RE-CALIBRATION OF HIGHER-ORDER CALIBRATION WEIGHTS

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1 SSC Aual Meet, May 00 Proceeds of te Surey Metods Secto RE-CALIBRATIO OF HIGHER-ORDER CALIBRATIO WEIGS Patrck Farrell ad Sarder S ABSTRACT A ew tecque for re-calbrat te er-order calbrated estators of te araces of arous estators of te populato total s proposed. Recet adaces prora tecques ad coputatoal speed ake te approac appeal for practcal use. Estators of te araces of te saple ea ad te rato ad reresso estators uder dfferet sapl scees are sow to be specal cases of te proposed tecque. A ew syste of predctors for te populato arace s sow to be a specal case of te approac as well. Te results of a eprcal study desed to estate te propertes of te proposed etodoloy uder sple sapl dess are also reported. Keywords: arace Estato; Calbrato; Rato ad Reresso Type Predctors; Aulary Iforato. RÉSUME O propose ue ouelle tecque pour recalbre les estateurs de rad ordre calbres des araces de ders estateurs du total d ue populato. Les prorès récets das les tecques de proraato et la tesse de calcul fot de cette approce ue soluto téressate e pratque. Les estateurs des araces de la oyee écatlloale et les estateurs de tau et de réresso sous dfféretes étodes d écatloae s aèret des cas spécau de la tecque proposée. U oueau systèe des prédcteurs pour la arace das la populato s aère éaleet u cas spécal de cette approce. Les résultats d ue étude eprque coçue pour étuder les proprétés de la étodoloe proposée sous des plas d écatlloae sples sot éaleet doés. Mot Clé: Calae, estato de la arace, forato aulare, prédcteurs de type rato et réresso.. ITRODUCTIO Estates of te araces of estators sere a portat role we draw ferece for fte populato paraeters. Suc estates ca be ore effcet we based o reresso, rato or product estato approaces, sce suc tecques allow for te corporato of aulary forato. Cosder a fte populato Ω = {,...,,... }, fro wc a probablty saple s ( s Ω s draw wt a e sapl des, p (.. I oter words, p ( s s te probablty tat s s selected. Te cluso probabltes = Pr( s = Pr( & s π ad π are assued to be strctly poste. I addto, let Θ = ( π π π. If y s te alue of te arable of terest for te -t populato ut, te te well-kow Hortz-Topso (95 estator of te populato total, Y, s Ŷ = d y (. s Delle ad Särdal (99 used calbrato o te aulary total X to odfy te basc sapl des wets, d = π, tat appear (. ad suested a ew estator Ŷ = w y (. G s w = d + dq d q X d are s s te calbrated wets obtaed by z c-square Patrck J. Farrell, Scool of Mateatcs ad Statstcs, Carleto Uersty, 5 Coloel By Dre, Ottawa, Otaro, Caada KS 5B6 ad Sarder S, Departet of Statstcs, St. Cloud State Uersty, St. Cloud, M , USA. 75

2 dstace fucto ( w d ( d q costrat s s = w subect to a lear X, ad leads to te GREG as: Yˆ = + G d y dq y dq X d (.3 s s s s Its for depeds upo te coce of q Te arace of te estator (.3 ca be approated as: ( ˆ ( = Θ de d e (. Ω e = y β suc tat E ( = 0, ( e σ ( e = ad E ( e e ρ σ ( ( = for, σ > 0. Here ρ s te correlato coeffcet betwee successe error ters tat are related accord to e = ρ e + u te u ~.. d. ( 0,.Follow Särdal, Swesso ad Wreta (99, te Yates- Grudy (953 for for te estator of te arace of (.3 s ( ( 0 Yˆ G = d weˆ w eˆ (.5 d ( π π π π s = ad eˆ = y βˆ. S, Hor, ad Yu (998 cosder a estator of te arace of te GREG as: ( ( Yˆ G = w weˆ w eˆ (.6 s w = d [ ] + d q ( d d q ( d d d s are te calbrated wets obtaed by z ( w d ( d q subect to s w s ( d d = estator of arace, as,, ad leads to a ew + Bˆ [ ] = (.7 G 0 G Bˆ = d q ( weˆ w eˆ ( d d s d q ( d d = ˆ µ ˆ µ 0 s ( say, = d ( d d s = Θ ( d d = s d Ω ad, X = X Ω,ad. Uder sple rado sapl wtout replaceet (SRSWOR, t s easy to sow tat te estators of s = s S s, S due to Isak (983, y I y ( ad s s + ( S s lr y ˆ = = γ, ( y y ( ( ˆ γ = are specal cases = of (.7. O te oter ad, Sa ad Patel (996 troduced a syste of predctors of S as y Q re ˆ ( S s + ˆ γ ( ( = s y + ˆ γ (.8 Ω s γ ad ˆ γ are saple-depedet costats.. RE-CALIBRATED ESTIMATOR OF THE ARIACE OF GREG We cosder ere a estator of te arace of GREG as o G = w ( weˆ w eˆ (. s o w are re-calbrated wets suc tat c-square type dstace fucto o o (. ( w w ( c w D = s s zed subect to te calbrato costrat or E w w o s = Θ d Ω ( w eˆ w eˆ = E o w s ( + w ρ w w ( ( ( + d ρd d G (.3 76

