Spline Regression in the Estimation of the Finite Population Total

Transcription

1 Scece Joral of Ale Matematc a Stattc 05; 3(5): 4-4 Pble ole Agt 3, 05 (tt: o: am ISSN: (Prt); ISSN: (Ole) Sle egreo te tmato of te te Polato otal Joe ego Cerot Deartmet of Comter a Stattc, Mo Uvert, loret, ea mal are: cerot77@aoo.com o cte t artcle: Joe ego Cerot. Sle egreo te tmato of te te Polato otal. Scece Joral of Ale Matematc a Stattc. Vol. 3, No. 5, 05, o: am Abtract: t ogt to etmate fte olato total g Sle regreo fcto. It comare te Sle regreo wt Samle Mea etmator, eg-bae a moel - bae etmator. o meare te erformace of eac etmator, te t coere average ba, te effcec b e of te mea qare error a te robte g te rate cage of effcec. I t reearc, fve olato were e. ree of tem were mlate accorg to te followg moel: lear omoceatc, qaratc omoceatc a lear eteroceatc a two atral olato. e erformace of te fve etmator were te er te fve olato. e fo tat Samle Mea(SM), Horvt- omo (H) a ato () etmator are ot robt wle Naaraa-Wato(NW) a Peroc Sle(PS) are robt we leart a omoceatct of te olato trctre are volate. ewor: Homoceatct, Polato, Samle, Sle egreo, obte, Smootg, tmator. Itrocto.. Itrocto ere are two geerall accete oto tg te caractertc of fte olato. e frt oto a t wc ever t of te olato eame calle a ce. Ue of a ce to t a olato tme comg, eeve, ofte moble a tragel eog, ofte accrate. e oter oto to t te caractertc of a olato b eamg a art of t. e teor of rve amlg a eveloe rg te at everal ecae rove wt varo of reaoable cetfc tool for rawg amle a mag val ferece abot te olato arameter of teret.... Ce Ver Samlg Meto Altog tere are avatage wt te ce meto, te cot, effort a te tme reqre to coct ce ma be eormo le te olato ver mall. I c a cae we reort to amlg tat volve eamato of a art of te olato. Altog a ce oerato gve a more relable ata, amlg more arorate we:. e cot of coctg ce wol be robtve.. e olato large, c tat t wol be moble to coct a ce.. e t volve etrcto of elemetar t er t, c tat t wol be arorate to coct amle tetg. v. Qc relt are reqre, c tat t wol be arorate to coct amle rve rater ta carrg ot a comlete cot.... Bac Iea of Samlg a tmato I te bac amlg et, te olato cot of a ow fte mber N of t c a eole or lot. Wt eac t aocate a vale of a varable of teret, ometme referre to a te -vale of tat t. e -vale of eac t te olato ow qatt. However, te t te olato are etfable a ma be labele wt mber,,. N. A amle of te t te olato electe a oberve. e ata collecte cot of te -vale for eac t te amle togeter wt te t label. e rocere b wc te amle t electe from te olato calle te amlg eg. Wt mot of te well- ow amlg eg, te eg eterme b agg to eac oble amle te robablt () of electg tat amle. or eamle, g te mle raom amlg eg, te t are electe wt eqal a eeet robablt ().

