Small area estimation under a Fay-Herriot model with preliminary testing for the presence of random area effects

Transcription

1 Catalogue no. -00-X ISSN Survey Methodology 4- Sall area estaton under a Fay-Herrot odel wth prelnary testng for the presence of rando area effects by Isabel Molna J.N.K. Rao and Gaur Sankar Datta Release date: June 9 05

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3 Survey Methodology June 05 Vol. 4 No. pp. -9 Statstcs Canada Catalogue No. -00-X Sall area estaton under a Fay-Herrot odel wth prelnary testng for the presence of rando area effects Isabel Molna J.N.K. Rao and Gaur Sankar Datta Abstract A popular area level odel used for the estaton of sall area eans s the Fay-Herrot odel. Ths odel nvolves unobservable rando effects for the areas apart fro the (fxed) lnear regresson based on area level covarates. Eprcal best lnear unbased predctors of sall area eans are obtaned by estatng the area rando effects and they can be expressed as a weghted average of area-specfc drect estators and regresson-synthetc estators. In soe cases the observed data do not support the ncluson of the area rando effects n the odel. Excludng these area effects leads to the regresson-synthetc estator that s a zero weght s attached to the drect estator. A prelnary test estator of a sall area ean obtaned after testng for the presence of area rando effects s studed. On the other hand eprcal best lnear unbased predctors of sall area eans that always gve non-zero weghts to the drect estators n all areas together wth alternatve estators based on the prelnary test are also studed. The prelnary testng procedure s also used to defne new ean squared error estators of the pont estators of sall area eans. Results of a lted sulaton study show that for sall nuber of areas the prelnary testng procedure leads to ean squared error estators wth consderably saller average absolute relatve bas than the usual ean squared error estators especally when the varance of the area effects s sall relatve to the saplng varances. Key Words: Area level odel; Eprcal best lnear unbased predctor; Mean squared error; Prelnary testng; Sall area estaton. Introducton A basc area-level odel called the Fay-Herrot (FH) odel s often used to obtan effcent estators of area eans when the saple szes wthn areas are sall. Ths odel nvolves unobservable area rando effects and the eprcal best lnear unbased predctor (EBLUP) of a sall area ean s obtaned by estatng the assocated rando effect. The EBLUP s a weghted cobnaton of a drect area-specfc estator and a regresson-synthetc estator that uses all the data. An estator of the ean squared error (MSE) of the EBLUP was obtaned frst by Prasad and Rao (990) usng a oent estator of the rando effects varance and later by Datta and Lahr (000) for the restrcted axu lkelhood (ML) estator of the varance. Rao (003 Chapter 7) gves a detaled account of EBLUPs and ther MSE estators for the FH odel. Soetes the observed data do not support the ncluson of the area effects n the odel. Excludng the area effects leads to the regresson-synthetc estator. Usng ths dea Datta Hall and Mandal (0) proposed to do a prelnary test for the presence of the area rando effects at a specfed sgnfcance level and then to defne the sall area estator dependng on the result of the test. If the null hypothess of no area rando effects s not rejected the odel wthout the area effects s consdered to estate the sall area eans.e. the regresson-synthetc estator s used. If the null hypothess s rejected the usual EBLUP under the FH odel wth area effects s used. Datta et al. (0) rearked that the above prelnary test estator (E) could lead to sgnfcant effcency gans over the EBLUP partcularly. Isabel Molna Departent of Statstcs Unversty Carlos III de Madrd C/Madrd Getafe (Madrd) Span and Insttuto de Cencas Mateátcas (ICMAT) Madrd Span. E-al: sabel.olna@uc3.es; J.N.K. Rao School of Matheatcs and Statstcs Carleton Unversty Ottawa Canada; Gaur Sankar Datta Departent of Statstcs Unversty of Georga Athens U.S.A.

