On Robust Small Area Estimation Using a Simple. Random Effects Model

Transcription

1 On Robust Small Area Estmaton Usng a Smple Random Effects Model N. G. N. PRASAD and J. N. K. RAO 1 ABSTRACT Robust small area estmaton s studed under a smple random effects model consstng of a basc (or fxed effects) model and a lnkng model that treats the fxed effects as realzatons of a random arable. Under ths model a model-asssted estmator of a small area mean s obtaned. Ths estmator depends on the surey weghts and remans desgnconsstent. A model-based estmator of ts mean squared error (MSE) s also obtaned. Smulaton results suggest that the proposed estmator and Kott s (1989) model-asssted estmator are equally effcent, and that the proposed MSE estmator s often much more stable than Kott s MSE estmator, een under moderate deatons of the lnkng model. The method s also extended to nested error regresson models. KEY WORDS: Desgn consstent; Lnkng model; Mean squared error; Surey weghts. 1 N. G. N. Prasad, Department of Mathematcal Scences, Unersty of Alberta, Edmonton, Alberta, T6G 2G1; J. N. K. Rao, Department of Mathematcs and Statstcs, Carleton Unersty, Ottawa, Ontaro, K1S 5B6.

2 1. INTRODUCTION Unt-leel random effects models are often used n small area estmaton to obtan effcent model-based estmators of small area means. Such estmators typcally do not make use of the surey weghts (e.g., Ghosh and Meeden 1986; Battese, Harter and Fuller 1988; Prasad and Rao 1990). As a result, the estmators are not desgn consstent unless the samplng desgn s self-weghtng wthn areas. We refer the reader to Ghosh and Rao (1994) fro an apprasal of small area estmaton methods. Kott (1989) adocated the use of desgn-consstent model-based estmators (.e., model asssted estmators) because such estmators prode protecton aganst model falure as the small area sample sze ncreases. He dered a desgn-consstent estmator of a small area mean under a smple random effects model. Ths model has tow components: the basc (or fxed effects) model and the lnkng model. The basc model s gen by y j =θ +e j, j=1,2,...,n ; =1,2,...,m (1) where the y j are the sample obseratons and the e j are uncorrelated random errors wth mean zero and arance σ 2 for each small area (=1,2,...,m). For smplcty, we take θ as the small area mean Y = y / N, where N s the number of populaton unts n the -th area. Note that Y j j = θ + E and E = e 0 f N s large. j j

3 the model The lnkng model assumes that θ s a realzaton of a random arable satsfyng θ =µ+ (2) where the are uncorrelated random arables wth mean zero and arance σ 2. Further, { } and {e j } are assumed to be uncorrelated. Assumng that the model (1) also holds for the sample {y j, j=1,2,...,n ; =1,2,...,m} and combnng the sample model wth the lnkng model, Kott(1989) obtaned the famlar unt-leel random effects model y j =µ+ +e j, j=1,2,...,n ; =1,2,...,m (3) also, called the components-of-arance model. It s customary to assume equal arances σ 2 =σ 2, although the case of random error arances has also been studed (Kleffe and Rao 1992; Arora and Lahr 1997). Assumng σ 2 =σ 2, Kott (1989) dered an effcent estmator $ θ k of θ whch s both model-unbased under (3) and desgn-consstent. He also proposed an estmator of ts mean squared error (MSE) whch s model unbased under the basc model (1) as well as desgn-consstent. But ths MSE estmator can be qute unstable and can een take negate alues, as noted by Kott(1989) n hs emprcal example. Kott (1989) used hs MSE estmators manly to compare the oerall reducton n MSE from usng $ θ k n place

