Conditional and unconditional models in modelassisted estimation of finite population totals

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1 Unversty of Wollongong Research Onlne Faculty of Informatcs - Papers Archve) Faculty of Engneerng and Informaton Scences 2011 Condtonal and uncondtonal models n modelasssted estmaton of fnte populaton totals Davd G. Steel Unversty of Wollongong, dsteel@uow.edu.au Robert Graham Clark Unversty of Wollongong, rclark@uow.edu.au Publcaton Detals Steel, D. G. & Clark, R. Graham. 2011). Condtonal and uncondtonal models n model-asssted estmaton of fnte populaton totals. Pakstan Journal of Statstcs, 27 4), Research Onlne s the open access nsttutonal repostory for the Unversty of Wollongong. For further nformaton contact the UOW Lbrary: research-pubs@uow.edu.au

2 Condtonal and uncondtonal models n model-asssted estmaton of fnte populaton totals Abstract The well known Godambe-Josh lower bound for the antcpated varance of desgn unbased estmators of populaton totals treats the auxlary varables as constants. We extend the result to models where these varables are random and show that the generalzed dfference estmator usng the expected values condtonal on all auxlary values s optmal. Ths has several mplcatons ncludng the fact that collectng multple survey varables does not reduce the lower bound. Dscplnes Physcal Scences and Mathematcs Publcaton Detals Steel, D. G. & Clark, R. Graham. 2011). Condtonal and uncondtonal models n model-asssted estmaton of fnte populaton totals. Pakstan Journal of Statstcs, 27 4), Ths journal artcle s avalable at Research Onlne:

3 Pak. J. Statst Vol. 274), CONDITIONAL AND UNCONDITIONAL MODELS IN MODEL-ASSISTED ESTIMATION OF FINITE POPULATION TOTALS Davd G. Steel and Robert G. Clark Centre for Statstcal and Survey Methodology Unversty of Wollongong Australa ABSTRACT The well known Godambe-Josh lower bound for the antcpated varance of desgn unbased estmators of populaton totals treats the auxlary varables as constants. We extend the result to models where these varables are random and show that the generalzed dfference estmator usng the expected values condtonal on all auxlary values s optmal. Ths has several mplcatons ncludng the fact that collectng multple survey varables does not reduce the lower bound. KEY WORDS Cluster samplng; Generalzed regresson estmator; Optmal estmaton; Probablty samplng AMS Classfcaton 62D05 1. INTRODUCTION 1.1 Inferental Frameworks for Survey Samplng Ths artcle extends a well known result relatng to the best possble estmator of a fnte populaton total accordng to a partcular crtera, the antcpated varance. Consder a populaton U consstng of N unts, where unts may be people, households, busnesses or other enttes. A sample s U s selected from U usng some probablty samplng method, wth = P s denotng the probablty of selecton for unt U. A varable of nterest Y s defned for each unt U. The am s to estmate Y = Y usng data on Y for sampled unts s only. We denote the vector of Y for s by Y s. A vector of auxlary varables X may also be avalable and observed for all unts U. We wrte X U and Y U to denote the collecton of all of these varables for all unts n the populaton. For example, the am may be to measure the total employment for workng age adults n a country. In ths case, the unts would be people and the populaton U would be all workng age adults. The varable Y would be defned to equal 1 f person s employed and 0 otherwse. A survey would be admnstered to determne the employment status of all people n a sample s. There s often external nformaton on age and sex for all people n the populaton, so the elements of X would be ndcator varables ndcatng the age and sex of person. The objectve s to combne nformaton on Y for s wth X U to gve the best estmator of Y. c Pakstan Journal of Statstcs 529

