A UNIFIED APPROACH TO ESTIMATION AND PREDICTION UNDER SIMPLE RANDOM SAMPLING

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1 A UIFIED APPROACH TO ETIMATIO AD PREDICTIO UDER IMPLE RADOM AMPLIG Edward J. taek III Departmet of Biostatistics ad Epidemiology, PH Uiversity of Massachusetts at Amherst, UA Julio da Motta iger Departameto de Estatística, IME Uiversidade de ão Paulo, Brazil Viviaa Beatriz Lecia Departameto de Ivestigació, FM Uiversidad acioal de Tucumá, Argetia ABTRACT We cosider a probability model where the desig based approach to iferece uder simple radom samplig of a fiite populatio ecompasses a simple radom permutatio superpopulatio model. The model cosists of a expaded set of radom variables followig a radom permutatio probability distributio that keeps track of both the uits labels ad positios i the permutatio. I particular, sice we keep track of the labels, the model allows us to attack the problem of estimatio of a uit s parameter. While some liear combiatios of the expaded set of radom variables correspod to liear combiatios of the uit parameters, other liear combiatios correspod to radom variables kow as radom effects. Usig a predictio techique similar to that employed uder the model-based approach, we develop optimum estimators of the liear combiatios of the uit parameters ad optimum predictors of the radom effects. The ubiased miimum variace estimator of the populatio mea is the sample mea ad of a uit parameter is the Horvitz-Thompso estimator if the uit is icluded i the sample, ad zero otherwise. The predictor of the radom variable at a give positio i the permutatio is i

2 the realized uit s parameter for positios i the sample, ad the sample mea for other positios. For other liear fuctios, uique miimum variace ubiased estimators may ot exist. Key words: Bias, fiite populatio, optimal estimatio, predictio, radom effects, mixed models, super-populatio, desig based, model based, iferece. Ruig Title: UIFIED IFERECE I IMPLE RADOM AMPLIG ii

3 . ITRODUCTIO We propose a probability model iduced by a simple radom sample desig for a fiite populatio that ecompasses a simple radom permutatio super-populatio model. Model based predictio tools are used to optimally estimate liear combiatios of radom variables i the model. Appropriate liear combiatios of the radom variables may be costructed to represet fiite populatio parameters, icludig the parameter for a idividual uit. Other liear combiatios correspod to radom variables that are aalogous to radom effects. The model provides a commo cotext for comparig results o predictio ad estimatio. ice the parameters ca be estimated ad the radom variables ca be predicted i a commo maer, the results lead to iterestig iterpretatios. The probability model we propose was motivated by the desire to costruct iferece for a uit parameter i simple radom samplig. While this problem is ot of compellig iterest, it is closely related to a similar commo problem i two stage samplig where there is iterest i predictig the parameter for a realized uit (or cluster). The ivestigatio of that more complicated problem led us to focus o this simpler settig which still retais some essetial aspects of the two stage problem. We explore the simpler settig here, deferrig further commets o the two stage settig to the discussio. Iferece about a parameter for a idividual labeled uit is ot possible uder the classical desig-based approach sice idividual labeled uits are ot idetifiable i the probability models geerally used to lik the sample to the populatio. I fact, the probability models used for such purposes are typically based o the distributios of exchageable radom variables which igore labels. We overcome this problem by itroducig a discrete probability model where parameters correspod to the values of the labeled uits. The model is based o

4 idicator radom variables geerated by a radom permutatio of uits, as would occur i a simple radom samplig desig. These radom variables keep track of both the uit s label ad the uit s positio i the permutatio. Rather tha characterizig a permuted fiite populatio by radom variables, the expaded framework icludes 2 radom variables. The model we propose does ot rely o the cocept of a super-populatio cosidered uder the model-based approach. However, estimators/predictors of liear combiatios of radom variables are costructed usig the predictio approach commo i model-based iferece. Furthermore, liear combiatios of the radom variables reproduce the simple radom permutatio super-populatio model. The problem we cosider is particularly simple, ad hece is related to a broad literature. The geeral modelig framework for survey samplig is give by Cassel, ärdal ad Wretma (977), with desig based ad model based iferece widely discussed (Bolfarie ad Zacks (992); Hedayat ad iha (99); Mukhopadhyay (200); ärdal, wesso, ad Wretma (992); Thompso (997); Valliat, Dorfma, ad Royall (2000)). Recet reviews of iferece i survey samplig are give by Rao (997, 999a). Brewer, Haif, ad Tam (988), Brewer (999) ad Brewer (2002) have discussed recocilig model-based ad desig based iferece. The radom permutatio super-populatio model has bee discussed by Rao ad Bellhouse (978), Mukhopadhyay (984) ad Rao (984), ad i the cotext of two stage samplig, by Padmawar ad Mukhopadhyay (985) ad Bellhouse ad Rao (986). Model-based approaches to the two-stage problem have bee studied by cott ad mith (969) ad Fuller ad Battese (973), ad recetly reviewed i the cotext of small area estimatio by Rao (999b). Of particular relevace are the fudametal results of Godambe (955) ad Godambe ad Joshi (965) that o uiform miimum variace ubiased liear estimator of the populatio total 2

