Estimation of Population Total using Local Polynomial Regression with Two Auxiliary Variables

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1 J. Stat. Appl. Pro. 3, o., ) 9 Joural of Statstcs Applcatos & Probablty A Iteratoal Joural Estmato of Populato Total usg Local Polyomal Regresso wth Two Auxlary Varables EL-Houssey A. Rady ad Dala Zeda Isttute of Statstcal Studes & Research, Caro Uversty, Caro, Egypt Receved: 9 Sep. 03, Revsed: Mar. 04, Accepted: 6 Mar. 04 Publshed ole: Jul. 04 Abstract: I ths paper, the estmato for fte populato total of a study varable wll be cosdered, ad the local lear regresso wll be used. The study varable s avalable for the sample ad s supplemeted by two auxlary varables, whch are avalable for every elemet the fte populato. Also, the resamplg methods wll be combed wth the local lear regresso method to estmate the total. The comparsos betwee dfferet methods wll be performed based o the mea squared error MSE), mea absolute error MAE), ad mea absolute percetage error MAPE). A smulato study s carred out to assess the effects. Keywords: survey samplg, auxlary varables, local lear regresso, bootstrap ad jackkfe Itroducto Survey samplg ofte supples formato about a study varable oly for sampled elemets. However, auxlary formato s ofte avalable for the etre populato. The relatoshp of the auxlary formato wth the study varable across the sample allows fereces about the o-sampled porto of the populato. Thus, the use of auxlary formato at the estmato stage of a survey mproves the precso of the estmates parameters studed. Oe approach to usg ths auxlary formato estmato s to assume a workg model descrbg the relatoshp betwee the study varable of terest ad the auxlary varables. Estmators are the derved o the bass of ths model. Usually a parametrc approach s used to represet the relatoshp betwee the auxlary varables ad the study varable. But some stuatos, the parametrc model s ot approprate, ad the resultg estmators do ot acheve ay effcecy ga over pure estmators. A atural alteratve was frst suggested by Kuo 988) for the dstrbuto fucto, that adopts a oparametrc approach, whch does ot place ay restrctos o the relatoshp betwee the auxlary data ad the study varable. Other mportat works ths topc are Chambers et al. 993), Drofma 993), Drofma ad Hall 993) ad Rueda ad Arcos 998). Bredt ad Opsomer 000) used the tradtoal local polyomal regresso estmator for the ukow regresso fucto mx). They assume that mx) s a smooth fucto of x ad obtaed a asymptotcally desg-ubased ad cosstet estmator of the fte populato total. The local polyomal regresso estmator has the form of the geeralzed regresso estmator, but s based o a oparametrc superpopulato model applcable to a much larger class of fuctos. Bredt, Claeskes, ad Opsomer 995) cosdered a related oparametrc model-asssted regresso estmator, replacg local polyomal smoothg wth pealzed sples. Km, Bredt, ad Opsomer 009) exteded the local polyomal oparametrc regresso estmato to two-stage samplg, whch a probablty sample of clusters s selected, ad the subsamples of elemets wth each selected cluster are obtaed. I ths paper, we cocered wth the estmato the fte populato total the presece of the two auxlary varables usg the local polyomal regresso. Multple Regresso Suppose ow that the covarate s d-dmesoal, where X =x,x,...,x d ) Correspodg author e-mal: dala dala444@yahoo.com