2 Scece Joral of Ale Matematc a Stattc 05; 3(5): tmato Aroace o etmate fte olato total ( ) rve, were, We ee to ave (,, 3,., N) te rve varable a (,, 3,., N) eg varable ( Alar varable). e followg terefore a lt of aroace tat are coere t t te etmato of fte olato total.... Deg - Bae Aroac alo ow a clacal aroac. I t aroac, te varable of teret of te target olato are vewe a fe qatte. Alo te eg troce electo robablte tat eterme te roerte of etmator tat are e to obta eecte vale, varace, bae etc. e amle are geerate b amlg eg () wt te vale,,...,,,...,, el fe. e reetto of amle rawg rocere form te ba of raomato framewor. e aroac ame tat moel ave o relevace to te feretal framewor. I eermetal eg, raomato emloe to rotect te eermeter agat bectve bae. Scott a Smt (975) etee relt of Blacwell. Accorg to er, raomato wa relevat before te ata were collecte bt ot te aal of ata wc agreemet wt mot tattca te eermetal cece. aomato terefore a race agat electo ba.... Moel Bae (Precto) Aroac rom oall (976) te cocet of te er olato troce t: e fte olato ol telf be regare a a raom amle from ome fte olato. Hece fte olato ame to be geerate a a raom amle from a er olato. Alo ote tat varable of teret are vewe a raom varable a roerte of etmator ee o te ot trbto of tee raom varable. A amle electe from te fte olato g a ow amlg ceme. e obervato are mae o te amle vale a are te e to mae recto abot te o amle vale. I t cae te moel coect a varable of teret wt a et of alar varable X, Co (995). However, ote tat te coce of a moel a t robte to mecfcato te maor e. Small evato from a coe moel ma lea to ero error a ferece. Sometme te moel become matematcall comle wle tll ot beg tabl realtc (omo, 99). or eamle, were moel amto of te varable beg te tat of eeece t gore te teec ma olato for earb or relate t to be correlate...3. No-Parametrc Aroac e arametrc meto of etmato e we t ame tat te ata raw or geerate from oe of te ow arametrc faml of trbto. I ma cae owever, te eermeter oe ot ow te form of te bac trbto a ee tattcal tecqe wc are alcable regarle of te form of te trbto. ee tecqe are referre to a o arametrc or trbto free meto. e al to ver we famle of trbto rater ta ol to famle ecfe b a artclar fctoal form. e o ot reqre te varo amto abot te trbto of olato from wc te amle wa obtae. e ma ea be t cla of moel tat te effect of a elaator (eg) varable a eeet varable of teret ot moele a arametrc, all lear fcto bt et fleble. e ol amto eee tat te effect of te elaator varable are moele a moot.e. fferetable fcto. e fctoal ae te to be etmate from te ata b eter g: erel bae meto or Sle bae meto. erel Bae Meto. e erel etmator eree term of a erel fcto wc atfe te coto; 0 < Uall, bt ot alwa, wll be a mmetrc robablt et fcto, te ormal et for tace. erefore, accorg to Slverma (986) te erel etmator of te et fcto wt erel efe b, () f () were te bawt. It clearl oberve tat te erel etmator a m of bm lace at te obervato. ac val bm create b { ( ) } a te etmate f a reltat m obtae b ag tem. Sle Bae Meto. e ame, Sle fcto wa gve b I.J Scoeberg (946) to te ecewe olomal fcto ow a varate olomal Sle. wa becae of ter reemblace to te crve obtae b ter raftme g a mecacal Sle a t fleble ro wt a groove a a et of wegt calle c e to oto te ro at ot trog wc t wa erve to raw moot terolato crve ag trog recrbe ot. e bac ea ate bac at leat to Wttaer (93). More reet aer o te bect cle Waba (975), Smt (979), a Slverma (985) amog oter. or erel regreo etmato a wegtg ceme e to Naaraa (964) Wato (964) a bee aocate wt raom eg, a a covolto te wegtg ceme wt fe eg bae o mea qare error; oe of te etmator forml otmal eter