4 of a drect desgn-based estmator y w gen by (4) below. He remarked that more stable MSE estmators are needed. The man purpose of ths paper s to obtan a pseudo emprcal best lnear unbased predcton (EBLUP) estmator of θ whch depends on the surey weghts and s desgn-consstent (Secton 2). A stable model-based MSE estmator s also obtaned (Secton 3). Results of a smulaton study n Secton 4 show that the proposed estmator s often much more stable than the MSE estmator of Kott, as measured by ther coeffcent of araton, een under moderate deatons of the lnkng model (2). Results under the smple model (3) are also extended to a nested error regresson model (Secton 5). 2. PSEUDO EBLUP ESTIMATOR Suppose wj ~ denotes the basc desgn weght attached to the j-th sample unt (j=1,2,...,n ) n the -th area ( =1,2,...,n). A drect desgn-based estmator of θ s then gen by the rato estmator y = w~ y / w~ = w y (4) w j j j j j where wj = w~ j / w~ j. The drect estmator yw s desgn-consstent but fals to borrow strength from the other areas. j j j j To get a more effcent estmator, we consder the followng reduced model obtaned from the combned model (3) wth σ 2 =σ 2 :

5 y = w e w j (µ + + ) =µ + + e j j w (5) where the e w are uncorrelated random arables wth mean zero and arance = σ j w j. The reduced model (5) s an area-leel model smlar to the well-known Fay-Herrot model (Fay and Herrot 1979). It now follows from the standard best lnear unbased predcton (BLUP) theory (e.g., Prasad and Rao 1990) that the BLUP estmator of θ =µ+ for the reduced model (5) s gen by where ~ θ = µ ~ + ~, (6) ~ = γ ( y µ ~ ) w w w wth ~ µ w = γ w y / γ w w and γ w = σ / ( σ + δ ). Note that θ ~ s dfferent from the BLUP estmator under the full model (3). We therefore denote θ ~ as a pseudo-blup estmator. The estmator (6) may also be wrtten as a conex combnaton of the drect estmator y w and µ ~ w : ~ θ = γ y + ( 1 γ ) µ ~. (7) w w w w The estmator ~ θ depends on the parameters σ 2 and σ 2 whch are generally unknown n practce. We therefore replace σ 2 and σ n (7) by model-consstent estmators σ$ and $ σ 2 under the orgnal unt-leel model (3) to obtan the estmator where θ$ = γ $ y + ( 1 γ $ ) µ $, (8) w w w w

6 γ $ = σ$ / ( σ$ + σ$ w j ) j w and µ $ γ $ / γ $ w = w yw w. The estmator $ θ wll be referred to as pseudo-eblup estmator. We use standard estmators of σ 2 and σ 2, based on the wthn-area sums of squares and the between-area sums of squares Q = y y w ( j ) 2 j Qb = n ( y y) 2, where y = n y / n s the oerall sample mean. We hae and $ σ 2 =max( ~ σ 2,0) where wth σ$ 2 = Q / ( n m) w ~ σ 2 =[Q b -(m-1) $ σ 2 ]/n * n * = n n 2 / n. It may be noted that σ 2 and σ 2 are ether not estmable or poorly estmated from the reduced model (5) due to dentfablty problems. Followng Kackar and Harlle (1984), t can be shown that the pseudo-eblup estmator θ s model-unbased for θ under the orgnal model (3) for symmetrcally dstrbuted errors { } and { e j }, not

7 necessarly normal. It s also desgn consstent, assumng that n 2 w j j s bounded as n ncreases, because $γ w conerges n probablty to 1 as n regardless of the aldty of the model (3), assumng $ σ 2 and $ σ 2 conerge n probablty to some alues, say, σ 2 and σ 2. Kott s (1989) model-based estmator of θ s obtaned by takng a weghted ( ) combnaton of y w and c y, that s, l l l ( ) ( ) f ( α, c ) = ( 1 α ) y w +α cl y, l l ( ) and then mnmzng the model mean squared error (MSE) of f ( α, c ) wth respect to ( ) α ι and c l subject to model-unbasedness condton: ( ) c l l =1. Ths leads to θ$ ( ) k = f ( α $,$ c ) (9) wth ( ) 2 ( ) 2 α$ / { $ / ( $ )( σ$ / σ$ = wj wj + cl n + 1+ cl )} j j l l and c$ ( ) =[( σ$ 2 / σ$ 2 )+n -1 1 ]/ [( σ$ / σ$ ) + n h ]. h The estmator $ θ k s also model-unbased and desgn-consstent. In a preous erson of ths paper, we proposed an estmator smlar to (9). It uses the best estmators of µ under

