PARAMETERS

Transcription

1 STATISTICA, anno LXIX, n. 4, 009 A COMPARISON OF ADJUSTED BAYES ESTIMATORS OF AN ENSEMBLE OF SMALL AREA PARAMETERS Enrco Fabrz 1. INTRODUCTION In recen years saple surveys have been characerzed by a growng deand for esaes of populaon descrpve quanes for doans (or 'areas') obaned classfyng he arge populaon accordng o geography or oher crera. As he saple poron peranng o doans s ofen oo sall o allow for relable esaon usng sandard desgn-based esaors, sall area esaon ehods have becoe a relevan research opc (see Rao, 003 or Lahr and Jang, 006 for a general nroducon). Eprcal and herarchcal Bayes ehods are an poran chaper of sall area esaon heory and are also wdely appled n pracce (see Rao, 003 Chapers 9 and 10 and he references heren). The basc dea behnd hese ehods s o rea doan descrpve quanes of neres (e.g. eans, oals, proporons) as rando and o esae he usng soe suary of her poseror dsrbuon, ypcally he poseror ean, ofen referred o as 'Bayes esaor' (Ghosh, 199). Bayes esaors ay be very effecve n provng he precson (saplng Mean Square Error) of 'drec' desgn-unbased (or desgn-conssen) esaors, bu hs proveen s ofen acheved a he cos of shrnkng he esaes oward a synhec esaor whch s obaned poolng ogeher daa fro all areas under sudy. For hs reason, Bayes esaors ay be proven o be poor for esang he acual dsrbuon funcon of a populaon ('enseble') of sall area paraeers (Lous, 1984, Heady and Ralphs, 004). The neres n he dsrbuon funcon ay be crucal when sall area esaes are used n subsanve applcaons such as he analyss of regonal dspares n he dsrbuon of econoc ndcaors of povery and ncoe nequaly (Fabrz e al. 005). A proper represenaon of he dsrbuon of he enseble of he paraeers s, for hese purposes, as poran as o dspose of relable esaes for each sub-naonal regon. In hs paper we dscuss he popular Fay-Herro odel (Fay and Herro, 1979) and a se of adjused esaors assocaed o. Wh adjused esaors we ean esaors of he sall area paraeers ha enjoy accepable properes wh respec o he esaon of eprcal dsrbuon funcon (EDF) or oher nonlnear funconals of he populaon ('enseble') of sall area paraeers.

2 70 E. Fabrz The an goal of he paper s o revew adjused esaors whn he fraework of herarchcal Bayesan odelng and o copare her frequens properes by eans of a Mone Carlo exercse. In parcular we focus on: ) he dsance beween he esaed and he rue dsrbuon funcon; ) effcency as easured by ean square error; ) robusness wh respec o he falure of he assupon of noraly of he rando effecs. We ephasze frequens properes snce hese are usually relevan for praconers and ore falar o fnal daa users. Moreover, we wll use he sae sulaon exercse o evaluae wheher poseror ean square errors, a naural easure of uncerany assocaed o adjused Bayes esaors, are also good frequens easures of varably. As ancpaed, we consder he effecs of sspecfcaon concernng he rando effecs: n parcular we focus on wrong dsrbuonal assupons. Ths ype of sspecfcaon s que lkely n pracce snce rando effecs are no drecly observable and deparures fro noraly are dffcul o deec (see Snharay and Sern, 003). We explore he properes of he adjused esaors when he odels assue noraly bu he rando effecs are acually generaed by an alernave dsrbuon. The paper s organzed as follows. In secon, we shorly dscuss he falure of Bayes esaors assocaed o he unvarae Fay-Herro odel as esaors of he varance (and hus of he EDF) of he 'enseble' of paraeers. Aong he any adjused esaors dscussed n he leraure we focus on consraned Bayes esaors (Ghosh, 199), consraned lnear Bayes esaors (Spjøvoll and Thosen, 1987) and a sulaneous esaon ehod proposed by Zhang (003). These esaors are revewed n secon 3. The sulaon exercse and he ools used n coparsons are nroduced n secon 4. Alhough all sulaons use populaons generaed under noraly, we wll focus on he accuracy of EDF esaon and no jus on ean and varance (as n Judkns and Lu, 000) snce wh a fne nuber of areas, he EDF of he populaon of area paraeers ay show soe slgh devaon fro noraly. Secon 5 focuses on sulaon s resuls; he behavour of adjused esaors when he noraly of rando effecs fals s dscussed n Secon 6. Concludng rearks are provded n Secon 7.. FAILURE OF BAYES ESTIMATORS AS ENSEMBLE ESTIMATORS IN THE FAY-HERRIOT MODEL The Fay-Herro odel ay be descrbed by he followng se of assupons: y e (1) v nd x () e N(0, ) (3)