3 6 Joe ego Cerot: Sle egreo te tmato of te te Polato otal eg. e mltte of o arametrc regreo etmator a e of coerable ractcal a teoretcal mortace. A we cla of etmator te b Jee Stemet a Gaer (988) cle fe wt erel etmator, mootg le a earet egbor etmator a artclar cae. No etmator forml bet term of tegrate mea qare error, bt te erel etmator tr ot to be te mma otmal. Sce o arametrc meto are all tee to be alcable to a broa varet of tato te mma roert a mortat afegar. wo efto of erel wegt eo artclar olart, te Naaraa Wato te (Naaraa 964, Wato 964) a te covolto te etmator (Pretl a Cao97, Gaer a Mller 979). e Naaraa-Wato meto ttvel motvate a a etmator of a cotoal eectato wc gget a cotet were te eeet varable raom. Hece t meto eem te for a tato of raoml electe eg ot, woe trbto eterme b te eg et. A le fcto a ecewe efe fcto wt certa moote coto. e mot commol e form te cbc le. ere are two ort of le; orar le a B-le. e two le fcto ave te ame geeral trctre regarg te ecewe efe fcto c a f 3 ( ) a a a a (3), 0,,,3 a te mootg coto. e fferece tat te orar le go trog all te ata ot eactl were a B -le o ot ecearl ft te ata eactl. or orar le, te crve a to go trog all te ot ece te eqato all te ot (, ) f a to be atfe for. e le fcto a to el te vale for. e mootg coto too ave to be flflle. B-le are ecewe efe fcto all olomal wt te ame moote coto a orar le. e are owever ot force trog te ata ot eactl, te fcto a ml to come cloe to te ata ot. I etmato of fte olato total, te callege to etf a etmator tat effcet we te olato trctre ot ow. I t t, tr to comare te le regreo wt te ow etmator of oarametrc (Naaraa-Wato), Samle Mea etmator, Deg-bae Horvt-omo etmator a Moelbae ato etmator. e callege to obta a etmator wc robt to te volato of bot leart a omoceatct of te olato trctre.. Metoolog.. No-Parametrc tmato of te Polato otal Ug erel coere. It ame tat te alar formato avalable for te etre olato a te alar varable X a te t varable are relate a more geeral wa. Coer te moel m ( ) ε I (4) were m( ) te mea fcto a ε a raom error term. It ame tat te fctoal form of m () ow bt ame to be moot a coto. Let w(),,,, be te wegt fcto ow a erel fcto. e erel a coto, boe a mmetrc fcto wc tegrate to oe. at () B tag () - to be te erel wt ba wt. e wegt eqece for te erel mooter a gve b Naaraa (964) - Wato (964) ( ) w() ( ) e Naaraa -Wato etmator of m() (3.) m () Sbtttg we ave m ( ) ( ) (5) w (6) e ae of te erel wegt eterme b, were a mmetrc robablt et fcto tat atfe coto eqato. Oe qe featre of te e of te bawt tat te maller t te more cocetrate are te wegt aro. However, te oarametrc regreo bae etmator for te olato total gve b were (7) m (8) m te Naaraa-Wato etmator gve (7). Hece b btttg (7) (8) Naaraa Wato etmator of te olato total become: w ( ) ( ) were rereet te Naaraa-Wato etmator of te w olato total. (9) I t ecto, te Naaraa-Wato erel etmator

4 Scece Joral of Ale Matematc a Stattc 05; 3(5): Proerte of Naaraa-Wato erel tmator of te Polato otal e Naaraa-Wato erel regreo etmator gve a (8) a (7) I orer to f a taar meare of etmato error, te Mea Sqare error (MS), te t looe at te cotoal mea a varace of er te moel ε m. m a So tat m m X were X te olato vector of X-vale. Bt w m were te taar Naaraa- Wato etmator of te et. Hece m X { } Sce m er te moel, we ave; X { } m m. Net, we loo at te cotoal error varace; Var X V Var Sce Var er te moel, we ave X V w were w..3. Sle egreo tmator of te Polato otal Waba (975) a ow tat erel mootg etmator ) ( m cloel relate to mootg Sle etmator we t rereete aromatel a a lear fcto of te ata vale. Hece tere et a wegt fcto (,) c tat () m (, ) were te fcto (,) efe a (,) f() (0) Hece we ave m f() Sbtttg ) ( m ) m( We get te mootg le etmator of te olato otal a. f() () were () efe a () 0.5 e(-.4)((.4) π4) a te fcto () a te followg roerte; < a ()

5 8 Joe ego Cerot: Sle egreo te tmato of te te Polato otal We ca ee tat te roerte of te fcto (.) above are mlar to toe gve for te erel fcto bt ca tae egatve vale a well. Hece te mootg le etmator correo aromatel to a erel te etmator of orer 4. ba (988) a ow tat f te fcto m(.) ame to be eroc te () m correo to a le etmator wt a fe bawt arameter a wegt (,) w() were 4 λ a w() λ λw te etmator correog to te eroc le, m were (.) a efe (0),ece gvg. f. m (3) Sce - (,) oe ot m to oe, we ve te wegt b ter m a we eote te mofe wegt b (, ), te (,) ) ( ) ( (4) terefore te m (, ) eroc le etmator of te fcto m() gve b; m (5) Sbtttg (5) (6) We ave te Peroc Sle tmator of te olato total a (6) ece m X X Let, e X { } { } m m Net, we coer te cotoal error varace; Var X V P X [ ] Var let W ece X V W. 3. mrcal elt a Dco o comare te erformace of te fve etmator, tat, Horvt omo, te ato etmator, Samle Mea etmator, te Naaraa - Wato erel etmator a eroc le etmator a le regreo etmator o a to etf a robt etmator, te t mlate tree olato bae o te followg moel; lear Homoceatc moel, Qaratc Homoceatc moel a Lear Heteroceatc moel. Alo te t e two real olato. e crtera for comarg tee etmator are average ba, mea qare error a te rate of cage of effcec a a meare of robte. 