8 the unt-leel model, based on the unweghted means y l, rather than $ µ w, the best estmator of µ under the reduced model (4), based on the surey-weghted means y w. 3. ESTIMATORS OF MSE It s straghtforward to dere the MSE of the pseudo-blup estmator ~ θ under the unt leel model (3). We hae MSE( θ ~ ) = E( θ ~ θ ) 2 = g ( σ, σ ) + g ( σ, σ ) 1 2 (10) wth 2 g 1 ( σ, σ ) = ( 1 γ ) σ and g1 ( σ, σ ) σ ( 1 γ ) / γ =. The leadng term, g 1 ( σ, σ ) s of order O(1), whle the second term, g 2 ( σ, σ ) due to estmaton of µ s of order O(m -1 ) for large m. A nae MSE estmator of the pseudo-eblup estmator $ θ s obtaned by estmatng MSE( $ θ ) gen by (10): mse N ( θ $ )= g 1 ( σ$, σ $ ) + g 2 ( σ$, σ $ ) (11)

9 But (11) could lead to sgnfcant underestmaton of MSE( $ θ ) because t gnores the uncertanty assocated wth $ σ 2 and $ σ 2. Note that MSE( θ $ ) =MSE( θ ~ )+ E( $ ~ θ θ ) 2 (12) under normalty of the errors { } and {e j } so that MSE( ~ θ ) s always smaller than MSE( $ θ ); see Kackar and Harlle (1984). To get a correct estmator of MSE( $ θ ), we frst approxmate the second order term E( $ ~ θ θ ) 2 n (12) for large m, assumng that { } and {e j } are normally dstrbuted. Followng Prasad and Rao(1990), we hae ( $ ~ ) 2 g 3 ( σ, σ ) (13) E θ θ where the neglected terms are of lower order than m -1, and g 3 ( σ, σ ) = γ ( 1 γ ) 2 σ 2 { ( σ ~ 2 ) 2( σ 2 / σ 2 ) co( σ ~ 2, σ$) + ( σ 2 / σ 2 ) 2 ar( σ$ 2 )}; ((14) w w see the Appendx 1. The arances and coarances of $ σ 2 and $ σ 2 are also gen n the Appendx 1. It can be shown that g 1 ( σ$, σ $ ) + g 2 ( σ$, σ $ ) s approxmately unbased for g ( σ, σ ) n the sense that ts bas s of lower order than m -1 (see Appendx 2). 1 Smlarly, g 2 ( σ$, σ $ ) and g 3 ( σ$, σ $ ) are approxmately unbased for g 2 ( σ, σ ) and ( σ, σ ), respectely. It now follows that an approxmately model-unbased g 3 estmator of MSE( $ θ ) s gen by mse( θ $ ) = g 1 ( σ$, σ $ ) + g 2 ( σ$, σ $ ) +2 g 3 ( σ$, σ $ ). (15) For the estmator $ θ k gen by (9), Kott(1989) proposed an estmator of MSE as mse( θ $ * 2 ( ) 2 k ) = ( 1 2α$ ) ( y ) + α$ ( y c y ), (16) w w l l l

10 where * ( y w ) s both a desgn-consstent estmator of the desgn-mse of y w and a model-unbased estmator of the model-arance of y w under the basc model (1). Snce α$ conerges n probablty to zero as n, t follows from (16) that mse( θ $ k ) s also both desgn-consstent and model unbased assumng only the basc model (1). Howeer, mse( $ θ k ) s unstable and can een take negate alues when $ α exceeds 0.5, as noted by Kott (1989). Note that our MSE estmator, mse( $ θ ), s based on the full model (3) obtaned by combnng the basc model (1) wth the lnkng model (2). Howeer, our smulaton results n Secton 4 show that t may perform well een under moderate deatons from the lnkng model. 4. SIMULATION STUDY We conducted a lmted smulaton study to ealuate the performances of the proposed estmator $ θ, gen by (8), and ts estmator of MSE, gen by (15), relate to Kott s estmator $ θ k, gen by (9), and ts estmator of MSE, gen by (16). We studed the performances under two dfferent approaches: () For each smulaton run, a fnte populaton of m=30 small areas wth N =200 populaton unts n each area generated from the assumed unt-leel model and then a PPS (probablty proportonal to sze)