4 530 Condtonal and Uncondtonal Models A number of frameworks have been used n sample survey theory, ncludng the desgnbased, model-based and model-asssted frameworks. In the desgn-based framework, X U and Y U are regarded as constants, and the only source of randomness s the selecton of the sample s. Ths framework avods any explct modellng of X U and Y U. The desgn expectaton of an estmator s defned to be ts expected value over all possble samples, wth X U and Y U treated as constants. Estmators of Y are requred to be exactly or approxmately desgn-unbased, that s ther desgn expectaton expectaton over all possble samples) must be exactly or approxmately equal to Y. For a dscusson of the desgnbased framework, see for example Cochran 1977) and Sarndal, Swensson, and Wretman 1992, ch.1-2). Desgn-varances are smlarly defned to be the varances over the possble values of s, wth X U and Y U treated as constants. In the model-based, or predcton framework, a model s postulated for Y U condtonal on X U. Estmators are usually evaluated n terms of the expectaton condtonal on X U and on the partcular sample s selected, so that the only sources of randomness are the populaton values Y U. Expectatons condtonal on s and X U are called model-expectatons; model-varances are defned smlarly. Predctors of Y are generally requred to be modelunbased, n the sense that the model expectaton of the estmator mnus Y should equal zero, and to have low model varance. For a dscusson of the model-based framework, see for example Brewer 1963), Royall 1970) and Vallant, Dorfman, and Royall 2000). In the model-asssted approach to estmatng a fnte populaton total, both probablty samplng methods and populaton models have a role Sarndal et al., 1992, pp.227, ). Both s and Y U are regarded as random, wth a model of some knd beng assumed for Y U. X U are regarded as constants. Estmators are requred to be at least approxmately) desgn unbased, that s unbased over repeated samplng condtonal on Y U and X U. Subject to ths constrant, t s desrable to reduce the varance over both repeated samplng and repeated realsatons of Y U based on an assumed model. The most commonly used estmator n the model-asssted framework s the generalzed regresson estmator. Most estmators of populaton totals used n practce are specal cases of the generalzed regresson estmator. Sarndal et al. 1992) contans an extensve dscusson of ths estmator and the model-asssted framework. 1.2 Optmal Estmaton The exstence of optmal estmators has been nvestgated for all of these frameworks. In the desgn-based framework, an optmal estmator of Y should deally mnmze the desgn-varance n the class of all desgn-unbased estmators. However, t has been shown that there s no such estmator Godambe, 1955). One proof of ths result Basu, 1971) shows that for every populaton, there s n theory a desgn-unbased estmator of a total whch has zero varance for that partcular populaton, but there s no estmator whch has the lowest varance for all populatons. As noted n Basu, 1971), ths result s not usable n practce because t requres perfect knowledge of the populaton values, but t nevertheless provdes a useful benchmark.) The desgn-based framework s therefore not sutable to gve gudance on best estmators wthout restrctng the range of populatons beng consdered. Ths can be done nformally, for example rato estmators have lower desgn varance than the smpler number-rased estmator for populatons satsfyng a smple condton, for smple random samplng Cochran, 1977, p.157).