4 Molna Rao and Datta: Sall area estaton under a Fay-Herrot odel wth prelnary testng when the nuber of sall areas s only odest n sze. For prelnary testng they consdered a noralty-based test as well as a bootstrap test that avods the noralty assupton. When the estated area effects varance s zero the EBLUP becoes autoatcally the regressonsynthetc estator. However the estated MSE obtaned by Prasad and Rao (990) or Datta and Lahr (000) does not reduce to the estated MSE of the regresson-synthetc estator. Thus the usual MSE estators are based for sall rando effects varance. For ths reason we propose MSE estators of the EBLUP based on the prelnary testng procedure. If the rando effects varance s not sgnfcant accordng to the test we consder the MSE estator of the synthetc estator. Otherwse we consder the usual MSE estators of the EBLUP. The EBLUP attaches zero weght to the drect estates for all areas when the estated area effects varance s zero. On the other hand survey practtoners often prefer to attach a strctly postve weght to the drect estates because the latter ake use of the avalable area-specfc unt level data and also ncorporate the saplng desgn. L and Lahr (00) ntroduced an adjusted axu lkelhood () estator of the varance of rando effects that s always postve and therefore leads to EBLUPs gvng strctly postve weghts to drect estators. As we shall see a prce s pad n ters of bas when usng the EBLUP based on the estator. We propose here alternatve sall area estators that always gve a postve weght to the drect estators but wth a saller bas. Ths paper studes eprcally the propertes of Es of sall area eans n coparson wth the usual EBLUPs and other proposed estators. In partcular we study the choce of the sgnfcance level for the area estates and for the MSE estates based on the prelnary test (). EBLUPs based on the estator of the rando effects varance of L and Lahr (00) whch gve non-zero weghts to the drect estators n all areas are also studed and copared to versons of (-). Dfferent MSE estators of these - estators are also studed wth respect to relatve bas. Based on sulaton results the EBLUPs and the assocated MSE estators that perfored well are recoended. Fnally coverage and length of noralty-based predcton ntervals obtaned usng the EBLUPs and the assocated MSE estators are exaned. The paper s organzed as follows. Secton descrbes the FH odel and the EBLUPs of sall area eans. Secton 3 coents on MSE estaton. Es of sall area eans and MSE estators based on the are ntroduced n Secton 4. Secton 5 descrbes sall area estators and assocated MSE estators under estaton of the area effects varance. Alternatve estators that also attach postve weghts to drect estators together wth proposed MSE estators are ntroduced n Secton 6. Secton 7 reports the results of the sulaton study. Fnally Secton 8 gves soe concludng rearks. Estaton of sall area eans Consder a populaton parttoned nto areas and let be the ean of the varable of nterest for area =. We assue that a saple s drawn ndependently fro each area. Let y be a desgnunbased drect estator of obtaned usng survey data fro the sapled area. Drect estators are very neffcent for areas wth sall saple szes. We study sall area estaton under an area level odel n whch the values of area level covarates are avalable for all areas. The basc odel of ths type s the Fay-Herrot odel ntroduced by Fay and Herrot (979) to estate per capta ncoe for sall Statstcs Canada Catalogue No. -00-X

5 Survey Methodology June 05 3 places n the Unted States. Ths odel conssts of two parts. The frst part assues that drect estators y of sall area eans are desgn unbased satsfyng nd y = e e N 0 D =. (.) Here the saplng varance D =Vary s assued to be known for all areas =. In practce the D s are ascertaned fro external sources or by soothng the estated saplng varances usng a generalzed varance functon ethod (Fay and Herrot 979). In the second part the Fay-Herrot odel treats as rando and assues that a p- vector of area level covarates x lnearly related to s avalable for each area.e. d = xβ v v N 0 A = (.) where v s the rando effect of area assued to be ndependent of e and A 0 s the varance of the rando effects. Observe that argnally nd y N xβ D A =. (.3) Lettng y = y y X = x x and D =dag D D odel (.3) ay be expressed n atrx notaton as y NXβΣ A wth Σ A = D AI where I denotes the dentty atrx. If A s known the coponentwse best lnear unbased predctor (BLUP) of θ = s gven by where θ A = A A = Xβ A AΣ A y Xβ A (.4) β A XΣ A X XΣ Ay A D xx A D xy = = s the weghted least squares (WLS) estator of β. In practce however A s not known. Substtutng a consstent estator  for A n the BLUP (.4) we get the EBLUP gven by (.5) θ = = Xβ Σ y Xβ (.6) A where β = β A and Σ = D A I. For the th area the EBLUP of can be expressed as a convex lnear cobnaton of the regresson-synthetc estator x β and the drect estator y as = B A xβ B A y (.7) Statstcs Canada Catalogue No. -00-X