5 exists if coefficiets are allowed to deped o the sequece of labels i the sample. Royall (969) coutered this result with the observatio that if radom variables represetig the samplig were reduced to their usual represetatio, where oe radom variable is associated with each selected uit, optimal estimators could be obtaied. Other approaches to overcome the o-existece result of Godambe have bee suggested by Hartley ad Rao (968,969). Our approach is i the same spirit as that of Royall s 969 result, where we reduce the most geeral set of radom variables defied by Godambe to a set of 2 radom variables. Defiitios ad otatio are developed i ectio 2 ad the expaded model is fully defied i ectio 3. Iterest is focused o liear combiatios of the radom variables defied i the expaded model. Certai liear combiatios simplify to o-stochastic fiite populatio parameters; other liear combiatios are radom variables. ice both parameters ad radom variables ca be defied by the liear combiatios, the methods we develop i ectio 4 are appropriate for both estimators (of parameters) ad predictors (of radom variables). For simplicity, we use the term estimator i referece to geeral liear combiatios of radom variables. The expaded model eables estimatio of the populatio mea, as well as parameters for labeled uits. The sample mea is the best liear ubiased estimate of the populatio mea. For a sigle uit, the best liear ubiased estimator is uique ad of the Horvitz-Thompso (952) type if the uit is icluded i the sample, ad zero otherwise. imultaeous estimatio of all uit parameters i the populatio does ot i geeral lead to uique estimators. However, with differet additioal restrictios, differet uique estimators arise. The predictor of the radom variable correspodig to the i th positio i a ordered permutatio, while ot of ay obvious 3

6 iterest, turs out to be aalogous to the widely used predictor of a realized radom effect i a mixed model. These results are discussed further i ectio DEFIITIO AD OTATIO We cosider the problem of estimatig certai characteristics of a fiite populatio of uits uder simple radom without replacemet samplig. We defie a fiite populatio as a collectio of a kow umber,, of idetifiable uits labeled =,,. Associated with uit is a parameter y. We summarize the set of parameters i the vector = ( ) y y,, y ad assume that whe uit is observed, the parameter y is kow without error. Typically, there is iterest i a p vector of parameters of the form β = Gy where G is a matrix of kow costats. For example, if G = I, with I deotig the -dimesioal idetity matrix, the β is the set of idividual parameters. If G = e, where e deotes a -dimesioal colum vector with ull elemets i all positios except for the th positio for which the value is assiged, the parameter β correspods to the value y associated with the uit labeled i the populatio. Whe G =, where deotes a -vector with all elemets equal to, β correspods to the populatio mea, µ. We defie a probability model that liks the populatio parameters to a expaded vector of radom variables which is essetially iduced by a simple radom samplig desig, ad develop estimators of liear fuctios of these radom variables. The proposed estimators are liear fuctios of the radom variables that defie a sample. We use the predictio approach that is commo i model-based iferece to develop the estimators. Before itroducig the 4

7 expaded model, we first review the predictio approach used i the cotext of super-populatio models. The predictio approach is based o a uderlyig probability model for a vector of radom variables * Y = (,, ) that characterizes a super-populatio. The populatio uder Y Y study, y = ( y,, y ), is cosidered to be a realizatio of these super-populatio radom variables. The vector of radom variables is partitioed ito a subset which we call the sample, Y = ( Y,, Y ) ad the remaider, * Y, such that * R = ( Y+,, Y) Y = ( Y, Y ). Iferece is * * * R solely based o liear models of the form * * * * Y = X β + E (2.) where * X is a kow o-stochastic matrix, parameters ad * β is a p-dimesioal vector of super-populatio * E is a -dimesioal vector of radom errors govered by the probability model uder which * E ξ ( ) = E 0, where ξ deotes expectatio with respect to the superpopulatio. Although the super-populatio parameters appear i the model, they are ot of primary iterest. Istead, the parameters of iterest are liear combiatios realizatio of β= Gy of a * Y. The populatio mea ad the populatio total are typical examples of β. Assumig that * Y is realized, the estimator of β is based o the predictor of fuctios of it, i such a way that it satisfies some optimality criteria (see Royal (976) or * Y R or some Bolfarie ad Zacks (992), for example). More specifically, Valliat et al. (2000, pp.29-30) poit out that the target parameters may be writte as β= β + β R, where β deotes the part of the liear combiatio observed i the sample ad β R deotes the part associated with the osampled uits. After selectig the sample, the problem of estimatig β is equivalet to 5