11 sample wthn each small area s drawn ndependently, usng n =20. () A fxed fnte populaton s frst generated from the assumed unt-leel model and then for each smulaton run a PPS sample wthn each small area s drawn ndependently, employng the fxed fnte populaton. Approach () refers to both the desgn and the lnkng model whereas approach () s desgn-based n the sense that t refers only to the desgn. The errors { } and {e j } are assumed to be normally dstrbuted n genertng the fnte populatons {y j, =1,2,,30;j=1,2,,200}. We consdered two cases: (1) The lnkng model (2) s true wth :=50. (2) The lnkng model s olated by lettng : arry across areas: : =50, =1,2,,10; : =55, =11,12,,20; : =60, =21,22,,30. To mplement PPS samplng wthn each area, sze measures z j (=1,2,...,30;j=1,2,...,200) were generated from an exponental dstrbuton wth mean 200. Usng these x- alues, we computed selecton probabltes pj = zj / j zj for each area and then used them to select PPS wth replacement samples of szes n =n, by takng n=20 and 40, and the assocated sample alues { y j } were obsered. The basc desgn weghts are gen by ~ w j = n 1 p 1 so that w = p 1 p 1 /. Usng j j j j j these weghts and the assocated sample alues y j we computed estmates $ θ and $ θ k and assocated estmates of MSE, and also the rato estmate y w for each smulaton run; the formula for * ( y w ) under PPS samplng s gen n Appendx 3. Ths process was repeated R=10,000 tmes to get from each run r(=1,2,,r) $ θ (r) and $ θ k (r) and assocated MSE estmates mse ( $ θ (r)) and mse ( $ θ k (r)) and also the drect estmate y w (r).

12 Usng these alues, emprcal relate effcences (RE) of $ θ and $ θ k oer y w were computed as and RE( θ $ ) = MSE ( y ) / MSE ( θ $ ) * w * RE( θ $ ) = MSE ( y ) / MSE ( θ $ ), k * w * k where MSE * denotes the MSE oer R=10,000 runs. For example, MSE*( θ $ ) = [ θ $ ( r) Y ( r)] / R r 2, where Y r ( ) s the -th area populaton mean for the r-th run. Note that Y r ( ) remans the same oer the runs r under the desgn-based approach because the fnte populaton s fxed oer the smulaton runs. and Smlarly, the relate bases of the MSE estmators were computed as RB[ mse( θ $ )] = MSE ( θ $ ) E mse( θ $ ) / MSE ( θ $ ) * * * RB[ mse( θ $ )] = MSE ( θ $ ) E mse( θ $ ) / MSE ( θ $ ), k * k * k * k where E * denotes the expectaton oer R=10,000 runs. For example, E* mse( θ $ ) = mse( θ $ r R r ( )) /. Fnally, the emprcal coeffcent of araton (CV) of the MSE estmators were computed as and CV [ mse( $ )] [ MSE { mse( $ 1/ 2 θ = θ )}] / MSE ( θ $ ) * * CV [ mse( $ )] [ MSE { mse( $ 1/ 2 θ = θ )}] / MSE ( θ $ ). k * k * k