3 A coparson of adjused Bayes Esaors of an enseble of sall area paraeers 71 nd v v N(0, ) (4) where { y } 1 s a collecon of 'drec' desgn-unbased (or approxaely desgn-unbased) esaors of a se of sall area populaon paraeers { }; { } s he se of assued known desgn-based varances assocaed o drec esaors and x a k 1 vecor of auxlary nforaon accuraely known for area. Moreover s assued ha Eev ( ) 0. (5) Sall area analyses are soewha dosyncrac as hey x randozaon and odel based probably spaces. More precsely, once denoed E D( ), V D( ) he expecaon and varance wh respec o he randozaon (desgn) dsrbuon and E M (), V M () he oens relaed o he odel or daa generang process, assupons (1) (4) ply ha EM( y ) ED( y) and VM( y ) VD( y), ha s he frs wo oens of y accordng o he odel (condonal on ) and randozaon dsrbuons are he sae. To be conssen n noaon le's also wre EM( ) x, VM( ) v and EM( ev) 0. Assung and v as known, he poseror eans of under odel (1) (5) are gven by ˆ B 1 y (1 ) x, where v ( v ) s a shrnkage facor ha gves o he drec esaor a wegh ha s a decreasng funcon of s saplng varance. We have ha drec esaors { y } are overdspersed wh respec o he underlyng { } and he sall area esaors are under-dspersed. The followng proposon holds. Proposon 1 Under odel (1) (5) and assung 1) we have ha: 1 EM ( ) v xx 1 (6) 1 1 wh xx ( 1) ( x x)( x x ), x x, ) 3) 1 v EMED ( y y) xx v xx 1 1 (7) 1 ˆ ˆ EMED ( ) v xx v xx 1 1 (8)

4 7 E. Fabrz A proof ay be found n he appendx. Noe ha does no represen a resrcve condon bu s helpful o oban sple forulas. We ay observe ha he shrnkage facor rules boh overdsperson of drec esaes and underdsperson of poseror eans. 3. ADJUSTED BAYES ESTIMATORS We consder hree dfferen adjused Bayes esaors assocaed o he Fay- Herro odel : ) he consraned Bayes, ) he consraned lnear Bayes (Spjøvoll and Thosen ehod), ) he esaors based on a sulaneous esaon ehod proposed by Zhang (003). These esaors wll be revewed whn a herarchcal Bayes fraework; ha s, we do no assue he hyperparaeers, v as known bu specfy a pror dsrbuon for he. In wha follows we denoe he daa on whch he analyss s condoned as z{ y,, x } 1. We ancpae ha all adjused esaors ˆ are suares of he poseror dsrbuons p( z ) dfferen fro he poseror ean ˆ HB E( z ) and are herefore subopal wh respec o quadrac loss. Uncerany assocaed o hese alernave poseror suares ay be easured by he poseror ean square error: PMSE( ˆ ) ( z ) ( ˆ ˆ ) (9) * * HB V We denoe E ( ) he expecaon wh respec o he odel ong he subscrp M, snce randozaon oens are no longer nvolved. Of course, PMSE( ) V( z ); he beer represenaon of he Eprcal Dsrbuon Funcon of he populaon of Sall Area paraeers s pad a he prce of soe loss of effcency. 3.1 The Consraned herarchcal Bayes esaor Consraned Bayes esaors have been nroduced and dscussed by Lous (1984) under noraly and by Ghosh (199) under less resrcve dsrbuonal assupons. The a s o oban a se of esaors { }, 1,...,, opal under quadrac loss and sasfyng he followng consrans: 1) ˆ HB 1 ) ( 1) ( ) E ( ) z 1 1 The consraned herarchcal Bayes (CHB) esaors (see Rao, 003, secon 10.13) are gven by:

5 A coparson of adjused Bayes Esaors of an enseble of sall area paraeers 73 ˆCHB ˆ HB ˆHB ˆ HB a( z )( ) (10) where V( 1 ) z az ( ) 1 ( 1) 1 ˆ ˆ 1 ( HB HB ) The poseror ean square error ( ˆCHB PMSE ) ay be used o evaluae he uncerany assocaed o hs esaor. 3. The consraned herarchcal lnear Bayes Esaor Le's suppose, for he oen, ha he hyperparaeers, v are known. The Consraned Lnear Bayes esaor of s a suary of he poseror dsrbuon n he for ˆL a y b sasfyng he consrans: 1) ( ˆL E ) x, 1 ) E( ˆ x ). L v Noe ha when he poseror ean ay be expressed n lnear for, he dsrbuon enjoys poseror lneary (Goldsen, 1975), a condon ha holds for a varey of conjugae odels. The consraned lnear Bayes esaor s gven by: ˆ CLB 1/ 1/ y (1 ) x (11) 1 wh v ( v ) (see Spjøvoll and Thosen, 1987 and Rao, 003, secon 9.8). Ths esaor owes s populary o s slary o ˆB : s sll a lnear cobnaon of y and x ha, wh respec o ˆB, pus ore wegh on he 'drec' esaor y, whereby leadng o a se of esaes less shrunken oward he synhec coponen. The esaor (11) ay be hough as condonal on (, v ). We sply propose o defne he Consraned Herarchcal Lnear Bayes (CHLB) esaor as ˆ CHLB E v (, ) ( ˆCLB ) (1) z where E (, v z s he expecaon aken wh respec o he poseror dsrbuon ) of, v. An explc forula for (1) depends on he chosen pror dsrbuons

6 74 E. Fabrz and ay be n general dffcul o work ou. Noneheless (1) ay be easly approxaed usng he oupu of Markov Chans Mone Carlo (MCMC) algorhs of coon use for he analyss of herarchcal odels. 3.3 The sulaneous esaon ehod proposed by Zhang Gven he se { } of he area paraeers of neres, le { ( )} be he assocaed ordered se ( (1) ()... ( )). Then E( ( ) z ) s he bes predcor of under quadrac loss and { } s he bes 'enseble' esaor of { } under quadrac loss. The se of esaors { } s no area specfc n ha s sngle eleens are no assocaed o specfc areas. To ach he { } wh he sall areas Zhang (003) proposes, under an eprcal Bayes esaon approach, o esae he ranks of { } usng hose of he { E( z )} se. By he way, he ranks of he poseror eans ay be poor esaors of acual ranks, especally f here s consderable varably n he poseror varances. Followng Ghosh and Ma (1999) we propose rˆ E( rank( z )), he poseror expecaon of ranks, as he esaor of ranks requred n order o ach he enseble esaor { } wh he areas. In he conex of herarchcal Bayes odelng, hs esaor of ranks ay be easly approxaed fro he oupu of MCMC algorhs. More specfcally, we can rank he ( s ) z fro any draw s of he Markov Chan afer convergence. Then we can approxae r ˆ averagng he ranks rank( ( s ) z ) MC 1 S over all draws, obanng rˆ S rank( ( ) ) s 1 s z where S s he nuber of eraons of Markov Chan afer convergence used for he esaon of he poseror dsrbuon. To suarze, he esaor based on Zhang deas pleened n he conex of herarchcal Bayes odelng s gven by: ˆ ZHB (13) rˆ wh r ˆ approxaed by r ˆMC when he poseror dsrbuons are obaned usng MCMC algorhs. 4. A SIMULATION EXPERIMENT UNDER THE ASSUMPTION OF NORMALITY FOR THE RAN- DOM EFFECTS In hs secon we nroduce a sulaon experen whose a s o copare he effecveness of he adjused esaors dscussed above n correcng he overshrnkage. We copare also her effcency and consder wheher he PMSE are adequae esaors of her frequens ean square errors. More