8 predictig β R ad the best liear ubiased estimate (BLUE) of β is obtaied by addig the optimal predictor of βr to β. The predictio process relies o the probability model for the super-populatio ad does ot ecessarily deped o the physical process used to select the sample. 3. THE EXPADED MODEL Our mai obective is to express a expaded set of radom variables iduced by the desig-based approach i the form of model (2.). We show that this model allows the costructio of estimators of liear combiatios of the correspodig radom variables based o the same optimality criteria cosidered uder the predictio approach. ome of these liear combiatios correspod to populatio parameters, while others are radom variables. We restrict ourselves to the case where the sample is selected by simple radom samplig without replacemet. We first describe the typical desig-based radom permutatio model, ad the itroduce the expaded model. A advatage of the expaded model is the ability to idetify a parameter associated with a labeled uit. Assumig simple radom without replacemet samplig, the typical radom permutatio probability model assigs equal probability to all permutatios of the fiite populatio uits. We idex each uit s positio i the permutatio by i=,...,. The value i positio i for a radomly selected permutatio is defied by the realizatio of the radom variable Y = U y i i = where U i = if uit is i positio i ad U i =0 otherwise. The radom vector ( 2 ) Y = Y Y Y is the radom permutatio super-populatio (Cassel et al., 977), ad the 6

9 radom variables Y, i =,...,, correspod to a sample. This represetatio of radom variables i does ot allow uits to be idetified ad hece does ot permit iferece about uit parameters. The expaded model is based o represetig the radom variables i the sum = U y i as idividual radom variables of the form Y = U i i y, which we summarize i a 2 vector ( 2 ) Y= Y Y Y where = ( 2 ) be defied compactly as = ( ) vec ( ) (earle, 982), y Y Y Y Y. The vector of radom variables ca Y D I U, where deotes the Kroecker product D y is a diagoal matrix with the elemets of y alog the mai diagoal, vec ( U ) is a vector represetig the colum expasio of U, ad U U2 U U 2 U 22 U 2 U =. U U U Give the radom structure of U, the expected value ad the variace of the expaded radom vector are respectively give by ad E ( Y) = Xy (3.) var ( ) Y = P (3.2) where X I, = P = I a J with J =, ad a a a a a a = DPD y y. (3.3) 7

10 The selectio of a simple radom sample of size from the populatio will result i the realizatio of of the expaded radom variables i the vector Y. We gather these radom variables for the sample i the vector = ( ( ) ) Y I 0 Y by rearragig the elemets i = the vector Y ; similarly, the remaiig ( ) radom variables are defied by the vector ( ( - ) ( - )) YR = 0 I Y, where = A deotes a block diagoal matrix, with blocks give = by A (earle, 982). The variace of the rearraged expaded radom vector is partitioed as Y V VR Var = YR VR VR V J, with J ( ) =. where ( = V I J ) ad R = ( ( ) ) As a illustratio, cosider a fiite populatio with = 4 uits from which we select a simple radom sample without replacemet of size 2 follows that Y = =. Lettig = ( ) y y y y y it ( y ( U U2 U3 U4 ) y2 ( U2 U22 U32 U42 ) y3 ( U3 U23 U33 U43 ) y4 ( U4 U24 U34 U44 )) ( ( 2 ) 2 ( 2 22 ) 3 ( 3 23 ) 4 ( 4 24 )) Y = y U U y U U y U U y U U ( ( 3 4 ) 2 ( ) 3 ( ) 4 ( )) YR = y U U y U U y U U y U U ad,. upposig that the first ad secod selected uits i a sample are uits 3 ad, respectively, the realized value of ( ) y y. 3 Y is 4. ETIMATIO A characteristic of the proposed model is that the vector of parameters y may be defied as liear combiatios LY of the expaded radom variables. For example, settig 8