6 4 Molna Rao and Datta: Sall area estaton under a Fay-Herrot odel wth prelnary testng where the weght attached to the regresson-synthetc estator x β s gven by B A where B A = D A D. Observe that the weght ncreases wth the saplng varance D. Thus when the drect estator s not relable.e. D s large as copared wth the total varance A D ore weght s attached to the regresson-synthetc estator x β. On the other hand when the drect estator s effcent D s sall relatve to A D and then ore weght s gven to the drect estator y. Several estators of A have been proposed n the lterature ncludng oent estators wthout noralty assupton ML estator and restrcted (or resdual) ML estator (ML) estator. The ML estator of A s * ML * A =ax 0 A ML where  ML can be obtaned by axzng the profle lkelhood functon gven by where c denotes a generc constant and = exp LP A c Σ A yp A y P A = Σ A Σ A X XΣ A X XΣ A. The ML estator of A s * restrcted/resdual lkelhood gven by A =ax 0 A where * L A = c XΣ A X Σ A exp yp A y.  s obtaned by axzng the In ths paper we focus on the ML estator  whch s frequently used n practce and we denote by θ = the EBLUP gven n (.6) obtaned wth A = A. 3 Mean squared error Note that the BLUP A of the sall area ean s a lnear functon of y. Hence ts MSE can be easly calculated and t s gven by the su of two ters: MSE A = g A g A where g A s due to the estaton of the rando area effect v and the regresson paraeter β wth g g A D B A g A B A x XΣ AX x. A s due to the estaton of However the EBLUP gven n (.7) s not lnear n y due to the estaton of the rando effects varance A. Usng a oents estator of A Prasad and Rao (990) obtaned a second order correct approxaton for the MSE of the EBLUP. Later Datta and Lahr (000) and Das Jang and Rao (004) Statstcs Canada Catalogue No. -00-X

7 Survey Methodology June 05 5 obtaned second order correct MSE approxatons under ML and ML estaton of A. When usng the ML estator of A ther approxaton to the MSE for large s gven by where g A g A g A o MSE = (3.) 3 V A g 3 A = B A and V A =. A D A D Note that as g A = O g A = O and g 3 A = O so g A s the leadng ter n the MSE for large. However for sall A g A s approxately zero and then g A ght be the leadng ter for sall. For exaple takng only one covarate p = wth 3 constant values x = and constant saplng varances D = D = and lettng A =0 we obtan g 0 = 0 g 0 = D and g 3 0 = D ; that s g 3 0 s twce as large as g 0. Datta and Lahr (000) obtaned an estator of the MSE of the EBLUP gven by = g A g A g A se =. (3.) 3 The MSE estator (3.) s second-order unbased n the sense that E o se = MSE. In the case that A =0 the BLUP of becoes the regresson-synthetc estator = x β 0. But surprsngly the approxaton to the MSE of the EBLUP gven n (3.) can be SYN very dfferent fro the MSE of the synthetc estator. Note that the latter s g g g MSE = 0 < 0 0 SYN 3 because g 3 0 s strctly postve even for A =0. In fact n the sple exaple wth only one covarate p wth constant values x = and constant saplng varances D = D = we have MSE = g 0 = D whereas the approxaton to the MSE of the EBLUP gven n (3.) wth SYN A =0gves MSE g 0 g 0 = 3 D three tes larger. It turns out that (3.) s not a 3 good approxaton of the MSE of the EBLUP when A =0 and nstead we should use MSE = g 0. Moreover snce for A =0 ths quantty does not depend on any unknown paraeter we can take t also as MSE estator.e. we can take se = g 0. In practce the true value of A s not known but we have the consstent estator A =0 the EBLUP becoes the regresson-synthetc estator for all areas that s In ths case = SYN = xβ 0 =. g A =0for all areas and the MSE estator gven n (3.) reduces to A. When Statstcs Canada Catalogue No. -00-X