13 Note that MSE [ mse( $ )] [ mse( $ ( r)) MSE ( $ 2 θ = θ θ )] / R and a smlar expresson for MSE * [ mse( θ $ k )]. * r * Table 1 reports summary measures of the alues of percent RE, RB and CV for cases (1) and (2) under approach (). Summary measures under approach () are reported n Table 2. Summary measures consdered are the meant and the medan (med) oer the small areas =1,2,..,30. It s clear from Tables 1 and 2 that $ θ k and $ θ perform smlarly wth respect to RE whch decreases as σ 2 when σ / σ / σ ncreases. Under approach (), RE s large for both cases 1 and 0.4, where as decreases sgnfcantly under approach () f the lnkng model s olated (case 2); the drect estmator y w s qute unstable under approach (). Turnng to the performance of MSE estmators under approach (), Table 1 shows that RB of mse( $ θ ) s neglgble (<4%) when the lnkng model holds (Case 1) and that t s small (<10%) een when the lnkng model s olated, although t ncreases. The estmator mse( $ θ ) has a larger RB but t s less than 15%. The CV of mse( $ θ ) s much smaller than the CV of mse( $ θ k ) for both Cases 1 and 2. For example, when the model holds (Case1) the medan CV s 25% for mse( $ θ ) compared to 148% for mse( $ θ k ) when σ =1; the medan CV decreases to 8% for mse( $ θ ) compared to 48% for mse( $ θ k ) when σ =2. Ths pattern s retaned when the model s olated (Case 2). It may be noted that

14 the probablty of mse( $ θ k ) takng a negate alue s qute large (>0.3) when σ / σ 0.4. Under approach (), Table 2 shows that RB of mse( $ θ ) s larger than the alue under approach () and ranges from 15% to 25%. On the otherhand, RB of mse( $ θ ) s smaller and ranges from 4% to 15%. The CV of mse( $ θ k ), howeer, s much larger than under approach (). For example, the medan CV for Case 1 s 295% compared to 37% for mse( $ θ ) when σ =1 whch decreases to 122% compared to 24% when σ =2. A smlar pattern holds for case 2 where the fxed fnte populaton s generated from the model wth aryng means To reduce RB of mse( $ θ ) under approach (), one could combne t wth mse( $ θ k ) by takng a weghted aerage, but t appears dffcult to chose the approprate weghts. The weghted aerage wll be more stable than mse( $ θ k ). 5. NESTED ERROR REGRESSION MODEL The results n Sectons 2 and 3 can be extended to nested error regresson models ' y = x β + + e, j = 1, 2,.., n ; = 1, 2,..., m (17) j j j

15 usng the results of Prasad and Rao (1990), where x j s a p-ector of auxlary arables wth known populaton mean X and related to y j, and β s the p-ector of regresson coeffcents. The reduced model s gen by y = x' β + + e (18) w w w w j j j wth x' = w x. Model-consstent estmates σ$ 2 and σ$ 2 are obtaned from the untleel model (17), employng ether the method of fttng constants (Prasad and Rao, 1990) or REML (restrcted maxmum lkelhood) estmaton (Datta and Lahr, 1997). where The pseudo-eblup of θ = X ' β + s gen by θ$ = γ $ y + ( 1 γ $ ) X ' β$, (19) w w w w $ 1 β ( γ $ x x' ) ( γ $ x y ). w = w w w w w An approxmate model-unbased estmator of MSE( $ θ ) s gen by (15) wth w as before, ( $, $ ) ( 1 $ ) $ 2 g 1 σ σ = γ w σ 2 1 ( σ$, σ $ ) = σ$ ( X γ $ x )'( γ $ x x' ) ( X γ $ x ) g 2 w w w w w w w

16 and g 3 ( σ$, σ $ ), obtaned from (14), noles the estmated arances and coarances of ~ σ 2 and $ σ 2. The latter can be obtaned from Prasad and Rao (1990) for the method of fttng constants and from Datta and Lahr (1997) for REML. ACKNOWLEDGEMENTS Ths work was supported by research grants from the Natural Scences and Engneerng Research Councl of Canada. We are thankful to the Assocate Edtor and the referee for constructe comments and suggestons. APPENDIX 1 Proof of (13): From general results (Prasad and Rao 1990) we hae E( $ ~ θ θ ) 2 tr [ A ( σ 2, σ 2 ) B ( σ 2, σ 2 )], where B ( σ, σ ) s the 2x2 coarance matrx of σ ~ 2 and σ$ 2, and A ( σ, σ ) s the 2x2 coarance matrx of θ σ θ, σ. Now, notng that * * θ σ * θ σ γw γ w( 1 γ w) = y = y w σ σ w, 2 * γw γ w( 1 γ w) = y = y w σ σ w, 2