11 L = I, (4.) LY = y, while the value for uit i the populatio, y, is defied by settig L = e. (4.2) The populatio mea µ is defied by settig 2 L =. (4.3) More geerally, we ca defie other liear combiatios of Y which are stochastic. For example, a radom variable correspodig to the value that will appear i the i th positio i a permutatio is defied by settig L = e. (4.4) i I geeral, for liear combiatios defied i terms of the expaded radom variables, we ca discuss estimatig a parameter or predictig a radom variable. The specificatio of L is ecessary to determie whether LY is fixed or radom. We ca ecompass both the estimatio ad the predictio problems i the same framework. As previously oted for simplicity, we use the term estimatio i referece to a geeral liear combiatios of radom variables. It is ot ecessary to use the expaded radom variables to develop estimators for all liear combiatios of Y. To see this, we evaluate the liear combiatio usig the expasio give by Y= I Y + ( P I) Y. (4.5) For example, usig (4.3), the liear combiatio defiig the populatio mea simplifies to LY = Y. imilarly, usig (4.4), the liear combiatio defiig the radom variable 9

12 correspodig to the value that will appear i the i th positio i a permutatio simplifies to LY = e Y. For the liear combiatios defied by (4.3) ad (4.4), the optimal estimator ca be i developed by solely cosiderig the radom variables Y sice the first ad secod terms i (4.5) are orthogoal, ad the secod term has expected value equal to zero (Rao ad Bellhouse (978), Theorem.). Usig the predictio approach, we develop the solutio to the problem of estimatig LY based o a sample. First, we partitio LY ito a sample compoet, LY, ad a remaiig compoet, LY. R R We require the predictors of LY R R to be liear i the sample, ad represet them by LRY. Defiig C= L + L R, the class of estimators of LY is give by E { : is a matrix of costats} C = CY C p. We require the estimators to be ubiased (such that ( ) geeralized mea squared error give by ( ) GME = Var p CY LY (4.6) (Bolfarie ad Zacks, 992). Usig (3.), we may write ( CY ) E CY LY = 0 ), ad have miimum E = CX y, where X I (4.7) = so that the ubiased coditio reduces to CX y = CX = LX. (4.8) LXy for all y, or equivaletly We solve (4.8) for C i terms of a arbitrary matrix, ad the miimize the GME with respect to that matrix. Whe LY is o-stochastic, the result is give by ˆ J C= L I + p p ( ) P T I P (4.9) 0

13 where T is a arbitrary matrix resultig from use of geeralized iverses to obtai the solutio (see Appedix A). olutios to the problem of estimatig a liear fuctio of LY are developed i a similar maer. We briefly outlie the solutio that was first give by Royall (976). First, ote that ( Y) E = X µ, where X=, ad ( ) 2 2, where ( ) 2 Var Y = σ P σ = y µ Y ito the sample, Y = ( Y,, Y ), ad the remaider, Y = ( Y,, Y ), results i Y V V R Var = Y R V R V R where 2 V ( = σ I J) ad 2 ( V ) R = σ J ( ) = R +. Partitioig. We partitio X ad L i a similar maer resultig i X =, X = ad LY = L Y + L Y. We require the predictor of R R R to be a liear fuctio of the sample, R R L Y R LY E, to be ubiased, i.e. to satisfy ( L R Y ) = E( L RY R ), ad to have miimum GME (give by var ( ) p CY LY, where C = L + L ). The resultig estimator is R where ( ) ( ˆ α) ˆ CY ˆ = L Y + L R Rα + X V RV Y X (4.0) = XV X XV Y. αˆ 4. Estimatig y value We obtai the estimator of LY with L defied by (4.2) correspodig to a particular y associated with the uit labeled. ice p =, P 0 ad (4.9) simplifies to p =

14 ˆ C= ( e ). (4.) This correspods to y whe uit is icluded i the sample, ad zero otherwise, a Horvitz- Thompso type estimator of the uit s value. For such a estimator, 2 GME = y. 4.2 Estimatig y We develop simultaeous ubiased estimators of all the idividual parameters, y, i a fiite populatio ext. These parameters are defied by settig L equal to (4.). ice p the solutio give by (4.9) simplifies to ˆ C= ( I ) + P T ( I P ) (4.2) =, where T is a arbitrary matrix. I geeral, the secod term i (4.2) is ot zero, ad hece there are multiple solutios, each of which has ( ) GME = σ 2. Uique estimators ca be obtaied by imposig restrictios o the structure of the coefficiets, C. For example, if we assume that C= I v, where v is a vector of ukow costats, followig the same strategy, we may show that the uique estimator of y is CY ˆ = ( I ) Y. (4.3) This restrictio forces the coefficiets to be the same for differet parameters, but allows the coefficiets to differ with positio. However, ot all structures for C lead to ubiased estimators. For example, there are o solutios for C= J v. 2