8 6 Molna Rao and Datta: Sall area estaton under a Fay-Herrot odel wth prelnary testng se = g 0 g 0 > g 0 = MSE =. 3 SYN Thus the MSE estator gven n (3.) can be serously overestatng the MSE for the overestaton we consder a odfed MSE estator of gven by 0 se = where g = g 0 = x X D X x =. g f A = 0 g 3 f A g A g A A > 0 A =0. To reduce In fact for A close to zero t ay happen that g s closer to the true MSE than the full MSE estator se but the queston of when s A close enough to zero arses. Ths queston otvates the use of a prelnary testng procedure of A = 0to defne alternatve MSE estators of the EBLUP n Secton 4. (3.3) 4 Prelnary test estators The estator of A used n the EBLUP of ntroduces uncertanty whch ght not be neglgble for sall. Indeed the ter g 3 n the MSE estator (3.) arses due to the estaton of A. However when the value of A s sall enough relatve to the saplng varances ths uncertanty could be avoded by usng the regresson-synthetc estator xβ 0 nstead of the EBLUP. Datta et al. (0) proposed a sall area estator based on a prelnary testng procedure of H : A = 0 aganst H : A > 0. When 0 H 0 s not rejected the regresson-synthetc estator s taken as the estator of ; otherwse the usual EBLUP s used. They proposed the test statstc T = y Xβ D y Xβ where β = XD X XD y s the WLS estator of β obtaned assung that H : = 0 0 A s true. The test statstc T s dstrbuted as X p wth p degrees of freedo under H. 0 Then for a specfed sgnfcance level the E of θ defned by Datta et al. (0) s gven by where X p f ; = Xβ T X p θ = θ f T > X p s the upper - pont of odest nuber of sall areas say 5. X. p The E s especally desgned to handle cases wth a Here we propose to use the procedure for the estaton of MSE of the EBLUP by consderng only the MSE of the synthetc estator g whenever the null hypothess s not rejected and the full MSE estate otherwse. But observe that the test statstc T n the procedure does not depend on the estator of A. Ths eans that even when H 0 s rejected t ay happen that A =0. Thus here we defne the estator of the MSE of the EBLUP as Statstcs Canada Catalogue No. -00-X

9 Survey Methodology June 05 7 f or = 0 g T X p A se = 3 f > g A g A g A T X p and A > 0. (4.) 5 Adjusted axu lkelhood The estaton ethods for A descrbed n Secton ght produce zero estates. In ths case the EBLUPs wll gve zero weght to the drect estators n all areas regardless of the effcency of the drect estator n each area. On the other hand survey saplng practtoners often prefer to gve always a strctly postve weght to drect estators because they are based on the area-specfc unt level data for the varable of nterest wthout the assupton of any regresson odel. For ths stuaton L and Lahr (00) proposed the estator that delvers a strctly postve estator of A. Ths estator denoted here A s obtaned by axzng the adjusted lkelhood defned as L A = A LP A. The EBLUP gven n (.6) wth = θ =. Note that θ assgns strctly postve weghts to drect estators. L and Lahr (00) proposed a second order unbased MSE estator of gven by A A wll be denoted hereafter as g A g A g A se 3 B A b A (5.) where b A s the bas of  and t s gven by b A trace P A Σ A A =. trace Σ A 6 Cobned estators The strctly postve estator of A has typcally a larger bas than ML or ML estators for A sall relatve to the D s. Thus f we stll wsh to obtan a sall area estator that attaches a strctly postve weght to the drect estator to reduce the entoned bas t wll be better to use the estator only when strctly necessary; that s ether when data does not provde enough evdence aganst A =0 or when the resultng ML estator of A s zero. Ths secton ntroduces two sall area estators of θ that gve a strctly postve weght to the drect estator whch are obtaned as a cobnaton of the EBLUP based on the ethod and the EBLUP based on ML estaton. In the frst cobned proposal the ethod s used to estate A when the prelnary test does not reject the null hypothess and n the second cobned proposal when the ML estate s non postve. Specfcally the frst cobned estator called hereafter - s defned by Statstcs Canada Catalogue No. -00-X