7 A coparson of adjused Bayes Esaors of an enseble of sall area paraeers 75 specfcally, we copare he drec esaors ˆ { y }, he poseror eans ˆHB { ˆHB } and he varous adjused esaors ˆCHB { ˆCHB }, ˆCHLB { ˆCHLB } and ˆ ZHB { ˆ ZHB }. The sulaon s based on R=1,000 Mone Carlo (MC) saples, and all coparsons are referred o he eprcal dsrbuon of he varous esaors n hs replcaon space. Daa are generaed accordng o he Fay-Herro odel (1) (5) seng for splcy x 0. We consder boh he case of oderae and large nuber of areas seng 30, 100. We se v 1 and consder hree dfferen confguraons of desgn varances. They are chosen n he followng way: we dvde he se of areas n fve groups. Varances vary across groups bu are consan whn he. All confguraons are llusraed n Table 1. They dffer accordng o nforaveness of 1 drec esaors, as easured by v ( v ). We evaluae he nforaveness of drec esaors n coparson o he dsperson of he values around he synhec coponen. Populaon 1 descrbes a suaon where drec esaors show a wde range of nforaveness ( [0.11,0.91] ); Populaon descrbes a suaon n whch drec esaors are poorly nforave ( [0.11, 0.5] ), whle n Populaon 3 we sudy he case of raher srongly nforave drec esaors ( [0.5, 0.91] ). DIR TABLE 1 Dfferen confguraons of desgn varances for he sulaon Desgn varances ,..., 1,..., 1,..., 1,..., 1,..., Populaon Populaon Populaon We do no consder equal saplng varances,.e. snce hs s seldo he case n pracce; oreover ay be proven ha, for x and, v known, ˆ CHB ˆ CHLB for (Rao, 003, secon 9.6). As a consequence, even f we do no assue and v as known and work wh a fne nuber of areas, we ay expec he wo esaors o show close perforances. When odelng, s assued ha and v are unknown wh pror dsrbuons p(, v ) p( ) p( v ) wh N(0, K), v Unf (0, L ) where K 100 and L 0 are large consans wh respec o he scale of he daa.

8 76 E. Fabrz Ths prors guaranee properness of he poseror dsrbuons, very ld pac on poseror dsrbuons and good behavor (fas convergence and good xng) of MCMC algorhs (Gelan, 006). To copare esaors, we consder he ndcaors descrbed below. ) Overshrnkage correcon. Le's defne R * 1 r r 1 AV( ˆ ) R v ( ) (14) ˆ ˆ ˆ, and * { DIR, HB, CHB, CHLB, ZHB}., we expec hs ndcaor o be larger han 1 for, less han 1 for ˆHB and close o 1 for he reanng esaors. ) Kologorov-Srnov dsance. For each eraon r, he Kologorov-Srnov dsance beween he EDF of he adjused esaor ˆ * { ˆ r, r} and ha of he rue values { r, } denoed respecvely as F * and F T s calculaed as D (, ) ax F ( u ) F ( u ) where j 1,..., and u (, ) s he * 1 * * where v ( r ) ( 1) ( 1, r r ) Snce we se x and v 1 ˆDIR r r r j * j, r T j, r r r r vecor obaned poolng ogeher he rue values and hose obaned wh he adjused predcor for he r-h MC replcaon. The dsances calculaed a each eraon are hen averaged over MC replcaons o oban 1 R D(, ) R D (, ). For he ease of coparson we repor r 1 ˆ r r r * D(, ) (, ) (15) HB D D( ˆ, ) whereby assung he non adjused herarchcal Bayes esaors as a benchark. ) Anderson-Darlng dsance. Anderson and Darlng (1954) nroduced a goodness-of-f sasc ha can be used o evaluae he dsance of an EDF fro a connuous reference dsrbuon. Copared o he Kologorov-Srnov dsance, s known o be nfluenced o a greaer exen by he dscrepances n he als of he dsrbuon. In our parcular conex he 'eprcal' Anderson-Darlng dsance s defned as follows 1 F ( u ) F ( u ) A (, ) ( F ( u )) * jr, T jr, r 1 (0,1) T j, r 4 j 1 FT( u j, r )[1 FT( u j, r )] The copued dsances are hen averaged over all R replcaons, and ˆ * A (, ) (, ) (16) HB A s repored n resuls. A ( ˆ, )