5 D.G. Steel and R.G. Clark 531 The non-exstence of an optmal desgn-based estmator of Y was one of the motvatons for the development of the model-asssted framework. In ths framework, estmators are stll requred to be exactly or approxmately desgn-unbased. Unlke the desgn-based framework, however, the am s to mnmze the model expectaton of the desgn varance, under an assumed model for Y U. Ths quantty, whch has been called the antcpated varance Isak & Fuller, 1982), can be mnmzed for desgn-unbased estmators Godambe & Josh, 1965). An estmator wth ths optmalty property would be desgn-unbased regardless of the correctness of the populaton model, and would be optmal n the sense of mnmsng the antcpated varance provded the model s true. Consder the followng model for Y U : E Y = µ vary = σ 2 Y,Y j ndependent j Desgn-unbased estmators Ŷ satsfy the nequalty var Ŷ Y 1 ) σ 2 2) where varance s over both repeated samplng and repeated realsatons of Y U Godambe & Josh, 1965). The estmator whch meets the lower bound s Ŷ µ = µ + Y µ ). 3) The estmator Ŷ µ generally cannot be calculated n practce, because the values of µ and ther sum over the populaton would be unknown n almost all cases. The usual strategy n practce s to assume that µ are known functons of X U and some unknown parameters. These parameters can then be estmated from sample data, and estmates ˆµ can then be used n place of µ n 3). Estmators formed n ths way may have varance whch tends to the lower bound 2) as the sample sze and populaton sze both tend to nfnty, gven a correct model for {µ }. For example, the generalzed regresson estmator has ths property Sarndal, 1980). Estmators wth ths property can be called asymptotcally optmal. Ths paper extends result 2) to the case where the auxlary varables X U are random varables, for certan types of desgns. Secton 2 contans a theorem for ths case. The theorem allows the possblty that multple survey varables are collected and also allows a model wth dependences between dfferent values of the survey varable, subject to a restrcton. A dfference estmator smlar to 3) s shown to be optmal, but wth µ replaced by E Y X U. Hence the apparently arbtrary treatment of X U as constants by Godambe and Josh 1965) s reasonable, as the same estmator s optmal under a more general model. Secton 3 gves four mplcatons of ths result and Secton 4 summarses the conclusons. 2. A THEOREM ON THE AV UNDER AN UNCONDITIONAL MODEL Ths secton contans a theorem gvng a lower bound for the antcpated varance AV) of desgn-unbased estmators of Y, and an optmal estmator of Y. The theorem allows for the possblty that other survey varables, Z, are observed for s. 1)

6 532 Condtonal and Uncondtonal Models The theorem requres that the samplng scheme s non-nformatve. A samplng scheme s sad to be non-nformatve f ps) = Ps s selected Y U,Z U,X U = Ps s selected X U,.e. the samplng process depends only on X U, whch s known at the tme of desgn, and not on the survey varables Y U and Z U whch are collected durng the survey. The populaton X U,Y U,Z U wll be assumed to be generated by a model such that E Y X U = µ = µ X U ) vary X U = σ 2 = σ 2 X U) Note that the expectatons n 4) are model-expectatons. Ths model s extremely general because t allows the means and varances of Y condtonal on X U to be any functons of X U. In practce, {µ } would be assumed to be known functons of X U and some unknown parameters. The unknown parameters would need to be estmated from sample data, so that the lower bound n the Theorem could not be perfectly acheved n practce. Theorem 1 Suppose that a varable Y and a number of other varables Z are observed for s. It s assumed that samplng s non-nformatve. The populaton X U,Y U,Z U s assumed to be generated by model 4). It s further assumed that Y,Z ) s condtonally ndependent of Y j,z j ) gven X U for all j such that there s at least one sample s S wth ps) > 0 where s and j s. Let Ŷ = Ŷ s,x U,D s ) be a desgn unbased estmator of Y so that E Ŷ X U,Y U,Z U = Y where D s denotes the sample data Y s,z s ). Defne Ŷ µ by Then Ŷ µ = } 4) Y µ X U )+ µ X U ) 5) var Ŷ Y var Ŷ µ Y. 6) If Y and Y j are condtonally ndependent gven X U for all j, then var Ŷ µ Y XU = 1 ) σ 2 X U ) 7) var Ŷ µ Y 1 ) σ 2 X U). 8)