3 c are real costats. Optzato (. subect to (.3 yelds te re-calbrated wets o = w + { } a cw( w ( + w ( ρ ww ( ( c w( w ( + w ( ρ ww ( ( s a (. Substtuto of (. to (. prodes a ew estator of te arace of GREG as Bˆ = Dˆ = ˆ ( ˆ ˆ[ ( ˆ ˆ ( ˆ G = 0 + B X X ] + Dˆ [ ] s a a d q ( d d ( weˆ w eˆ s d ( q d d s c d s c w ( w ( + w ( ρ ww ( ( ( we w e ( w ( + w ( ρ ww ( ( = d ( d d, a ad ρ d d s ( d = Θ ( + d ( ( (, ρ w w (, ( + w ( ˆ a w w s = ( (. (.5 Te class (.5 s wder ta Wu (98, Delle ad Sardal (99, De ad Wu (987, Sa ad Patel (996, S, Hor ad Yu (998 ad Wu ad Stter (00 ad we aed t er order odel asssted calbrato. 3. RE-CALIBRATIO USIG OPTIMAL DESIGS FOR THE GREG-ESTIMATOR Cosder te stuato of ρ ot ecessarly zero ad ote π, te (.3 reduces to tat f ( s ( + ρ = ( ρ o w Θ (3. + q d q [ X ] = s Ω. Te recalbrated estator of arace takes te for G = w ( w eˆ weˆ + s wc ( + ρ ( w e we s cw ( + ρ s ( ρ Θ w ( + ρ (3. s s Ts result llustrates tat te proposed tecque works to re-calbrate te Yates ad Grudy (953 for of te estator of te arace of GREG uder te codto of u arace for te estator of te total uder te true odel. It s ofte te case tat c =. If ts s so, te (3. s a ew estator of arace of te GREG estator. Tree oter specal cases of (3. are also worty of ote. Case 3.. If ρ = 0 ad c = ( +, te te estator (3. reduces to = ( Θ G w w eˆ we w ( + ˆ (3.3 s Ω s Case 3.. If ρ,0 0, ad = ( + = w ( w eˆ w eˆ [ ( c ρ te (3. takes te for G s ( ρ Θ w ( + ρ Ω s (3. Case 3.3. If ρ = +, te for c = ( te estator of arace (3. becoes ˆ ˆ = 0 (3.5 (. Y G ote also tat te codto ( π correspods to te Godabe ad Jos (965 lower boud of arace, so te arace for fed saple des uder te true odel ay be equal to zero. Ts deostrates te usefuless of te proposed re-calbrato etod of te estator of te arace of te GREG estator.. EMPIRICAL STUDY: O AUTOCORRELATIO 77