10 8 Molna Rao and Datta: Sall area estaton under a Fay-Herrot odel wth prelnary testng f or = 0 θ T X p A θ = f T > X p and A θ > 0. (6.) The second cobned estator called ML- s gven by θ f A = 0 θ = θ A f > 0 (6.) see Rubn-Bleuer and Yu (03). For the estaton of MSE of Usng se because MSE estator se = se f A = 0 se f A > 0. θ these authors proposed when A =0leads to substantal overestaton f the true value of A s sall wll be closer to the regresson-synthetc estator. Hence we propose the alternatve Agan snce for sall se also the followng estator se = 0 g f A = 0 se f A > 0. A ght stll be overestatng the true MSE of (6.3) (6.4) we consder f or = 0 g T X p A se = se f > T X p and A > 0. (6.5) 7 Sulaton experents A sulaton study was desgned wth the followng purposes n nd: (a) To study the propertes n ters of bas and MSE of the estators as vares for fxed A and as A vares for fxed. We would lke to see whch values of are adequate for a gven A. (b) To copare the Es wth the EBLUPs based on ML and wth the EBLUPs based on. (c) To study the perforance of the proposed MSE estators n ters of relatve bas and also n ters of coverage and length of predcton ntervals. (d) To copare the three ntroduced sall area estators that gve strctly postve weght to the drect estator for all areas naely EBLUP based on - and ML- estators. To accoplsh the above goals data were generated fro the Fay-Herrot odel gven by (.)-(.) wth a constant ean that s wth p = β = and x = =. We let =0wthout loss of Statstcs Canada Catalogue No. -00-X

11 Survey Methodology June 05 9 generalty nuber of areas = 5 and D = =. The sulaton study was repeated for ncreasng values of the odel varance A and also for sx sgnfcance levels of the test of H : A = 0 aganst H : A > 0 naely 0 0 = For each cobnaton of A and the followng steps were perfored for each sulaton run = L wth L = 0000 runs:. Generate data fro the assued odel wth constant zero ean;.e. nd v v N 0 A nd y e e N 0 D =.. Calculate the followng estators of θ : the EBLUP based on ML estaton of A θ the estate θ the EBLUP based on estaton of A θ the cobned - estate θ and the ML- estate θ. 3. For each area = calculate: the three estates of the MSE of the EBLUP gven n (3.) (3.3) and (4.) denoted respectvely by se se and 0 se and the three estates (6.3) (6.4) and (6.5) of the MSE of the cobned sall area estator respectvely. denoted se se and 0 se 4. For each area = obtan the noralty-based predcton ntervals for the sall area ean based on the three consdered MSE estators of the EBLUP: CI Z se CI Z se 0 0 CI Z se where Z s the upper - pont of a standard noral dstrbuton. 5. Repeat Steps -4 for = L for L =0000. Then for each sall area estator = copute ts eprcal bas and MSE as B = MSE =. L L L = L = Then obtan the average over areas of absolute bases and MSEs as AB θ = B AMSE = MSE θ. = = Statstcs Canada Catalogue No. -00-X

12 0 Molna Rao and Datta: Sall area estaton under a Fay-Herrot odel wth prelnary testng 6. Calculate the relatve bas of each MSE estator se as follows L RB se = se MSE MSE. L = Calculate the average over areas of the absolute relatve bases as ARBse θ = RB se. = 7. For each type of predcton nterval CI = L U for CI CI CI CI 0 n Step 4 calculate the eprcal coverage rate (CR) and the average length (AL) as gven L # CI U L L L = CR(CI ) = AL CI =. Fnally average over areas the coverage rates and average lengths as CR CI = CR CI AL CI = AL CI. = = Fgures 7. and 7. plot the average MSEs of the Es for each A together wth the average MSE of the EBLUPs based on ML and aganst the sgnfcance level. Note that when A s sall for large the procedure s rejectng H 0 ore often and therefore the E becoes ore often the usual EBLUP whereas for sall the procedure rejects H 0 less often and the regresson-synthetc estator s then ore often used. In contrast for a large value of A the E becoes the EBLUP ore frequently regardless of. The absolute bases of the estators are not shown here because they are roughly the sae for all the Es across values. The reason for ths s that when the odel holds both coponents of the E the synthetc estator and the EBLUP are unbased for the target paraeter. Note that the synthetc estator s unbased even when A >0. The frst concluson arsng fro Fgures 7. and 7. s that the MSE of the E s practcally constant across 0.. See also that the average MSE of the E for a gven ncreases wth A because the E reduces to the EBLUP ore often as A ncreases and the MSE of the EBLUP ncreases wth A. Observe also that the E and the EBLUP based on ML perfor very slarly for 0.. However for <0. the E becoes ore effcent than the EBLUP as soon as A oves close to the null hypothess A <0. whch agrees wth the reark of Datta et al. (0). Turnng to the EBLUP based on Fgures 7. and 7. show that ts average MSE s sgnfcantly larger than that of the other two estators but the dfferences wth the other ones decrease as A ncreases. Ths s due to bas of the estator of A for sall A. We shall study later the cobned sall area estators - and ML- whch use the EBLUP based on only when null hypothess s not rejected or when the realzed estate of A s zero. Statstcs Canada Catalogue No. -00-X