15 A more geeral class of estimators ca be cosidered if we replace the requiremet of uit ubiasedess by average ubiasedess, E ( ) 0 CY LY =. With this requiremet, ad proceedig i a maer similar to that used to obtai (4.2), the estimator of y simplifies to ˆ CY = J Y + P T Y. (4.4) This estimator is ot uique sice T is arbitrary. If C= I v, a uique solutio results ad is give by (4.3). If C is restricted to be of the form C= J v, it follows that the uique solutio is CY ˆ = y, the sample mea for each elemet. We illustrate these results via a simple example. Let us assume that = 4, = 2, y = ( y y y y ) ad that the realized value of Y is ( ) y y, i.e., the 3 third uit was selected i the first positio ad the first uit was selected i the secod positio i the sample. If we require the estimator (4.2) to be liear ad ubiased, oly the estimator for the uselected uit (ad also for uits for which y = 0 ) is uique, ad equal to zero. The estimate for uit = is ay, while the estimate for uit = 3 is cy 3 with a ad c deotig fuctios of elemets i the arbitrary matrix, T. If we require the estimator to be liear ad ubiased, ad restrict the coefficiets to be of the form C= I v, the the uique estimates for uits = ad = 3 are give by the Horvitz-Thompso type estimate, 4 y. The estimates 2 for uit = 2 ad = 4 are zero. Usig the average ubiased costrait, ad requirig estimators to be liear i the sample with coefficiets of the form C= J v, the uique estimate for all uits is give by the sample mea, y. 3

16 4.3 Estimatig µ ad predictig radom variable(s) i the i th positio i a permutatio based o Y. The liear combiatio LY with L give by (4.3) defies the populatio mea; settig L equal to (4.4) defies the radom variable that will appear i the i th positio i a permutatio. Usig (4.5), both liear combiatios are equal to liear fuctios of LY with i L =, ad L = e, respectively. Usig the coefficiets that defie the populatio mea, ad otig that J V = 2 I + ad ˆ α = y, the estimator (4.0) simplifies to y, the sample mea. σ e e e where e i is a vector of dimesio to predict the We partitio = ( ) i i ir radom variable i the i th positio i a permutatio. Whe i, CY ˆ = e Y which will i correspod to the value of the uit that is i the i th positio i a realized permutatio, i.e. uy i (where i = u represets the realized value of ( ˆ α) U i ). Whe i >, LY = L RY R, ad ˆ CY ˆ = e ir Rα + X V RV Y X which simplifies to y. The GME of the predictor is zero whe i, ad equal to σ 2 + whe i>. imultaeous predictors of the uits realized i all positios are defied by settig L = I ad result i the same predictors as those obtaied for the idividual positios. The predictors correspod to the realized uit s values whe i, ad to the sample mea whe i >. For the vector of predictors, estimator of y i ectio 4.2. ( ) 2 GME = σ, which is equal to the GME of the 4

17 5. DICUIO Desig based ad model based methods are usually discussed as separate approaches for estimatio ad iferece i fiite populatio samplig. We have preseted a expaded probability model iduced by the possible physical process of simple radom samplig. ice o super-populatio model is required ad the probability model arises solely from samplig, we cosider the resultig estimators to be desig-based. o additioal assumptios or cocepts are required for estimatio, which is accomplished by developig predictors of liear fuctios of the uobserved radom variables. Liear fuctios of the expaded probability model lead to a set of radom variables referred to by others as a simple radom permutatio super-populatio model (Cassel, ärdal ad Wretma (977)). Thus, the expaded model ecompasses both desig ad model based frameworks. Although we feel that the expaded model uifies aspects of survey samplig methodology for simple radom samplig, it has ot yet bee exteded to the broad class of super-populatio models, icludig the more geeral radom permutatio superpopulatio models. Others have ivestigated a radom permutatio model i the cotext of a superpopulatio framework, ad cocluded that the sample mea is the uiform miimum variace ubiased estimator of the populatio mea (Rao ad Bellhouse, 978). I such a framework, the likelihood is uiformative for uit parameters, ad estimatio has focused o the mea. Iclusio probabilities for labeled uits, as opposed to the basic idicator radom variables uderlyig uit selectio are used. Although the idicator radom variables used to defie the expaded probability model are ot ew (see, for example, eyma (934, 935), ad Kempthore (952)), their use i developig estimators of uit parameters appears to be ovel. 5