3.. e Coce of te erel a Bawt t e te Gaa erel Naaraa - Wato etmator of te olato total wc efe a < <, π e were. Ame tat te erel fcto atfe te coto gve eqato. A otmal bawt for Naaraa-Wato mooter wa coe wt te terval were te

6 Scece Joral of Ale Matematc a Stattc 05; 3(5): taar evato of ', ( Slverma, 986). erefore, te bawt e wa coe to be te cetre ot e erel fcto e te eroc le () 0.5 e(-.4)((.4) π4) (Waba, 975) 3.. Decrto of te St Polato a tmator e artfcal olato wa mlate te followg maer. a) I artfcal olato I, 76 ata ot were geerate accorg to te moel; α ( ε ) were ε N( 0, ), U[ 0, ] 4 α 0.5. a X gre. e olato lear wt omoceatc varace trctre.. b) I artfcal olato II, we aga geerate 76 ata ot accorg to te moel α ε. 4 c) I artfcal olato III, oce more 76 ata ot were geerate accorg to te moel, 4 are te ame a olato I α ε. Were α a ε b a c ) e eal olato IV, wa obta from te ea Natoal Brea of Stattc (NBS) for te olato ce oe ea 009. I t olato, coere te Alar varable X to be te mber of oeol te t Dtrct a t varable te total olato b Dtrct ecet for Narob rovce were Dvo are e tate of Dtrct, were,,., 76. Or varable of teret te olato total. e) Polato V, t olato a varable X ecrbg are a ctomer alrea oee (Acqre) ver are ale for (Booe) a toc ecage broerage farm, varable, bot eree. Aga electe 76 ata ot t olato. e average ba a Mea Sqare rror of te olato total were comte for eac of te followg fve etmator: Samle Mea, SM Horvt-omo, ato etmator H, Naaraa-Wato a eroc le w. Below a mmar of te formlae e comtg ter reectve olato total. e followg are catter agram owg te trbto of te fve olato metoe above. X gre. e olato qaratc wt omoceatc varace trctre X gre 3. Lear olato wt eteroceatc varace trctre

7 0 Joe ego Cerot: Sle egreo te tmato of te te Polato otal gre 4. ea olato ce- 009 ( mllo). olato aear to be lear wt eteroceatc varace trctre. N olato, we comte te tre olato total. Defe a te olato total etmator, were r r r SM,, H, NW, a PS. e were r r olato total etmate of te t amle a rt etmator wle te mber of amle relcate.hece te ba of eac etmator of olato total were comte a Ba r r te average ba for eac etmator for bot te real a artfcal olato total are Ba r 30 r olato( ) 30 were,, 3, 4, 5. We efe te mea qare error to be MS r Var r Ba r were Var r te cotoal varace of te etmator over te 30 relcate for te artfcal a atral olato. erefore, te Mea Sqare rror te etmato of bot te artfcal a atral olato gve b: MS 30 r r Ba r olato( ) 30 olato( ) gre 5. Stoc cage Sare ( mllo ). e acqre are a booe are t olato trctre aear to be correlate Decrto of te Comtato Procere or eac artfcal olato of e 76, amle of e 40 were geerate b mle raom amlg wtot relacemet a 30 relcate amle were electe a etmate comte. Smlarl, for te real olato of e 76, amle of eac e 40 were relcate 30 b SSWO a te etmator of te olato total comte. or te cae of Horvt-omo, te amle t are electe wt eqal robablte. o elect a amle wt eqal robablte wt Horvt-omo wegt, we aveπ, te robablt of te t beg cle te amle c tat π S a S were. Hece te etmate of te olato total obtae a H. or eac of te π e elatve Cage ffcec (C) for eac etmator wa gve b C MS Were,,3,4. r o MS 3.4. elt a Iterretato ( ) r MSro o e relt of t t are mmare able to 5. O eac olato te erformace of eac etmator aale g te average ba a mea qare error. e average ba a cato of te meare of ow cloe a etmator from te tre vale, wle te MS e to ae effcec of a etmator. or eamle a etmator wll be a to be more effcec ta aoter, f t MS comarabl maller.e f MS ( ) < MS ( ), were a are etmator, te a to be more effcet ta.