9 A coparson of adjused Bayes Esaors of an enseble of sall area paraeers 77 v) Frequens effcency. We evaluae he average pac of he adjusen on he uncondonal frequens MSE of sall area predcors defned as MSE( ) E( ) (see Rao, 003, secon 6.). We esae he MSE( ) * 1 * by eans of s Mone Carlo approxaon ˆ R se ( ) ( ˆ MC R 1 r, r, ). r These quanes are area-specfc. We jus focus on he ean of her dsrbuon across areas: ase ( ˆ ) se ( ˆ ) (17) * 1 * MC 1 MC v) Frequens evaluaon of PMSE. Frequens properes of (9) are evaluaed usng he followng easure of relave bas: apse( ˆ ) ( ˆ ) (18) 1 R * * 1 R pse 1 MC r r, 1 semc ( ) * where pse ( ˆ MC r, ) s he pse ( MC ) calculaed usng daa fro he r-h draw of he Mone Carlo exercse. Moreover noe ha ( ˆHB psemc ) error reduces o poseror varance. Codes are wren n R (R Developen Core Tea, 006). For MCMC calculaons we used he Brugs package (Thoas and O'Hara, 006) whch recursvely calls he MCMC dedcaed sofware OpenBUGS (Thoas e al. 006). As for echncal deals concernng MCMC calculaons, we generae saples of sze 0,000 for all chans deleng a conservave 'burn n' saple of sze 5,000. In fac, he relavely sple noral odels eployed n he sulaons have all shown very fas convergence raes. Convergence has been checked by eans of sandard convergence sascs (Cowles and Carln, 1996). 5. SIMULATION RESULTS For brevy, we repor resuls for he case 100 and dscuss he case 30 only when dfferences are rearkable or he coparson hghlghs poran pons. Resuls abou he shrnkage correcon are dsplayed n able. I s apparen ha drec esaes are overdspersed, wh overdsperson ncreasng wh he average varance of drec esaors; herarchcal Bayes esaors are underdspersed n all suaons, and ore serously so when he drec esaes convey lle nforaon (Populaon ). More poran, all adjused esaors approxaely elnae he overshrnkage. Flucuaons of values around 1 do no see o follow any sgnfcan paern. The correcon of overshrnkage does no see o be nfluenced by he nuber of areas.

10 78 E. Fabrz TABLE Overshrnkage correcon as easured by he ndcaor AV( ), 100 ˆDIR ˆHB ˆCHB ˆCLHB ˆZHB Populaon Populaon Populaon Table 3 shows resuls based on he Kologorov-Srnov dsance. All adjused esaors clearly prove he perforances of ˆHB and he proveen s larger when 100 wh respec o he case of 30. Aong adjused esaors, ˆZHB eerges clearly as he bes. Noe ha, gven he sze of Mone Carlo (MC) errors, all observed dfferences appearng n able 3 can be aken as 95% sgnfcan. Moreover, noe ha he advanage of ˆZHB over he oher adjused ehods s less pronounced for Populaon (poorly nforave drec esaes). Ths akes sense snce he esaon of ndvdual ( ) requres ore nforaon han he global adjusen on whch ˆCHB and ˆCLHB are based. ˆCHB and ˆCLHB perfor closely and none of he wo sees preferable. TABLE 3 Rao of he Kologorov-Srnov dsances beween esaed and acual EDF averaged over MC replcaons dvded by he sae quany calculaed for he ˆHB esaors,.e. D (, ), 100 Populaon ˆDIR ˆHB ˆCHB ˆCLHB ˆZHB Populaon Populaon Populaon Table 4 presens he resuls relaed o he Anderson-Darlng dsance. We noe ha ˆHB s n an neredae poson beween he drec esaors (ha are he wors perforers over all sengs) and he adjused esaors ha are beer. Aong adjused esaors ˆZHB s clearly beer han he oher wo excep for Populaon, characerzed by poorly nforave drec esaes where, by he way, sll perfors a lle beer. We ay hen conclude ha he esaon ehod proposed by Zhang urns ou o be he bes wh respec o boh consdered easures of dsance and gves s bes when he nuber of areas s large and he drec esaes are no oo precse. TABLE 4 Rao of he Anderson-Darlng dsances beween esaed and acual EDF averaged over MC replcaons dvded by he sae quany calculaed for he ˆHB * esaors,.e. A ( ˆ, ), 100 Populaon ˆDIR ˆHB ˆCHB ˆCLHB ˆZHB Populaon Populaon Populaon