18 The expaded model exteds the typical permutatio model to a broader set of radom variables, but falls short of the very geeral set of radom variables evisioed by Godambe (955) which spas a ( ) dimesioal space. The radom variables i a typical permutatio model spa a dimesioal space. The radom variables i the expaded model spa a ( ) 2 dimesioal space. Higher dimesioal radom variables may be postulated itermediate to Godambe s geeral model that may lead to ew isights. Our motivatio i developig the expaded permutatio model was to improve our uderstadig of realized radom effects i the cotext of a mixed model. I a mixed model, a realized radom effect is commoly defied as the differece betwee the parameter for a realized uit, ad the mea of a populatio. With this defiitio, the expected value of a radom effect is zero. To simplify the discussio, we defie a realized radom effect as the parameter for a labeled uit that is realized at a particular positio i a permutatio. Our defiitio is a reparameterizatio of the defiitio commoly used for mixed models. If a uit is icluded i a simple radom sample, the realized radom effect is simply the parameter for that uit. The value of the parameter (which is observed) is the best liear ubiased predictor. ice the predictor is the parameter for the realized uit, may we iterpret the predictor as a predictor of the parameter for a specified uit? The expaded model provides the aswer to this questio sice we ca predict a radom effect ad a specified uit as separate liear combiatios of radom variables i the same model. The liear combiatios that defie these two quatities differ, as do their estimators. A clearer statemet of the iterpretatio for what is commoly referred to as the predictor of a realized radom effect is the predictor of a positio i a permutatio. I fact, sice the expected value of the radom variable at a positio 6

19 is the populatio mea, the predictor of this parameter will almost ever equal the parameter beig predicted. I a aalogous maer, the predictor of a realized radom effect i a simple mixed model will carry the iterpretatio as the predictor of the expected value of uits that ca occur at a positio i a permutatio. imilar results based o a expaded model for cluster samplig, while outside the scope of this paper, have bee developed for equal size clusters both with ad without respose error (taek ad iger, 2002a) ad i a ubalaced settig (taek ad iger, 2002b). The expaded framework is particularly importat to retai the estig of secodary samplig uits i primary samplig uits i a ubalaced two stage samplig cotext. uch results share the basic awkward iterpretatio as predictors of positios, ot idetified uits, as illustrated i the expaded simple radom samplig model. While the results preseted here are for simple radom samplig, extesios to may other sample settigs appear to be feasible. uch extesios iclude addig measuremet error to simple radom samplig, stratified samplig, ad ubalaced cluster samplig settigs. trategies that accout for covariates have bee iitially addressed i dissertatios by Lecia (2002) ad Li (2002). Extesios also appear feasible for experimetal studies. There are also limitatios. The two stage sample results are limited by the curret lack of a optimal strategy for variace compoet estimatio. trategies for hadlig a cotiuous covariate are ot yet developed ad may ot be feasible. Extesios to uequal probability samplig, may be possible but have ot yet bee developed. From a differet perspective, i the expaded model, liear combiatios of radom variables that correspod to uit parameters ca be defied, ad have a clear iterpretatio. The ubiased estimator of a uit s parameter (which correspods to the Horvitz-Thompso estimator 7

20 whe the uit is icluded i the sample, or zero otherwise) suffers from the criticism of Basu s (97) elephat example. The estimator is ot ituitive, although it clearly satisfies the costrait for ubiasedess. May practitioers have used the predictor of a positio i a permutatio as a estimate of the parameter for a uit i the populatio. uch a estimator correspods to the value for the uit if it is icluded i the sample or to the sample mea if it is ot i the sample ad may be writte as ( ) yˆ s = I{ } Yi + I{ } Yi / = s = s i= = where I { = s} deotes a idicator fuctio. This ad hoc estimator may be expressed i terms of the elemets of the expaded radom vector Y but ot i terms of the collapsed radom variables Y i. However, it is a o-liear fuctio of Y, suggestig that beyod the eed of keepig track of both labels ad values attached to the uits i the populatio for which we wat to draw iferece, a broader class of estimators is eeded to obtai such a result. Oe way the oliearity ca be avoided is to defie a exteded set of radom variables, beyod those proposed i this paper. Curret research is uderway to ivestigate such expaded sets, ad use them to develop liear predictors of specific uits. 8