2 30 E. A. Rady ad D. Zeda: Estmato of Populato Total usg Local... I ths case, Y = mx,x,...,x d )+ε For local lear regresso, the kerel fucto K s defed as a fucto of d varables. Gve a osgular postve defte d d badwdth matrx H, we defe K H x)= ) H / K H / x. ) Ofte, oe scales each covarate to have the same mea ad varace ad the we use the kerel h d K x / h ) ) where K s ay oe-dmesoal kerel. The there s a sgle badwdth parameter h. At a target value x=x,x,...,x d ), the local sum of squares s gve by where, I ths case, the estmator s w x) Y a 0 d j= w x)=k x x / h ) a j x j x j )) 3) ˆmx)=â 0 4) where â=â 0,â,...,â d ) s the value of a=a 0,a,...,a d ) that mmzes the weghted sums of squares. The soluto â s â= X WX ) X WY 5) where X ths case s x x... x d x d x x... x d x d X = x x... x d x d ad W s the dagoal matrx whose, ) elemet. For more detals [see Casella, G. et al 006)]. 3 Estmato of Total the Case of Two Auxlary Varables I ths case ad x x j x x j x x j x x j X =..., j =,,..., x x j x x j... X x = x j x x j... x x j, j =,,..., Let j = x x j, ths case, w j = k h j = x x j j + j) ) ad x x j x x j... x x j w j w j 0 W = w j

3 J. Stat. Appl. Pro. 3, o., ) / 3 where w j = w j. So, we wll substtute the equato ˆmx) =X WX) X WY by X, W ad Y to get the estmato of the total. Hece X W = w j w j w j j w j j w j j w j, ad j w j j w j j w j X WX = l l l 3 l l l 3, where l 3 l 3 l 33 ote that:x WX) s a symmetrc matrx. Hece, the verse of the matrxx WX) s The secod term the estmato of â s where l = w j l 3 = j w j l 3 = j j w j l = j w j l = j w j l 33 = jw j X WX ) = Ad j X WX )) X WX X WY = w j y j w j y j w j y â= X WX ) X WY. Sce our prmary terest s to compute a estmate of Y, the ecessary computatos are lmted to the oes that estmate the parametera 0. Therefore, the estmator s smplfed to ŷ j = â 0 = e X WX ) X WY where e s a colum vector wth the frst elemet equal to oe, ad the rest equal to zero. The where 0 j =, ad s ŷ j = â 0 = 3 a a=s ) ) s = j w j j w j ) s = j j w j ) j w j ) s 3 = j w j ) j j w j a ) j w j y / X W X 6) ) j j w j ) j w j ) j j w j ) j w j ) j w j ow, our ma purpose s to estmate the total T). Therefore, accordg to Drofma 99) the estmate of the total ˆT = Substtute from equato 6) 7), the estmated total s where D= X W X ˆT = = y + + j D j j D j y + ŷ j 7) j=+ 3 a a ) j w j y a=s ) 3 s a a ) j w j y 8) a=

4 3 E. A. Rady ad D. Zeda: Estmato of Populato Total usg Local... 4 Bootstrappg Local Lear Regresso for Estmatg the Total Efro979) has developed a ew resamplg procedure amed as Bootstrap. Bootstrap resample cossts of elemets that are draw radomly from the orgal data observatos wth replacemet Fredl & Stampfer, 00). The all bootstrap samples are, but we choose B bootstrap samples. Bootstrappg ca be doe by ether resamplg the resduals, whch the regressorsx,x,...,x d ) are assumed to be fxed, or resamplg the y values ad ther assocated x values, whch the regressors are assumed to be radom. I our study, we deal wth the resduals resamplg, where the bootstrap techque wth oparametrc regresso to estmate the total of the populato wll be used ad the local lear regresso wll also be cosdered. Suppose we have a uvarate respose varable Y ad two auxlary varables X ad X, the the oparametrc regresso model s Y = mx,x )+ε,,..., ad the bootstrap procedure based o the resamplg errors ca be summarzed as follows: ) Let Y =Y,Y,...,Y ) deote the sample of observatos was selected from the geerated populato. The based o the sample Y the local lear regresso estmator ˆmx) s gve by ŷ = ˆmx,x )=e X WX ) X WY, ) Calculate the resduals as followg ˆε = Y ˆmx,x ), =,,...,. 3) Defe the cetered resduals by ε = ˆε ˆε. 4) Draw wth replacemet a radom sample of sze from the resduals, ε, ε,..., ε, were calculated step 3) gvg / probablty for each ε values. Ths gves -bootstrap sample of the resduals ε, =,,...,. [See Ste 985, 990) ad Wu 986)]. 5) The bootstrap sample of observatos s costructed by addg a radomly sampled resdual to the orgal predcted value for each observato. After resamplg, ew observatos s gve by Y = ˆmx,x )+ε. 6) Obta the local lear estmate from the frst bootstrap sample as follows: Ŷ ) = e X WX ) X WY 7) Repeat the steps 4, 5 ad 6, B tmes. The, the bootstrap estmate s Ŷ = B B r= Ŷ r) 9) ow, we wll estmate the total usg local lear regresso estmato wth bootstrap method, sce we have but j Y j s ukow, so we wll estmate t as: T = j= ˆT = Y j = Y + Y + Ŷ j j Y j j = Y + j B B r= D j 3 a a ) j w j Ŷ a=s r) 0)