13 Survey Methodology June 05 a) A = 0.05 Average MSE Average MSE ML alpha b) A = 0. ML alpha Fgure 7. Average MSEs of E EBLUP based on ML and EBLUP based on aganst for a) A =0.05and b) A =0.. Datta et al. (0 page 366) recoended 0. for the E. Moreover the lterature on estaton for fxed effects odels suggests that a good choce of n ters of bas and MSE s =0.(Bancroft 944; Han and Bancroft 968). But the above results suggest that for 0. the E s practcally the sae as the EBLUP and therefore one ght choose to always use the EBLUP. Statstcs Canada Catalogue No. -00-X

14 Molna Rao and Datta: Sall area estaton under a Fay-Herrot odel wth prelnary testng A = 0. Average MSE ML Fgure 7. Average MSEs of E EBLUP based on ML and EBLUP based on aganst for A =0.. Now we study the propertes of the for MSE estaton n ters of. Fgure 7.3 plots the average absolute relatve bas of the MSE estators se labelled aganst the sgnfcance level for each value A When s taken very sall <0. the null hypothess H : = 0 0 A s less often rejected and se becoes often g whch leads to underestaton. For large >0. the null hypothess s ore often rejected and se becoes the usual MSE estator of the EBLUP whch severely overestates the true MSE for sall A. The value =0.appears to be a good coprose choce wth an average absolute relatve bas around 0% for A 0. and 0% for A = alpha Average abs. rel. bas of MSE estator A = 0.05 A = 0. A = 0. A = Fgure 7.3 Average over areas of absolute relatve bases of the MSE estator alpha for A aganst sgnfcance level. se labelled Statstcs Canada Catalogue No. -00-X

15 Survey Methodology June 05 3 The above results suggest that = 0.s a good choce when usng the procedure to estate the MSE of the usual EBLUP. Ths has been ore thoroughly studed by lookng at the (sgned) relatve se for each area. These results are plotted n Fgures 7.4 and 7.5 wth four plots one bases of for each value of A The fgures appearng n the legends of these plots are the se. These plots confr our prevous sgnfcance levels for the MSE estator observatons: the MSE estator based on the se underestates MSE and overestates for large. It turns out that se wth =0. values of A. a) A = 0.05 for sall s a good canddate for all Relatve bas of MSE estator Relatve bas of MSE estator area b) A = Fgure 7.4 Relatve bases of se for each sgnfcance level area aganst area for a) A =0.05and b) A =0.. Statstcs Canada Catalogue No. -00-X

16 4 Molna Rao and Datta: Sall area estaton under a Fay-Herrot odel wth prelnary testng a) A = Relatve bas of MSE estator Relatve bas of MSE estator area b) A = Fgure 7.5 Relatve bases of se for each sgnfcance level aganst area for a) A =0.and b) A =. Let us now copare se MSE estators se and se 0 for the selected sgnfcance level =0. wth the other two gven by (3.3) and (3.) respectvely. Fgure 7.6 plots the average absolute relatve bases of the three MSE estators labelled respectvely ML0 and se se se ML. We note that 0 perfors better than for all areas but stll s better than se for all consdered values of A except for A = where the dfferences 0 between the three estators are neglgble. The dfferences decrease as A ncreases but observe that the se can be severely based for sall A wth an average absolute relatve usual MSE estator area Statstcs Canada Catalogue No. -00-X