7 D.G. Steel and R.G. Clark 533 See Appendx for proof. The proof s smlar to Godambe and Josh 1965) except that the condtonng on X U has been explctly stated. Wth X U a random varable, we have obtaned essentally the same result as Godambe and Josh 1965), provded that Ŷ = Ŷ s,x U,D s ). The condton regardng ndependence of Y,Z ) and Y j,z j ) allows for these to be dependent for some j provded that there s never a dependency between values n the sample and values not n the sample. Ths s s a generalzaton of the condton n Godambe and Josh 1965), where Y was assumed to be ndependent of Y j for every j. The generalzaton allows the result to be appled to one-stage cluster samplng where there are dependences between Y,Z ) and Y j,z j ) for and j n the same cluster, provded that observatons from dfferent clusters are ndependent. The result does not cover multstage samplng, because n ths case there may be correlatons between the values of the sampled and unsampled unts from each prmary samplng unt. However, t would cover most other desgns, ncludng stratfed sngle-stage samplng. There s no requrement n Theorem 1 or n Godambe and Josh 1965) for the estmator Ŷ to be lnear. 3. IMPLICATIONS OF THE THEOREM No Value n Modellng X U The values X U may follow some complex model, for example there may be nterestng dependences between the elements of X, or between the elements of X and X j. The theorem shows that Ŷ µ s the optmal estmator regardless of the dstrbuton of X U. The values of E Y X U are requred to calculate Ŷ µ, and generally modellng of Y condtonal on X U s needed to estmate E Y X U. There s no mprovement n the lower bound on the AV of estmators of Y from consderng the margnal dstrbuton. Hence no beneft from knowng ths margnal dstrbuton would be expected, at least for suffcently large samples. It should be noted that the Theorem states a lower bound whch s generally only attaned asymptotcally. In practce, the model for E Y X U must be estmated from sample data, so that for small samples there could be some beneft from consderng the margnal dstrbuton of X U. Multple Survey Varables It s clear from Theorem 1 that consderng several survey varables Y and Z together does not reduce the lower bound for estmates of Y. The same AV can be acheved by modellng one at a tme each varable collected n the survey. The practcal mplcaton s that there s no large sample reducton n the AV f multvarate models for several survey varables condtonal on X U are used. Agan, t should be noted that the lower bound s generally only attaned asymptotcally, so t s possble that multvarate models could gve some beneft for small samples. Ŷ µ s Stll Optmal under Cluster Samplng In one-stage cluster samplng, a sample of clusters of unts s selected, and all unts n these clusters are selected. In ths case, t would be usual to assume that there may be dependences between the values of the survey varables for unts n the same cluster,

8 534 Condtonal and Uncondtonal Models but no dependences across clusters. In ths stuaton, Theorem 1 shows that Ŷ µ s stll the optmal estmator. Hence modellng the dependences wthn the cluster cannot gve any mprovement n estmators of Y, at least for large samples. One practcal example of onestage cluster samplng mght be all people n selected households, where households are selected by telephone samplng. In mult-stage samplng, a sample of frst stage unts e.g. suburbs) s selected, followed by a sample of second stage unts e.g. households) from selected frst stage unts, followed by a sample of thrd stage unts e.g. from selected second stage unts, and so on. Onestage cluster samplng s a smple specal case of mult-stage samplng, but n practce more complex mult-stage samples are often used. Theorem 1 would not generally apply n ths case, because there could be correlatons between sampled and non-sampled unts. In some mult-stage household surveys, the fnal stage of selecton conssts of all people n selected households. In ths case, the correlatons across households may be neglgble compared to the correlatons between people n the same household, so that Theorem 1 may apply at least approxmately. Mukerjee and Sengupta 1989) derved the optmal lnear model-asssted estmator for correlated data, whch could potentally be appled to mult-stage samples n general. Auxlary Data for All Unts Should Be Used The lower bounds n Theorem 1 are acheved by the estmator Ŷ µ n 5) wth µ Y X U whch condtons on the values of the auxlary varables for all unts n the populaton. Ths may be dfferent from E Y X whch s wdely used n motvatng estmators, for example the generalzed regresson estmator Sarndal, 1980) and the nonlnear estmators of Frth and Bennett Frth & Bennett, 1998) and Lehtonen and Vejanen Lehtonen & Vejanen, 1998). In other words, t s usually assumed that the expectaton of Y condtonal on X U does not depend on X j for j. Ths may not be reasonable for populatons wth natural clusterng or a herarchcal structure Steel & Welsh, 2006). Cluster samplng or stratfed cluster samplng) of all people n selected households may be one case where E Y X U E Y X. In some household surveys, X U conssts of demographc data. A person s survey varable could depend on ther own demographc characterstcs and those of other household members, because the latter could ndcate famly or lvng arrangements. Models of ths knd are called contextual models; see for example Kreft and De Leeuw 1998). Consder a regresson estmator based on E Y X, Ŷ A = Y µ )+ µ where µ Y X. Ths s not the same as Ŷ µ, because Ŷ µ uses µ Y X U. If t can be assumed that there are no correlatons across dfferent households, then Theorem 1 means that the AV of Ŷ A must be greater than or equal to the AV of Ŷ µ. To llustrate the possble loss from usng Ŷ A nstead of the optmal estmator, consder the specal case