8 Scece Joral of Ale Matematc a Stattc 05; 3(5): 4-4 tmator Samle Mea(SM) able. Smmar of te formlae e comtg ter reectve olato total. ormla Horvt-omo(H) H H ato() Naaraa-Wato(NW) w w Peroc-Sle(PS) SM SM N π X ( ) ( ( )) ( ) able. Polato I (Lear a omoceatc). SM H ato NW PS tmate Ba Var MS Polato otal I olato I, ote tat from te low vale of te ba tat all te fve etmator erform well er tee coto. However, Naaraa-Wato a te leat ba followe b SM, Horvt-omo, Peroc le a ato etmator tat orer. Loog at MS of t olato, SM etmator a te lowet MS, followe b Naaraa-Wato, eroc le, a ato. H- etmator a te get MS t olato. However, te vale of te MS of tee etmator o t olato are lowet a comare to toe obtae te oter olato. mle tat tee etmator ave g effcec lear a omoceatc olato trctre. og te amle mea wt te leat MS te mot effcet t olato. able 3. Polato II (Qaratc a omoceatc). SM H ato NW PS tmate Ba Var MS Polato otal I olato II, ote tat SM a te leat abolte ba followe b Naaraa-Wato, eroc le, Horvt-omo, a latl te ato etmator. Net, loog at MS, te Naaraa-Wato a eroc le bot ave low MS followe b SM, H- a latl ato etmator. Here we ote tat te Naaraa-Wato te bet etmator for a qaratc a omoceatc olato wle ato etmator a te get MS t mag t te leat effcet etmator for t olato. tre becae te rato etmator bae o te amto of leart wc we volate te etmator a eecte brea ow. able 4. Polato III (Lear a eteroceatc). SM H ato NW PS tmate Ba Var MS Polato otal I olato III, ote tat Naaraa-Wato a te leat abolte ba, followe b SM, Horvt-omo, Peroc le a latl ato etmator. Coerg te MS, Naaraa-Wato a SM ave a low MS followe b eroc le, ato a te H- etmator tat orer. Naaraa Wato a SM become te bet etmator of t olato wc lear a eteroceatc olato. able 5. Polato IV (ea olato ce-999). SM H ato NW PS tmate Ba Var MS Polato otal I olato IV, ote tat Samle Mea a te leat ba followe b Naaraa-Wato, H-, ato a latl eroc le. Loog at MS, SM a te leat MS, followe b Naaraa-Wato, Peroc le, ato a H- etmator., SM a rove to be te bet etmator for t real olato wc aear to be lear a wt eteroceatc varace from te catter agram.