17 Survey Methodology June 05 5 bas over 50% for A <0.and exponentally growng as A tends to zero. The concluson s that when H s not rejected even f the realzed estate of A s postve t sees better to ot the g ter n 0 the MSE estator and consder only g. 3 Average absolute relatve bas of MSE estator ML0 ML Fgure 7.6 Average over areas of absolute relatve bases of MSE estators se labelled se labelled ML and se wth =0. labelled ML0 aganst A. 0 We now turn to the sall area estators that attach strctly postve weght to the drect estator for all areas: EBLUP based on θ and the two cobned estators - gven n (6.) and ML- gven n (6.). Average MSEs are plotted n Fgure 7.7 for these three estators. In ths plot θ sees to be a lttle less effcent followed by -. The cobned estator ML- sees to perfor slghtly better than ts two counterparts for sall A although for A 0. the - estator s very close to t. For MSE estaton we focus on ML- because of ts better perforance. A Average MSE ML Fgure 7.7 Average over areas of MSEs of - estator wth =0. EBLUP based on and ML- estator aganst A. A Statstcs Canada Catalogue No. -00-X

18 6 Molna Rao and Datta: Sall area estaton under a Fay-Herrot odel wth prelnary testng For the cobned estator ML- Fgure 7.8 shows that the MSE estator based on the se whch uses only g whenever A =0 or the null hypothess s not rejected has average absolute relatve bas less than 0% for A 0. and t s saller than the correspondng values se se especally for A 0.4. for and 0 Average abs. relatve bas of MSE estator ML- ML Fgure 7.8 Average over areas of absolute relatve bases of the MSE estators se and se 0 aganst A. A se labelled respectvely ML- ML-0 and Fnally we analyze the average over areas of coverage rates and average lengths of noralty-based predcton ntervals for the sall area ean usng the EBLUP based on ML as pont estate and the three dfferent MSE estators of the EBLUP naely se 0 se and se. Fgure 7.9 shows the coverage rates of these three types of ntervals where the MSE estators based on the procedure were obtaned takng = It sees that the good relatve bas propertes of the MSE estator based on the se for sall A cannot be extrapolated to coverage based on noral predcton ntervals showng undercoverage especally for A =0.. In ths case takng a larger sgnfcance level = 0.3reduces a lttle the undercoverage of the predcton ntervals obtaned usng se. Stll the coverage rates of se 0 are better for all values of A. As expected the usual MSE estator se provdes overcoverage for sall values of A whch s due to the severe overestaton of the MSE. On the other hand the ntervals showng undercoverage also lead to shorter predcton ntervals as shown by Fgure 7.0. It s worthwhle to enton that the constructon of predcton ntervals for based on the Fay- Herrot odel wth accurate coverage rates s not an obvous task. Several papers have appeared n the lterature for ths proble. For exaple Chatterjee Lahr and L (008) proposed predcton ntervals wth second order correct coverage rate usng only the g ter as MSE estate and applyng a bootstrap procedure to fnd the calbrated quantles. Dao Sth Datta Mat and Opsoer (04) have recently Statstcs Canada Catalogue No. -00-X

19 Survey Methodology June 05 7 obtaned predcton ntervals wth second order correct coverage rate avodng the use of resaplng procedures and usng the full MSE estator. Obtanng predcton ntervals wth accurate coverage usng other MSE estates s stll a challenge and t s out of scope of ths paper. Coverage Noral Confdence Interval ML alpha = 0.3 ML0 alpha = Fgure 7.9 Average over areas of coverage rates of noralty-based predcton ntervals for usng the MSE estators se 0 se and se wth =0.0.3 labelled respectvely ML ML0 and aganst A. A Average Length Noral Confdence Interval ML alpha = 0.3 ML0 alpha = Fgure 7.0 Average over areas of average lengths of noralty-based ntervals for usng the MSE estators se 0 se and se wth =0.0.3 labelled respectvely ML ML0 and aganst A. A Statstcs Canada Catalogue No. -00-X