9 D.G. Steel and R.G. Clark 535 when: are fxed constants not dependng on X U ; X are ndependently and dentcally dstrbuted for U; {Y : U} are ndependent condtonal on X U ; {X,Y ) : U} are dentcally dstrbuted; the sample sze, n, s non-random so that n n =. It s further assumed that σ 2 = vary X U s equal to a constant, V y xu, for all U, and that vary X s equal to a constant V y x for all U. From Theorem 1, var Ŷ µ Y 1 ) σ 2 X U ) The AV of Ŷ A s gven by = V y xu 1 ). var Ŷ A Y var Ŷ A Y s + var E Ŷ A Y s var Ŷ A Y s + 0 var V y x { 1 ) Y µ ) Y µ ) s s 1 ) 2 + N n } = V y x { 1 ) } 2 + N n = V y x { = V y x Hence the neffcency of Ŷ A s 1 ) 1 ) } ) var Ŷ A Y /var Ŷ µ Y = V y x /V y xu. So the rato of the AVs s equal to the rato of the varance of Y condtonal on X to the varance of Y condtonal on X U, whch depends on the predctve power of X U for Y gven X. 4. SUMMARY The Godambe-Josh lower bound, whch defnes optmal model-asssted estmators of total, treats the auxlary varables as known constants. It s applcable when the values of the survey varable are ndependent for dfferent unts. Theorem 1 states a smlar lower bound whch allows the auxlary varables to be random, allows for multple survey varables, and allows for correlatons between values for dfferent unts provded that sample and non-sample values are always ndependent). Ths leads to four new conclusons: Modellng the margnal dstrbuton of the auxlary varables cannot gve any mprovement n the asymptotc AV of desgn-unbased estmators of Y as the sample and populaton sze tend to nfnty.

10 536 Condtonal and Uncondtonal Models If several survey varables are collected for the sample, t does not reduce the lower bound f all are consdered together. Therefore multvarate modellng of several survey varables has no beneft for the asymptotc AV. The optmal estmator s applcable to one-stage cluster samplng even f there are dependences between values of the survey varables for dfferent unts n the same cluster. However the estmator s not optmal for more complex cases of mult-stage samplng whch are often used n practce. Condtonal models for the varable of nterest for a unt should condton on the auxlary data for all populaton unts, not just the auxlary data for that unt. Ths can make a dfference for contextual models n cluster samplng. Acknowledgements Ths work was jontly supported by the Australan Research Councl and the Australan Bureau of Statstcs. REFERENCES 1. Basu, D. 1971). An essay on the logcal foundatons of survey samplng, part 1. In R. Holt & Wnston Eds.), Foundatons of Statstcal Inference pp ). Toronto. 2. Brewer, K. R. W. 1963). Rato estmaton and fnte populatons: some results deductble from the assumpton of an underlyng stochastc process. Australan Journal of Statstcs, 5, Cochran, W. G. 1977). Samplng Technques 3rd ed.). New York: Wley. 4. Frth, D., & Bennett, K. 1998). Robust models n probablty samplng wth dscusson). Journal of the Royal Statstcal Socety Seres B, 601), Godambe, V. 1955). A unfed theory of samplng from fnte populatons. Journal of the Royal Statstcal Socety Seres B, 17, Godambe, V., & Josh, V. 1965). Admssblty and Bayes estmaton n samplng fnte populatons 1. Annals of Mathematcal Statstcs, 36, Isak, C., & Fuller, W. 1982). Survey desgn under the regresson superpopulaton model. Journal of the Amercan Statstcal Assocaton, 77377), Kreft, I., & De Leeuw, J. 1998). Introducng Multlevel Modelng. London: Sage. 9. Lehtonen, R., & Vejanen, A. 1998). Logstc generalzed regresson estmators. Survey Methodology, 241), Mukerjee, R., & Sengupta, S. 1989). Optmal estmators of fnte populaton total under a general correlated model. Bometrka, 764), Royall, R. M. 1970). On fnte populaton samplng theory under certan lnear regresson models. Bometrka, 572), Sarndal, C. 1980). On nverse weghtng versus best lnear unbased weghtng n probablty samplng. Bometrka, 673),