9 Joe ego Cerot: Sle egreo te tmato of te te Polato otal able 6. Polato V (toc ecage are). OLS H ato NW PS tmate Ba Var MS Polato otal olato aear to be eter lear or omoceatc from te catter agram gre 5. I t olato, Peroc le a te leat abolte ba, et Naaraa-Wato, ato, SM, a latl Horvt-omo etmator. A cocer te MS, Peroc le a te leat MS t rovg to be te bet etmator for t olato woe trctre ot ow. It followe b Naaraa-Wato, SM, H- a ato etmator. able 7. Mea Sqare rror. SM H ato NW PS POP I POP II POP III POP IV POP V able 8. elatve Cage ffcec (C). SM H ato NW PS C I C II C III C IV all, te t comare te relatve Cage ffcec (C) amog te fve etmator. rt, wa te cae we leart amto of te olato trctre volate. Coerg te C I, able 7 tat te oarametrc etmator, Naaraa-Wato a Peroc Sle ave low C. ml tat te are te leat etve to te volato of te leart trctre of te olato a ece te mot obt amog te fve etmator. e are te followe b te SM, a ato etmator. Nevertele, Horvt-omo etmator te leat obt amog tem a far a te volato of leart amto of te olato trctre cocere. Secol C II, vetgate te volato of te Homoceatc amto a olato trctre. Coerg te C II, able 7 tat te Naaraa-Wato, Peroc Sle a SM ave te lowet C. ml tat te are te leat etve to te cage of trctre of te olato a ece te mot robt amog te fve we omoceatc amto volate. O te oter a te ato a Horvt-omo are leat robt to te volato of omoceatc coto o te olato trctre. Net we coer C. SM avg te leat vale. Net o te lt Naaraa-Wato, Peroc le, ato a Horvt-omo etmator. However, we ave alo ote tat all vale of C are qte low. mle tat tog SM te mot robt to te cage te olato trctre, te low vale ow tat te oter etmator are alo robt a we cocle tat olato I almot mlar trctre to olato IV, tog t eem tat omoceatc coto volate. Latl, C IV, e Peroc Sle etmator a te leat vale of C t becomg te mot robt etmator to te cage of olato trctre from lear a omoceatc to te trctre wc o lear a o omoceatc. Naaraa-Wato alo rove to be robt to te ame cage te trctre. However, ato a Horvt-omo etmator rove to be gl etve to te cage te olato trctre. ee two etmator are terefore le robt a comare to te oter two o arametrc etmator. e leat robt etmator o t lt a fr a t olato cocer SM. erefore, Peroc le a rove to be robt we bot leart a omoceatc coto are volate. 4. Coclo a ecommeato 4.. Coclo t a reveale tat te le regreo etmator erforme mrevel well all aect coere: ba, effcec a robte. We ote tat t erforme well lear omoceatc moel a qaratc omoceatc moel. However, eve we te omoceatct amto wa volate t tll erforme well. We terefore cocle tat Peroc Sle etmator a robt etmator. It terefore recommee to be e a a table etmator of te olato total we te trctre of te olato ow. It a alo bee ote tat te Naaraa-Wato etmator erform well te lear omoceatc moel a alo we te leart coto volate. It alo ffce to meto tat t erformace wa qetoabl mreve te lear eteroceatc moel clearl catg tat t robt to te volato of leart a omoceatc coto. 4.. ecommeato. rom te fg of or reearc, te Horvt- omo (eg-bae) etmator a te ato etmator (moel-bae) ol be e wt te cofe of a lear omoceatc moel. e are ot arorate for e we te trctre of te olato ot ow.. e two etmator, Naaraa-Wato a eroc le etmator; are table for e lear omoceatc moel a eve we te amto of te moel are volate etvt to te cage of olato trctre relatvel low a ece are clafe a gl robt. Notato. N e of te fte olato geerall ame to be ow.. amle e. 3. eg varable. It vale ca eter be mae