11 A coparson of adjused Bayes Esaors of an enseble of sall area paraeers 79 The resuls relaed o he repeaed saplng effcency, as easured by he eprcal uncondonal Mean Square Error are shown n Table 5, where he adjusen of ˆHB has, as expeced, a cos n ers of effcency. The ncrease n Mean Square Errors depends on he precson of drec esaors. When he precson s hgh (Populaon 3), he rse s around 10%, bu when s low, ase MC ( ) are 0% or even 30% (n he case of ˆCLHB ) hgher han n he case of ˆHB. Noneheless we ay noe ha he proveen wh respec o he drec esaors reans subsanal. Moreover ˆCHB and ˆZHB show slar perforances, whle ˆCLHB urns ou o be a lle less effcen. TABLE 5 Effcency of he consdered esaors as easured by ase MC ( ), 100 ˆDIR ˆHB ˆCHB ˆCLHB ˆZHB Populaon Populaon Populaon A ore dealed analyss of he dsrbuon of ase MC ( ) across areas (for whch we do no show ables) hghlghs he very dfferen behavor of ˆCLHB wh respec o ˆCHB and ˆZHB : perfors clearly beer when drec esaes are ore precse han he average and far worse n he case of areas characerzed by he os precse drec esaes. Ths behavor depends on he naure of he esaor. Fro (11) we ay noe ha, wh respec o poseror ean, ˆCHB gves less wegh o he synhec coponen and ore o he drec one. When y s very precse, hs leads o ore effcen esaors han oher adjused ehods; unforunaely y receves ore wegh even when s unrelable, hus producng a large loss n effcency wh respec o and he oher adjused ehods. TABLE 6 Average rao of he esaed PMSE o he MSE : apse( ), 100 ˆHB ˆCHB ˆCLHB ˆZHB Populaon Populaon Populaon Table 6 repors an evaluaon of frequens properes of he Poseror Mean Square Error defned n (9). I s apparen ha hs Bayesan uncerany easure, represens a sensble easure of varably also wh respec o repeaed saplng; n fac n all cases s approxaely unbased; alhough should be noed ha hs propery holds on average wh respec o he se of areas beng suded. The propery, ha was known o hold for he poseror varances as frequens var-

12 80 E. Fabrz ably easures of ˆHB under careful choce of he prors (Ganesh and Lahr, 008), s hen exensble o he case of poseror ean square errors, a leas for he pror dsrbuons chosen n our sulaon exercse. 6. PROPERTIES OF THE ADJUSTED ESTIMATORS UNDER FAILURES OF THE NORMALITY OF RANDOM EFFECTS In hs secon we copare he perforance of adjused esaors already consdered when he assued noraly of rando effecs does no hold, ha s, we evaluae wheher hey are robus wh respec o hs deparure fro he assupons under whch hey are obaned. We generae populaon daa accordng o he sae sulaon experen dscussed n Secon 4, changng he dsrbuon assued for he rando effecs. We consder: 1) v Laplace(0,1/ ) and ) v Exp(1). In boh cases Vv ( ) 1, so he nerpreaon of Populaon 1, Populaon, Populaon 3 n ers of nforaveness of he drec esaors reans unchanged. Case 1) represens a ld devaon fro noraly whle case ) represens a ore serous deparure fro hs assupon. We noe ha he perforances of herarchcal Bayes esaors of he area eans rean approxaely unchanged when rando effecs are generaed by a Laplace dsrbuon bu a noral odel s assued. Ths robusness wh respec o oderae falures of he assupons on he dsrbuon of he rando effecs s noed n Snharay and Sern (003) and Fabrz and Trvsano (009). When rando effecs are generaed fro an Exponenal dsrbuon, he deeroraon of he perforances of HB s subsanal, even hough no caasrophc. Under he assupon on he drec esaors varances of Populaon 1 he Kologorov-Srnov dsance grows by a 30%, he Anderson-Darlng by a 60%, whle he Mean Square Error averaged over he se of all he areas only by 10%. In Table 7 resuls abou he Kologorov-Srnov and he Anderson- Darlng dsance are repored for 100. Resuls relaed o oher ndcaors and 30 are no repored snce hey are subsanally slar. Fro Table 7 we ay noe ha all adjused predcors provde saller proveens wh respec o HB han n he case of noraly; hs eans ha he esaon of he Eprcal Dsrbuon Funcon of he area averages or soe of s feaures s ore sensve o wrong dsrbuonal assupons han he esaon of ndvdual area averages. The perforance of he adjused predcors are now closer han under noraly of rando effecs. ˆZHB reans slghly beer n ers of Kologorov- Srnov dsance, whle ˆCHB s a lle beer f we consder he Anderson- Darlng dsance and ase MC ( ). Snce ˆZHB urned ou o be beer under noraly, we ay conclude ha s perforances deerorae ore when hs as-