20 8 Molna Rao and Datta: Sall area estaton under a Fay-Herrot odel wth prelnary testng Ths sulaton study descrbed above was repeated for several patterns of unequal saplng varances D Although results are not reported here conclusons are very slar as long as the varance pattern s. not extreely uneven. 8 Conclusons The followng ajor conclusons ay be drawn fro the results of our sulaton study on the estaton of sall area eans based on the Fay-Herrot area-level odel when the nuber of areas s odest n sze say = 5 : ) Under the Fay-Herrot odel wth a value of rando effects varance A clearly away fro zero the E does not see to notceably prove effcency relatve to the usual EBLUP unless the sgnfcance level s taken sall ( 0. n our sulaton study). ) Our sulaton results ndcate that usng the procedure wth a oderate n partcular =0. to estate the MSE of the usual EBLUP leads to a reducton n bas as copared wth the usual MSE estator. Hence we recoend the use of se gven by (4.) to estate the MSE of the EBLUP. 3) Aong the estators that attach a strctly postve weght to the drect estator for all areas we recoend the cobned estator ML- gven by (6.) because t acheves slghtly hgher effcency than the EBLUP based on and the - gven by (6.). 4) For estatng the MSE of the recoended ML- estator the estator se gven by (6.5) perfors better than the alternatve ones. 5) Our results on predcton ntervals based on noral theory ndcate that the good perforance of the proposed MSE estators ay not translate to coverage propertes of these ntervals. Constructon of predcton ntervals that lead to accurate coverages usng the proposed MSE estates appears to be a dffcult task. Sooth alternatves to the prelnary test estates n the case of locaton paraeters have been proposed n the lterature usng weghted eans of the estates obtaned under the null and alternatve hypotheses wth weghts dependng on the test statstc see e.g. Saleh (006). Mean squared error estates of ths knd have not been studed and we leave ths subject for further research. Acknowledgeents We would lke to thank the edtor for very constructve suggestons. Gaur S. Datta s research was partally supported through the grant H fro the Natonal Securty Agency Isabel Molna s research by grants ref. MTM MTM C0-0 and SEJ fro the Spansh Mnstero de Educacón y Cenca and J.N.K. Rao s research by the Natural Scences and Engneerng Research Councl of Canada. References Bancroft T.A. (944). On bases n estaton due to the use of prelnary tests of sgnfcance. The Annals of Matheatcal Statstcs Statstcs Canada Catalogue No. -00-X

21 Survey Methodology June 05 9 Chatterjee S. Lahr P. and L H. (008). Paraetrc bootstrap approxaton to the dstrbuton of EBLUP and related predcton ntervals n lnear xed odels. The Annals of Statstcs Das K. Jang J. and Rao J.N.K. (004). Mean squared error of eprcal predctor. The Annals of Statstcs Datta G.S. and Lahr P. (000). A unfed easure of uncertanty of estated best lnear unbased predctors n sall area estaton probles. Statstca Snca Datta S. Hall P. and Mandal A. (0). Model selecton by testng for the presence of sall-area effects and applcaton to area-level data. Journal of the Aercan Statstcal Assocaton Dao L. Sth D.D. Datta G.S. Mat T. and Opsoer J.D. (04). Accurate confdence nterval estaton of sall area paraeters under the Fay-Herrot odel. Scandnavan Journal of Statstcs to appear. Fay R.E. and Herrot R.A. (979). Estaton of ncoe fro sall places: An applcaton of Jaes- Sten procedures to census data. Journal of the Aercan Statstcal Assocaton Han C.-P. and Bancroft T.A. (968). On poolng eans when varance s unknown. Journal of the Aercan Statstcal Assocaton L H. and Lahr P. (00). An adjusted axu lkelhood ethod for solvng sall area estaton probles. Journal of Multvarate Analyss Prasad N.G.N. and Rao J.N.K. (990). The estaton of the ean squared error of sall-area estators. Journal of the Aercan Statstcal Assocaton Rao J.N.K. (003). Sall Area Estaton. Hoboken NJ: Wley. Rubn-Bleuer S. and Yu Y. (03). A postve varance estator for the Fay-Herrot sall area odel. SRID--00E Statstcs Canada. Saleh A.K. Md. E. (006). Theory of Prelnary Test and Sten-type Estaton wth Applcatons. New York: John Wley & Sons Inc. Statstcs Canada Catalogue No. -00-X