21 Ackowledgemets The authors are grateful to the Coselho acioal de Desevolvimeto Cietífico e Tecológico (CPq), Fudação de Amparo à Pesquisa do Estado de ão Paulo (FAPEP), FIEP (PROEX), Coordeação de Aperfeiçoameto de Pessoal de ível uperior (CAPE), Brazil ad to the atioal Istitutes of Health (IH-PH-R0-HD36848), UA, for fiacial support. The authors also wish to thak Dalto Adrade, Heleo Bolfarie, Joh Buoaccorsi, ad Oscar Loureiro for helpful commets that lead to improvemets i the mauscript. The authors also gratefully ackowledge helpful commets by referees that have lead to improvemets i the paper. 9

22 REFERECE Basu, D. (97). A essay o the logical foudatios of survey samplig, Part. I: V.P. Godambe ad D.A. prott, Eds., Foudatios of tatistical Iferece, Holt, Riehart ad Wisto, Toroto, Bellhouse, D.R. ad Rao, J..K. (986). O the efficiecy of predictio estimators i two-stage samplig, J. tat. Pla. Iferece, 3, Bolfarie, H. ad Zacks,. (992). Predictio Theory for Fiite Populatios. priger-verlag, ew York. Brewer, K.R.W., Haif, M., ad Tam,.M. (988). How early ca model-based predictio ad desig-based estimatio be recociled?, J. Am. tat. Assoc. 83, Brewer, K.R.W. (999). Desig-based or predictio-based iferece? tratified radom vs stratified balaced samplig, It. tat. Rev., 67, Brewer, K.R.W. (2002). Combied urvey amplig Iferece. Oxford Uiversity Press, ew York. Cassel, C.M., ärdal, C.E. ad Wretma, J.H. (977). Foudatios of Iferece i urvey amplig. Wiley, ew York. Fuller, W. A. ad Battese, G.E. (973). Trasformatios for estimatio of liear models with ester-error structure. Am. tat. Assoc. 68, Godambe, V.P. (955). A Uified Theory of amplig from Fiite Populatios. J. Roy. tatist. oc. er. B 7,

23 Godambe, V.P. ad Joshi, V.M. (965). Admissibility ad Bayes estimatio i samplig from fiite populatios:i. A.Math.tatist. 26, Graybill, F.A. (983) Matrices with Applicatios i tatistics. Wadsworth Iteratioal Group, Belmot, Calif. Hartley, H.O. ad Rao, J..K. (968). A ew Estimatio Theory for ample urveys. Biometrika 55, Hartley, H.O. ad Rao, J..K. (969). A ew Estimatio Theory for ample urveys, II. I ew Developmets i urvey amplig, Godambe ad prott (Eds), ew York: Wiley Itersciece, Hedayat,A. ad iha, B.K. (99). Desig ad Iferece i Fiite Populatio amplig. Wiley, ew York. Horvitz, D.G. ad Thompso, D.J. (952). A geeralizatio of samplig without replacemet from fiite populatio. J. Am. tat. Assoc. 47, Kempthore, O. (952). Desig ad Aalysis of Experimets. Wiley, ew York. Lecia, V.B. (2002). Modelos de Efeitos Aleatórios E Populações Fiitas, Ph.D. dissertatio i the Departmet of tatistics, Uiversity of ao Paulo, ao Paulo, Brazil. Li, W. (2002). Use of Radom Permutatio Models i Rate Estimatio ad tadardizatio. Ph.D dissertatio i the Departmet of Biostatistics ad Epidemiology, Uiversity of Massachusetts, Amherst, Massachusetts. 2

24 Mukhopadhyay, P. (984). Optimum estimatio of a fiite populatio variace uder geeralized radom permutatio models. Calcutta tat. Assoc. Bull., 33, Mukhopadhyay, P. (200). Topics i urvey amplig, priger Lecture otes i tatistics 53, ew York. eyma, J. (934). O the two differet aspects of the represetatio method: the method of stratified samplig ad the method of purposive selectio. J. Roy. tatisti. oc. 97, eyma, J.K. Iwaszkiewicz, et al. (935). tatistical problems i agricultural experimetatio. Roy. tatisti. oc. 2 (upplemet) Padmawar, V.R. ad Mukhopadhyay, P. (985). Estimatio uder two-stage radom permutatio models. Metrika 25, Rao, J..K. ad Bellhouse, D.R. (978). Optimal estimatio of a fiite populatio mea uder geeralized radom permutatio models. J. tat. Pla. Iferece 2, Rao, J..K. (997). Developmets i ample urvey Theory: A Appraisal. Ca. J. tat. 25, - 2. Rao, J..K. (999a). ome Curret Treds i ample urvey Theory ad Methods. akhyã B, 6, -25. Rao, J..K. (999b). ome recet advaces i model-based small area estimatio, urv.meth. 25, Rao, T.J. (984). ome aspects of radom permutatio models i fiite populatio samplig theory, Metrika, 3,