13 A coparson of adjused Bayes Esaors of an enseble of sall area paraeers 81 supon fals wh respec o oher adjused esaors. Ths akes sense, snce ˆZHB akes an heaver use of he assupon of noraly n he esaon of { } han he oher predcors, whose correcons are based on he esaon of ( ) oens. Noe, however, ha he perforances of ˆZHB rean always coparable o hose of oher adjused predcors even under bg deparures fro noraly of rando effecs. TABLE 7 Coparson of adjused esaors under falures of he dsrbuonal assupon on rando effecs, 100 Acual dsrbuon of he v Populaon ˆDIR ˆHB ˆCHB ˆCLHB ˆZHB Laplace Exponenal Laplace Exponenal D (, ) Populaon Populaon Populaon Populaon Populaon Populaon A (, ) Populaon Populaon Populaon Populaon Populaon Populaon CONCLUSIONS In hs paper we dscuss and copare hree dfferen ehods for adjusng a se of sall area esaors n order o prove esaon of he EDF of he 'enseble' of sall area paraeers. Two of hese predcors (CHB and CLHB) are well known n he leraure, whle he hrd (ZHB) s ore recen. The evaluaon of he perforances of he hree esaors s based on a sulaon exercse coverng a range of suaons whch are poenally relevan for sall area praconers. A frs concluson fro hs analyss s ha all he ehods consdered ee he goal of correcng overshrnkage and leadng o ore realsc esaes of EDF of he 'enseble' of sall area eans. On he oher hand, adjused esaors are less effcen han poseror eans, bu her gan n precson wh respec o drec esaors reans subsanal. Zhang s predcor sees as beer han he oher wo when he noraly of rando effecs holds. We also consdered falures of hs assupon, generang populaons wh Laplace and Exponenally dsrbued rando effecs. In hs laer cases Zhang s predcor and he beer known CHB predcor show close perforances. Thereby Zhang s predcor s ore sensve o he falure of dsrbuonal assupons. We suded he frequens behavor of easures of uncerany assocaed o adjused herarchcal Bayes esaors, fndng ha, a leas for he pror dsrbu-

14 8 E. Fabrz ons chosen n he sulaon exercse, he poseror MSEs have good frequens properes and ay be assued by pracconers as accepable easures of frequens varably. Dpareno d Scenze Econoche e Socal Unversà Caolca del S. Cuore, sede d Pacenza ENRICO FABRIZI APPENDIX Proof of (6). Now 1 1 EM ( ) E ( x x)( x x) v v 1 as M( ) M( ) [ M( )] v xx E V E and 1 M( ) M( ) [ ( )] v E V E xx To prove (7) noe ha E 1 1 M E D ( ) M D 1 y y 1 1 E E y y 1. As D( ) D( ) [ D( )] E y V y E y and 1 D( ) D( ) [ D( )] E y V y E y we have ha 1 1 EMED y y EM ( 1) Wh he sae arguen used n he proof of (6) we oban 1 1 EM ( 1) v ( )( ) x x x x 1 Evenually, o prove (8) noe ha y (1 ) x and B