17 and Var( y w )=σ + δ = σ / γ, we get w A and hence the result (14). 1 σ / σ ( σ, σ ) =[ γ w( 1 γ w ) σ ], σ / σ ( σ / σ ) Coarance matrx of ~ σ 2 and $ σ 2 : Under normalty, we hae and where 2 4 Var( σ$ ) = 2σ / ( n m), Var( σ ~ 2 ) = 2n 2 [ σ 4 ( m 1)( n 1)( n m) 1 + 2n σ 2 σ 2 + n σ 4 ] * * ** Co( σ$ 2, σ ~ 2 ) = ( m 1) n 1 Var( σ$ 2 ), n** = n 2 n / n + ( n ) / ( n ) ; * see Searle, Cassell and McCulloch (1992, p428). APPENDIX 2 Proof of E[ g ( σ$, σ$ ) + g ( σ$, σ$ ) g ( σ, σ )

18 By a Taylor expanson of g 1 ( σ$, σ $ ) around ( σ, σ ) to second order and notng that E( σ$ σ$ ) = 0 and E( σ$ 2 σ$ 2 ) 0, we get 1 E[ g ( $, $ 1 σ σ ) g1 ( σ, σ )] tr[ D ( σ, σ ) B ( σ, σ )], 2 where D ( σ, σ ) s the 2x2 matrx of second order derates of g 1 ( σ, σ ) wth respect to σ 2 and σ 2. It s easy to erfy that 1 tr[ D ( σ, σ ) B ( σ, σ )] = g3 ( σ, σ ). 2 Now, notng that E[ g ( σ$, σ$ )] g ( σ, σ ) we get the desred result. 3 3 APPENDIX 3 The desgn-based estmator of arance of y w under PPS samplng s gen by ( y w )= m m 1 j w ( y y ). j j w Kott (1989) model-asssted arance estmator s * ( y ) = { V ( y ) / E( y )} ( y ) w w w w = j j ( w ) w ( y y ) 2 j j j w j w ( 1 2w + ) w 2 j j j j, where E and V denote expectaton and arance wth respect to the unt-leel model (3). REFERENCES

19 ARRORA,V., and LAHIRI,P. (1997). On the superorty of the Bayesan method oer the BLUP n samll area Estmaton problesms. Statstca Snca, 7, BATTESE, G.E., HARTER, R., and FULLER,W.A. (1988). An error component model for predcton of county crop areas usng surey and satellte data. Journal of the Amercan Statstcal Assocaton, 83, DATTA, G.S. and LAHIRI, P. (1999). A unfed measure of uncertanty of estmated best lnear unbased predctor n small-area estmaton problems. Techncal Report, Unersty of Nebreska-Lncoln. FAY, R.E. and HERRIOT, R.A. (1979). Estmates of ncome for small places: an applcaton of James-Sten procedures to census data. Journal of the Amercan Statstcal Assocaton, 74, GHOSH, M and MEEDEN, G. (1986). Emprcal Bayes estmaton n fnte populaton samplng. Journal of the Amercan Statstcal Assocaton, 81, GHOSH, M and RAO, J.N.K. (1994). Small area estmaton: an apprasal. Statstcal Scence, 9, KACKAR, R.N. and HARVILLE, D.A. (1984). Approxmatons for standard errors of estmators for fxed and random effects models. Journal of the Amercan Statstcal Assocaton, 79, KLEFFE, J and RAO, J.N.K. (1992). Estmaton of mean squarer error of emprcal best lnear unbased predctors under a random error arance lnear model. Journal of Multarate analyss, 43, 1-15.

20 KOTT, P. (1989). Robust small doman estmaton usng random effects modellng. Surey Methodology, 15, PRASAD, N.G.N and RAO, J.N.K. (1990). The estmaton of mean squared errors of small-area estmators. Journal of the Amercan Statstcal Assocaton, 85, SEARLE, S.R., CASELLA, G and McCULLOCH, C.E. (1992). Varance components. Wley (New York).