11 D.G. Steel and R.G. Clark Sarndal, C., Swensson, B., & Wretman, J. 1992). Model Asssted Survey Samplng. New York: Sprnger-Verlag. 14. Steel, D., & Welsh, A. 2006). Jont models for clustered data. Preprnt). Unversty of Wollongong School of Mathematcs and Appled Statstcs. 15. Vallant, R., Dorfman, A. H., & Royall, R. M. 2000). Fnte Populaton Samplng and Inference: A Predcton Approach. New York: Wley. APPENDIX: PROOFS Let Ŷ = Y. Ths s the Horvtz-Thompson or nverse probablty estmator whch s exactly desgn unbased e.g. Sarndal et al., 1992). Let S be the set of all possble samples s. Lemma: Under the assumptons stated n Theorem 1, ps)cov Ŷ Y,Ŷ Ŷ s,xu = 0. Proof of Lemma: ps)cov Ŷ Y,Ŷ Ŷ s,xu = ps)e Ŷ Y E Ŷ Y )Ŷ ) s,xu Ŷ s,xu = ps)e = ps)e = ps)e Ŷ ) Y µ ) Y µ )) Ŷ s,x U 1 ) Y µ ) 1 ) ) Y µ X U )) s Ŷ s,ds,x U ) Ŷ s,d s,x U ) ) s,xu Y µ )) Ŷ Ŷ ) s,x U ) ps)e Y µ X U )) s Ŷ s,ds,x U ) Ŷ s,d s,x U ) ) s,x U 9) The second term of 9) s zero because Z,Y ) and Z j,y j ) are ndependent for all s and j s condtonal on X U, so that Y s condtonally ndependent of Ŷ s,ds,x U ) Ŷ s,d s,x U ) )

12 538 Condtonal and Uncondtonal Models for s. So 9) becomes ps)cov Ŷ Y,Ŷ Ŷ s,xu = ps)e 1 ) Y µ X U )) ) Ŷ s,ds,x U ) Ŷ s,d s,x U ) ) s,xu 10) Now, ps) s the condtonal probablty that sample s s selected gven X U. Hence ps)e 1 ) Y µ X U )) ) Ŷ s,ds,x U ) Ŷ s,d s,x U ) ) s,xu 1 ) Y µ X U )) ) Ŷ s,ds,x U ) Ŷ s,d s,x U ) ) X U and so 10) becomes ps)cov Ŷ Y,Ŷ Ŷ s,x U E 1 ) Y µ X U )) ) s Ŷ s,ds,x U ) Ŷ s,d s,x U ) ) XU 1 ) Y µ X U )) ps) Ŷ s,ds,x U ) Ŷ s,d s,x U ) ) X U,Y U,Z U X U 1 ) Y µ X U )) Ŷ s,ds,x U ) Ŷ s,d s,x U ) ) XU ps) 1 ) Y µ X U )) Ŷ s,ds,x U ) Ŷ s,d s,x U ) ) X U 11) where denotes summaton over all samples s whch contan unt. By assumpton, Ŷ s s desgn unbased so ps) Ŷ s,d s,x U ) Ŷ s,d s,x U ) ) = 0

