OPTIMAL ESTIMATORS FOR THE FINITE POPULATION PARAMETERS IN A SINGLE STAGE SAMPLING. Detailed Outline

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1 OPTIMAL ESTIMATORS FOR THE FIITE POPULATIO PARAMETERS I A SIGLE STAGE SAMPLIG Detailed Outlie ITRODUCTIO Focu o implet poblem: We ae lookig fo a etimato fo the paamete of a fiite populatio i a igle adom amplig () Ou goal i to etimate liea combiatio of the idividual paamete: The uual appoach focue o the populatio mea ad the populatio total, hee we alo may etimate all the idividual paamete Thee ae two appoache i the liteatue: Claical Appoach, baed o Fixed Populatio Model, ad Pedictio Appoach, baed o Supepopulatio Model Ou wok, baed o the fome appoach, cotuct a model fom the amplig (Cael, Sädal ad Wetma, 977; Hedayat ad Siha, 99; Bolfaie ad Zack, 992; Valliat, Dofma ad Royall, 2000)At thi poit we may dicu a lik betwee amplig appoach ad model appoach Special et of liea etimato: We coide a pecial et of liea etimato with coefficiet that etai ifomatio about the ampled ubject ad the ode i the ample, we deote thi et by C E ote that C E i popely icluded i the Mot Geeal Type of Liea Etimato defied i Godambe(955) ad iclude popely thoe baed o the adom pemutatio model Optimality Citeio: I C E etimatio baed o optimality citeia ae poible Uiquee of the uifomly miimum vaiace etimato of the populatio mea i C E otice that i Godambe(955) the oexitece of the uifomly miimum etimato fo the

2 populatio mea i poved, i ou cae, impoig aditioal etictio o the etimato, we get the UMVE i C E 2 THE POPULATIO, PARAMETERS, SAMPLIG AD OTATIO Fiite populatio of j =,, ubject, with kow ad y = ( y y ) the vecto 2 y of o-tochatic idividual paamete of the fiite populatio Taget paamete ae liea fuctio of the idividual paamete, p = G p y Fix the pobability amplig deig, we wok with a imple adom amplig of idividual fom the fiite populatio, (,) Expaded populatio of adom vaiable, we defie Y = U y, whee fo i =,, ad j =,,, U ij idicate whethe a paticula ubject j i aiged to poitio i The expaded populatio vecto ca be epeeted compactly by Y ( Dy I ) vec(u) ij ij j =, whee y D i a diagoal matix whoe diagoal elemet ae the elemet of y ad U U2 U U 2 U 22 U 2 U = U U 2 U Momet of Y, ice (,), E( Y) = I y ad V ( Y ) = I, whee = Dy I D y ad =

3 Obeved ad o-obeved pat of Y, fom the (,) we obeve Y j= ( I 0 ) Y = ( ), ad we may eaage the expaded populatio vecto by Y K Y = Y j= = j= ( I 0 ) ( ) ( ) Y 0 I ( ) Relatig the taget paamete with the expaded populatio, otice that ( L Y + L Y ) = y = G, whee = I G L ad L = I 3 ESTIMATIO Itoductoy paagaph o Claical Appoach ad whee it beak dow whe etimatig a idividual paamete Actually we ae lookig fo a pedicto, ice = G L Y + G L Y we have to pedict G L Y Set of pedicto, = { Y L R } C E p L, whee L deped o y oly though thoe y j fo which the idividual j occu i the ample Set the Baic Popetie of the pedicto Ubiaede: We equie the etimato to atify eithe a paamete-wie ubiaede ˆ ˆ = cotait, ie E ( ) = 0, o a aveage ubiaede cotait, ie E( ) 0 p The

4 fome implie that evey paamete i ubiaedly etimated, ad the late implie that they ae ubiaedly etimated i meaat thi poit we may how, with pactical example, ituatio whee oe of the ubiaede cotait i moe appopiate to ue Veify optimal citeia: miimize the geealized mea quaed eo (GMSE), the tace of the mea quaed eo o it detemiat Some cotait o L, depedig o the epecific ifeece poblem we may aume a et of liea etictio o the coefficiet of L Fo example etimatig the idividual paamete (p=), we ca impoe oe of the followig liea etictio: = I v L, L o L = ( v v ) = v Each etictio allow diffeet type of etimato (dicu cotait o L o o the etimato of ) Exitece of the optimal etimato, thoe etimato which veify the baic popetie ae called optimal i C E Depedig o ad o the aditioal etictio o Eo! Bookmak ot defied thee may ot exit optimal etimato, may exit oly oe o may exit eveal Fo example, whe = y, ie we ae lookig fo etimato fo all the idividual paamete, the Table how how we may fid diffeet ituatio depedig o the ubiaede citeia ad o the etictio o whe we ae miimizig the GMSE L,

5 Table : Liea ubiaed etimato of the idividual paamete that miimize the GMSE ude diffeet etictio Retictio o L = I v L = v L = v v 2 v L Satifyig the paamete-wie ubiaede cotait βˆ P = I Y Doe ot exit Doe ot exit Satifyig the aveage ubiaede cotait βˆ A I = Y βˆ A I 2 = + Y ˆβ A I 3 = + + M Y with vec M = I I ( ) Etimatig Equatio, I eally do t kow if it i coveiet to wite the etimato equatio becaue a a matte of fact it i ot a etimatio equatio fo L but fo z, whee vec( L )=WA+WBz, ad the chage with the optimal citeia Maybe we ca give ome olutio fo pecial caei do t kow Uiquee fo etimate the populatio mea, the geeal of the etimatig equatio without ay pecial etictio o L i β = Y + ( I ) I Y ˆ We pove that fo be a pope etimato ( I ) I mut be ull, ad the the ample mea i the optimal liea ubiaed etimato fo the populatio mea i C E that miimize the GMSE

6 4 ADD RESPOSE ERROR Two tage poblem Supepopulatio Model, if add epoe eo i thik that ou fiite populatio ( y y2 y ) i a ealized outcome of a vecto adom vaiable ( Y Y2 Y ), i thi ituatio we ae pecifyig a upepopulatio model (Cael, Sädal ad Wetma, pp 80) 5 PROPERTIES FOR THE ESTIMATORS Ditibutio Popetie fo the mea ample, thee i a theoem fom Hájek (960) with aymptotic eult deived uig oly a pobability amplig pla to geeate the tatitical ditibutio 6 Bibliogaphy Bolfaie, H ad Zack, S (992) Pedictio Theoy fo Fiite Populatio ew Yok: Spige-Velag Cael, CM, Sädal, CE ad Wetma, H (977) Foudatio of Ifeece i Suvey Samplig, ew Yok: Wiley Godambe, VP (955) A Uified Theoy of Samplig fom Fiite Populatio oual of Royal Statitical Society B 7, Hájek, (960) Limitig Ditibutio i Simple Radom Samplig fom Fiite Populatio Publicatio of the Mathematical Ititute of Hugaia Academy of Siece 5,

7 Hedayat, AS ad Siha, BK (99) Deig ad Ifeece i Fiite Populatio Samplig ew Yok: Wiley Valliat, R, Dofma, AH ad Royall, RM (2000) Fiite Populatio amplig ad Ifeece: A Pedictio Appoach ew Yok: Wiley