15 A coparson of adjused Bayes Esaors of an enseble of sall area paraeers 83 1 ˆB ˆ B 1 ( ) ( y y) (1 ) xx (1 ) ( y y)( x x) 1 1 As a consequence 1 ˆB ˆ B EMED ( ) 1 1 (1 ) (1 ) xx v xx xx v xx REFERENCES T.W. ANDERSON AND D.A. DARLING (1954) A es of goodness of f, Journal of he Aercan Sascal Assocaon, 49, pp M.K. COWLES, B.P. CARLIN (1996) Markov Chan Mone Carlo convergence dagnoscs: a coparave revew, Journal of he Aercan Sascal Assocaon, 91, pp E. FABRIZI, M.R. FERRANTE AND S. PACEI (005) Esaon of povery ndcaors a sub-naonal level usng ulvarae sall area odels. Sascs n Transon, 7, pp E. FABRIZI AND C. TRIVISANO (009) Robus Lnear Mxed Models for Sall Area Esaon, Journal of Sascal Plannng and Inference, 140, R. FAY R. AND R.A. HERRIOT (1979) Esaes of ncoe for sall places: an applcaon of Jaes-Sen procedures o Census daa, Journal of he Aercan Sascal Assocaon, 74, pp N. GANESH AND P. LAHIRI (008), A new class of average oen achng prors, Boerka, 95, pp A. GELMAN (006) Pror dsrbuon for varance paraeers n herarchcal odels, Bayesan Analyss, 1, pp M. GHOSH (199) Consraned Bayes esaon wh applcaons, Journal of he Aercan Sascal Assocaon, 87, M. GHOSH AND T. MAITI (1999) Adjused Bayes esaors wh applcaons o Sall Area Esaon, Sankhya, Ser. B, 61, pp H. GOLDSTEIN (1975) A Noe on soe Bayesan Nonparaerc Esaes, Annals of Sascs, 3, pp P. HEADY AND M. RALPHS (004) Soe fndngs of he EURAREA projec and her plcaons for sascal polcy, Sascs n Transon, 6, pp D.R. JUDKINS AND J. LIU (000) Correcng he Bas n he Range of a Sasc across Sall Areas, Journal of Offcal Sascs, 16, pp P. LAHIRI, J. JIANG (006) Mxed odel predcon and sall area esaon, Tes, 15, pp T.A. LOUIS (1984) Esang a Populaon of Paraeers Values Usng Bayes and Eprcal Bayes Mehods, Journal of he Aercan Sascal Assocaon, 79, pp J.N.K. RAO (003) Sall Area Esaon, John Wley and Sons, New York. R DEVELOPMENT CORE TEAM (006) R: A language and envronen for sascal copung. R Foundaon for Sascal Copung, Venna, Ausra. ISBN , URL downloadable a hp://

16 84 E. Fabrz S. SINHARAY, H.S. STERN (003) Poseror predcve odel checkng n herarchcal odels, Journal of Sascal Plannng and Inference, 111, pp E. SPJØTVOLL, I. THOMSEN (1987) Applcaon of soe eprcal Bayes ehods o Sall Area Esaon, Bullen of he Inernaonal Sascal Insue,, pp A. THOMAS, B. O'HARA (006) The BRugs package, Sofware and Docuenaon, downloadable a hp://cran.r-projec.org/org/packages/brugs. A. THOMAS, B. O HARA, U. LIGGES, S. STURZ (006) Makng BUGS Open, R News, 6, pp L.C. ZHANG (003) Sulaneous esaon of he ean of a bnary varable fro a large nuber of sall areas, Journal of Offcal Sascs, 19, pp SUMMARY A coparson of adjused Bayes Esaors of an enseble of sall area paraeers Wh enseble properes of sall area esaors, we ean her ably o reproduce he Eprcal Dsrbuon Funcon (EDF) characerzng he collecon of underlyng sall area paraeers (eans, oals). Good enseble properes ay be relevan when esaon of non-lnear funconals of he EDF of sall area paraeers (such as her varance) s needed. Sall area esaors assocaed o he popular Fay-Herro odel are consdered. Bayes esaors,.e. poseror eans, do no enjoy of good enseble properes. In hs paper hree dfferen adjused predcors are copared, by eans of a sulaon exercse, under he assupon of correcly specfed odel. As he dsrbuonal assupons on he rando effecs are dffcul o assess, he consdered predcors are copared also wh respec o her robusness o he presence of falures n he dsrbuonal assupons on he rando effecs.