13 D.G. Steel and R.G. Clark 539 and therefore for any U so that for any U. Substtuton nto 11) gves 0 = ps) Ŷ s,d s,x U ) Ŷ s,d s,x U ) ) s + ps) Ŷ s,d s,x U ) Ŷ s,d s,x U ) ) s ps) Ŷ s,d s,x U ) Ŷ s,d s,x U ) ) s = ps) Ŷ s,d s,x U ) Ŷ s,d s,x U ) ) s ps)cov Ŷ Y,Ŷ Ŷ s,xu = E = E = E s E ps) 1 ) Y µ X U )) s s Ŷ s,ds,x U ) Ŷ s,d s,x U ) ) XU ps) 1 ) Y µ X U )) Ŷ s,ds,x U ) Ŷ s,d s,x U ) ) s,x U X U ps) 1 ) E Y µ X U )) Ŷ s,ds,x U ) Ŷ s,d s,x U ) ) s,x U X U. 12) From the assumptons of the theorem, Y and Y j are condtonally ndependent whenever s and j s for all samples s wth ps) > 0. Hence Y s condtonally uncorrelated wth Ŷ s,ds,x U ) Ŷ s,d s,x U ) ) for all samples s wth ps) > 0. Hence the rght hand sde of 12) s zero. Proof of Theorem 1: var Ŷ Y XU var Ŷ Y Ŷ s,xu XU + var E Y s,xu XU E var Ŷ Y s,x U X U = ps)var Ŷ Y s,xu

14 540 Condtonal and Uncondtonal Models = ps)var Ŷ Ŷ +Ŷ Y s,xu = ps) { var Ŷ Ŷ s,xu + var Ŷ Y s,xu +2cov Ŷ Ŷ,Ŷ Y } s,x U ps)var Ŷ Y s,xu +2 ps)cov Ŷ Ŷ,Ŷ Y s,xu 13) The Lemma states that the second term of 13) s zero. Also, Ŷ s equal to Ŷ µ plus terms dependng only on s and X U and hence constant condtonal on s and X U. Hence var Ŷ Y s,x U = var Ŷµ Y s,x U. Makng both of these substtutons nto 13) gves: Inequalty 14) mples that var Ŷ Y X U ps)var Ŷ µ Y s,x U 14) var Ŷ Y var Ŷ Y Ŷ XU + var E Y XU E var Ŷ Y XU E ps)var Ŷ µ Y s,xu. 15) Notce that Ŷ µ s unbased condtonal on s and X U : E Ŷ µ Y s,x U = Y µ )+ µ µ )+ µ Y s,x U µ µ = 0. Hence var Ŷ µ Y var Ŷ µ Y XU,s + var E Ŷ µ Y XU,s var Ŷ µ Y X U,s + var0 var Ŷ µ Y XU,s ps)var Ŷ µ Y s,xu 16) Substtutng 16) nto 15) gves whch s result 6) of the Theorem. var Ŷ Y var Ŷ µ Y

15 D.G. Steel and R.G. Clark 541 If Y and Y j are condtonally ndependent gven X U for all j, then the condtonal varance of Ŷ µ Y s var Ŷ µ Y XU var Y µ )+ µ Y s,x U X U +var0 X U = = var 1 ) Y Y s,x U X U s σ 2 X U 1 ) 2 σ 2 + s 1 ) 2 σ )σ 2 { 1 ) }σ 2 = 1 ) σ 2 whch s result 7) of the Theorem. The uncondtonal varance of Ŷ µ Y s var Ŷ µ Y var Ŷ µ Y XU + var E Ŷµ Y XU 1 ) σ 2 + var0 1 ) σ 2 whch s result 8).