4 Uder te assupto tat ρ = 0, te perforace of te re-calbrated estator eac case as bee assessed at a uber of alues for te odel paraeter wt ( =. To aod ay cofuso, we ae redefed te estators cosdered for coparso te eprcal study. ote tat S, Hor, ad Yu (998 coducted a slar coparso of low ad leel calbrato approaces... Rato estator We copare te estator of te arace of te rato estator, ( ˆ ( f Y X Rato = ˆ ( e ˆ = X wt te re-calbrated estator of arace ( ˆ ( ( = ˆ ˆ f Y Y + Rato Rato = = ˆ γ ( S s X, ( S s ˆ = ( + ( eˆ eˆ ( + γ. =.. Reresso Estator = Fally, we copare te estator of te arace of te reresso estator ( ˆ ( f Y ( ˆ re = ( eˆ + ˆ γ X X = + ˆ γ ( X + ˆ γ ( S s wt te re-calbrated estator of te arace of GREG ˆ = re re = 3 t = + cˆ 0 (, S s ( ( t + t ( cˆ = ad t 0 = ( X = +. = = 78 We ake use of a populato cosst of = 0 uts fro Hortz ad Topso (95, te study arable, y, s te uber of ouseolds o te -t block, wle te kow aulary caracter,, s te eye estated uber of ouseolds o te -t block. All possble saples of sze = 5 were selected by SRSWOR, wc results = = M 550 saples. For te k-t saple, te rato estator Y ˆ ad ts arace, Rato k M Yˆ Rato k k = = M [ Y ] Rato, were coputed, alo wt te er-order ad re-calbrated estators of arace ˆ Rato k, = 0, respectely, alues of ra fro to 3 steps of 0. were used. Te ea squared error was coputed as MSE M { ˆ Rato } = M [ Rato k Rato ] k= were te ealuated oer all possble saples. Te percet relate effcecy of ( Y ˆ Rato ˆ wt respect to ( Y ˆ ˆ Rato was calculated as RE = MSE{ Rato } 00 MSE{ Rato } wle te coerae by 95% cofdece terals CCI [ ˆ ( Y ˆ Rato ] was detered for te -t estator by deter te proporto of tes tat te true populato total, Y, fell betwee te lts defed by ( Rato k Y ˆ Rato k t α. Te relate bases te estators were also coputed ad are preseted Table. Ts etre process was te repeated for te reresso estators ˆ ˆ Y Y + y ( X X ˆ. k = = = re Te sulato study sows tat te proposed estator s always better ta te est estators. 5. EMPIRICAL STUDY: AUTOCORRELATED ERROR TERMS I order to cosder te stuato ρ s ot ecessarly zero, for dfferet pars of ad ρ as sow Table we eerated us IMSL subroutes FORTRA fte populatos of = 000 uts eac fro te odel y = β + e e = ρ e + u wt e 0 = 0 ad u ~ ( 0,. Te aulary arable was assued to follow a beta dstrbuto wt bot paraeters set at 0.. Te Y alues were obtaed fro te odel wt β =.. Fro eac populato of 000 uts, we selected 5000 saples of = 50. Te, for eac

5 populato defed by a dfferet par of ad ρ, we detered te effcecy of te proposed estator of te arace of te rato estator ( ˆ ( f Y = Rato + ˆ γ = ˆ γ = cf = = ( + = + = eˆ X X cf ρ ad cf ( ( ρ ( S s ( S s ( e e ρ relate to ( ˆ Y ˆ Rato. = = Te results, preseted Table, sow tat te relate effceces rae fro 09.3% to 0.8% wt a eda of 9.5%. Fally, ote tat te proposed estator s ot defed for ρ = ad = 0. ACKOWLEDGEMET Ts researc was supported by te atural Sceces ad Eeer Researc Coucl of Caada. Te secod autor was post doc at Carleto Uersty dur ts estato. REFERECES De, L.Y. ad Wu, CFJ.(987.Estato of arace of te reresso estator. J. Aer. Stat. Assoc.,8, Delle, J.C. ad Särdal, C.E.(99. Calbrato estators surey sapl. J. Aer. Statst. Assoc., 87, Godabe,.P. ad Jos,.M. (965, Adssblty ad Bayes estato sapl fte populatos I. A. Mat. Statst., 36, Hortz, D.G. ad Topso, D.J. (95.A eeralsato of sapl wtout replaceet fro a fte uerse. J.Aer.Statst Assoc., 7, Isak,C.T.(983.arace estato us aulary forato. J. Aer. Statst. Assoc., 78, 7-3. Särdal, C.E., Swesso, B. ad Wreta, J.H. (99. Model asssted surey sapl. Y : Sprer- erla. Sa, D.. ad Patel, P.A. (996. Asyptotc propertes of a eeralzed reresso-type predctor of a fte populato arace probablty sapl. Caada J. Statst., (3, S,S.,Hor,S.ad Yu, F.(998. Estato of arace of eeral reresso estator: Her leel calbrato approac. Surey Metodoloy,, -50. Wu, C. ad Stter, R.R. (00. A odel-calbrato approac to us coplete aulary forato fro surey data. J. Aer. Statst. Assoc., 96, Wu, C.F.J. (98. Estato of arace of te rato estator. Boetrka, 69, Yates, F. ad Grudy, P.M. (953. Selecto wtout replaceet fro wt strata wt probablty proportoal to sze. J. R. Statst. Soc., B, 5,

6 Table. Coparso of te estators of te arace of Rato ad GREG based o based o te Hortz ad Topso (95 populato for =5 ad =0. Rato GREG RB { } RB { } RE C { } C { } RB { } RB { } RE C { } C { } Table. Relate effcecy of te proposed estator of arace of te rato estator we autocorrelato ( ρ s ot ecessarly zero for =000 ad =50. ρ

Sample Allocation under a Population Model and Stratified Inclusion Probability Proportionate to Size Sampling

Sample Allocation under a Population Model and Stratified Inclusion Probability Proportionate to Size Sampling Secto o Survey Researc Metods Sample Allocato uder a Populato Model ad Stratfed Icluso Probablty Proportoate to Sze Sampl Su Woo Km, Steve eera, Peter Soleberer 3 Statststcs, Douk Uversty, Seoul, Korea,