10 Scece Joral of Ale Matematc a Stattc 05; 3(5): avalable before a or te core of ata collecto. 4. te rve varable or varable er t amle mea. 6. amle varace ( ) N 7. σ (gma) olato varace ( ) N 8. N N I N te fte olato mea. 9. te olato total. 0. Srwor Abbrevato of mle raom amlg wtot relacemet.. ea llg.. SMSamle Mea 3. H Horvt-omo 4. ato 5. NW Naaraa-Wato 6. PS Peroc-Sle Acowlegemet rt a foremot, I ta Almgt Go for H abat grace, fatfle, care, owlege a te tregt trogot t ero of t. Secol, I cerel ta m ervor, Dr. Nega, for rofeo gace, atece, avalablt for coltato a rovo of te referece materal tat I eee te core of t reearc roect. M gratte alo goe to te oter team of m lectrer te Deartmet of Matematc (tattc), eatta Uvert for ter ort a ver ecatve lectre. Secal ta alo go to J.P. Waala of NBS wo cloel ore wt er ttto bae rogram for ecoragemet. Latl, I am gratefl to Mr. el Wae of eatta Uvert for valable teccal atace. eferece [] Cael, C.M., Saral, C.. a Wretma, J.H. (977). oato of ferece rve amlg. New or: Wle. [] Camber,.L (003) wc amle rve trateg? A revew of tree fferet aroace. Sotamto Stattcal Scece eearc Ittte. Uvert of Sotamto [3] Cocra, W.G. (940). e etmato of te el of cereal eermet b amlg for te rato of gra to total roce. Joral of Agrcltral Scece, [4] Cocra, W.G. (977). Samlg ecqe, 3 r eto. New or: Wle. [5] Co, B. G (995) Be rve meto. New or: Jo Wle [6] Gaer,. a Mller, H.G. (979). erel etmato of regreo fcto. I mootg tecqe for crve etmato. Heelberg: Srger verlag [7] Hbbac, J.A (97). Samlg for rce el Bar a Ora. Imeral Agrcltral eearc Ittte, Pa, Bllet No. 66 (rereete Saa, 946, 7, [8] aer, A.N (897). e rereetatve meto of tattcal rve. (976 gl tralato of te orgal Norwega), Olo. Cetral Brea of Stattc Norwa. [9] m, J.-. (004). Noarametrc egreo tmato Srve Samlg. P.D. te, Iowa State Uvert [0] Lalace, P. S. (80). A loocal ea o robablte. gl tralato, Dover, 95 [] Moaa,P.(005). eor a meto of Srve Samlg, Pretce-Hall of Ia Prvate Lmte, New Del [] Naaraa,.A. (964). O tmatg egreo. eor of Probablt Alcato 9, 4-4 [3] Newma, J (934). O te two fferet aect of te rereetatve meto: te meto of tratfe amlg a te meto of rove electo. Joral of te oal Stattc Socet, 97, [4] Oambo,. O. A twa, W. No-reoe wegtg atmet aroac amle rve ble MSC Dertato (006) Jomo eatta Uvert of Agrcltre a ecolog. [5] Satme, P.V. a Satme, B.V. (970) Samlg eor of Srve wt Alcato. [6] Pretl, M.B. a Cao, M.. (97). No arametrc fcto fttg. Stattc Socet. B 34,385,39. [7] oall,.m. (97).Lear regreo moel fte olato amlg teor. I V.P Goambe a D.a. Sott. oato of Stattcal Iferece. Holt, eart a Wto, oroto [8] oall,.m. A Hero, J. (973a). obt etmato fte olato I. Joral of te Amerca Stattcal Aocato, 68, [9] oall,.m. A Hero, J. (973b). obt etmato fte olato II. Joral of te Amerca Stattcal Aocato, 68, [0] Scoeberg, I. J. (946). Sle fcto a te roblem of graato. Proc. Nat. Aca. Sc. USA 5, [] Slverma, B.W. (984). Sle mootg: te eqvalet varable erel meto. e Aal of Stattc,, 3, [] Slverma, B.W. (985). Some aect of te le mootg aroac to o- arametrc regreo crve fttg. Joral of te oal Stattcal Socet 3, 47, -5. [3] Slverma, B. W. (986).Det tmato for tattc a Data Aal Cama a Hall. [4] Smt, P. (979). Sle a a efl a coveet tattcal tool. Amerca Stattcal Joral 33, [5] Smt,.M.., a Nega,. (99). obt moel-bae meto for aaltcal rve. Srve Metoolog 8,,

11 4 Joe ego Cerot: Sle egreo te tmato of te te Polato otal [6] Waba, G. (975). Otmal covergece roerte of a varable ot, erel a Ortogoal ere meto for et etmato. A. Stattc. 3, 5-9. [7] Wttaer,. (93). O a ew meto of graato. Proc. brg mat. Soc 4, [8] Zarovc, S. (956). Note o te tor of amlg meto a. Joral of te oal Stattcal Socet Sere A, 9, [9] Zeg, H. a Lttle,.J. (004). Peale Sle Noarametrc Me Moel for Iferece abot a te Polato Mea from wo-stage Samle. Srve Metoolog 30, 09 8.