25 Royall, R.M. (969). A Old Approach to Fiite Populatio amplig Theory. J. Am. tat. Assoc. 63, Royall, R.M. (976). The Liear Least-quares Predictio Approach to Two-tage amplig. J. Am. tat. Assoc. 7, ärdal, C-E, wesso, B., ad Wretma, J. (992). Model Assisted urvey amplig. priger-verlag, ew York. earle,.r. (982). Matrix Algebra Useful for tatistics. Joh Wiley, ew York. cott, A. ad mith, M.F. (969). Estimatio i Multi-tage urveys. J. Am. tat. Assoc. 64, taek, E.J. III ad iger, J.M. (2002). Predictig Realized Radom Effects with Clustered amples from Fiite Populatios with Respose Error. Upublished: ( taek, E.J. III ad iger, J.M. (2002). Predictig Realized Cluster Parameters from Two tage ample of Uequal ize Clustered Populatios. Upublished: ( Thompso, M.E. (997). Theory of ample urveys. Chapma ad Hall, Lodo. Valliat, R., Dorfma, A.H. ad Royall, R.M. (2000). Fiite Populatio amplig ad Iferece: A Predictio Approach Joh Wiley, ew York. 23

26 Appedix A: Optimal estimators We solve (4.8) for C ad the miimize the GME with respect to that matrix. First, ote that (4.8) ca be re-expressed as XC = XL. I geeral for fixed matrices A ad B, the set of solutios to = AW B is give by = + ( ) W A B I A A Z where A is a specific g-iverse of A ad Z is a arbitrary matrix (as defied by Graybill (983)). We make repeated use of this result i obtaiig the solutio. ettig = X I, all solutios that satisfy the costrait for ubiasedess are give by J C = I + ( ) L I P Z, (A.) where Z p is a arbitrary matrix. Whe LY is o-stochastic, (as i (4.), (4.2) ad (4.3)), the GME i (4.6) simplifies to which is a fuctio of Z. Defiig ( ) GME = Var CY = CV C, p p p p c = Z (A.2) p ad J = a I L p, the GME simplifies to ( ) 2 ( ) GME = ava + c P c+ c P a. (A.3) Differetiatig (A.3) with respect to c ad settig the resultig derivatives equal to zero yields ( ) ˆ = ( ) P c P a. ice give as P is orthogoal to ( ) P a= 0 ad the solutios are J, ( ) c ˆ = I PP r, (A.4) 24

27 where r is a arbitrary vector. We replace c by (A.4) i equatio (A.2), ad solvig for ˆ Z, results i Z ˆ p = p ( ) + p p p r I PP PT, (A.5) where = p p p, ad T is a arbitrary matrix. ubstitutig (A.5) ito (A.) ad simplifyig, ˆ J C= L I + p ( ) + p p ( ) r p I P P T I P, (A.6) where both r ad T are arbitrary, ad is a g-iverse of (3.3). To defie, we first let m be the umber of values of y, =,..., that are ozero. Furthermore, let y represet a vector with elemets equal to y if y 0, ad zero otherwise. Fially, let i y represet a vector with elemets equal to oe if y 0, ad zero otherwise. With such defiitios, we defie a g-iverse of as = ( ) + D D y y m yy. (A.7) Pre-multiplyig this expressio by yields = D. ubstitutig this expressio ito (A.6), i y ˆ J C= L I + p + p p ( ) r p Dy 0 P P T I P (A.8) where D = I D, a diagoal matrix with diagoal elemets equal to zero for diagoal y0 iy elemets with y 0, ad oe for diagoal elemets with y = 0. The geeral result give by (A.8) ca be simplified by otig that for all Y, r D y 0 P Y = 0. otig that the GME will ot chage with differet choices of the 25

28 arbitrary vector r, elimiatig the term that depeds o r will ot alter the predictor or the GME, ad simplifies the result. As a result, optimal estimators ca be costructed usig ˆ J C= L I + p p ( ) P T I P. (A.9) 26