5 J. Stat. Appl. Pro. 3, o., ) / Jackkfg Local Lear Regresso for Estmatg the Total I ths Secto, the algorthm of estmatg the total usg local lear regresso method wth jackkfe techque wll be gve. The techque of deletg sgle case from the orgal sample delete oe jackkfe) sequetally wll be used. Suppose the dataset cossts of vectors Y,X,X ), where Y s the study varable ad X,X are cosdered auxlary varables. For smplcty, let x = x,x )ad d k = y k,x k ), k =,,..., deote the values assocated wth th observato. I ths case, the set of observatos s the vector d,d,...,d ). The, the jackkfe procedure based o delete-oe s as follows. ) Draw szed sample from populato radomly ad label the elemets of the vector d k =y k,x k ), k=,,...,. ) Omt frst observato of the vector d k = y k,x k ) ad label the remag - szed observato set Y J) ) =y,...,y ), ad X J) ) =x,...,x ) as delete-oe jackkfe sample d J) ). 3) Obta the local lear regresso estmate ˆm J) x j ) from d J) ). 4) Omt the secod elemet of the vector d = y,x ) ad label remag - szed observato set Y J) ) =y,y 3,...,y ), ad X J) ) =x,x 3,...,x ) as d J) ). 5) Obta the local lear regresso estmate ˆm J) x j ) from d J) ). 6) Smlarly, omt each oe of the observatos there s samples jackkfe each of them has observatos) ad estmate the local lear regresso ˆm Jk) x j ), where ˆm Jk) x j ) s the jackkfe local lear regresso estmate after deletg of k th observato from d k =y k,x k ). 7) The, the jackkfe estmate of ˆmx j ) s ˆm J) x j )= k= ˆmJk) x j )= k= D j 3 a= s a a ) jw j y k. 8) Usg the jackkfe estmate of ˆmx j ) estmatg the total ˆT J) = y + j ˆm J) x j )= y + j k= D j 3 s a a= a ) j w j y k ) 6 Performace Crtera of the Models The performace of the model s related wth how close are the predcto values to the observed values. Three dfferet cosstecy crtera are used order to compare amog dfferet methods. These are mea square error MSE), mea absolute error MAE) ad mea absolute percetage error MAPE) respectvely whch are defed as follows:.mse = y ŷ )..MAE = y ŷ. 3.MAPE = y ŷ y 00%). 7 Smulato studes Sometmes samplg, we do ot usually observe all the survey formato. That s, the survey varable Y s ot observable for all the populato uts. Auxlary varable X, s ofte used to estmate the uobserved survey varables. Oe way of overcomg the above problem s the super populato approach, whch a workg model relatg the two auxlary varables s assumed. I ths study, we smulate data from four models, whch troduced by Ye et al 006), each wth Y = mx,x )+δx )ε, where ε 0,). Model ): m x,x )=x x δ x,x )= x 0.04 ) I x >o.04) Model ): m x,x )=x exp x ) Model 3): m 3 x,x )=x + s.5x ) δ x,x )=.5 x 0.04 ) I x >o.04) δ 3 x,x )= x 0.04 ) I x >o.04) + 0.0

6 34 E. A. Rady ad D. Zeda: Estmato of Populato Total usg Local... Table MSE, MAE, ad MAPE of the total estmato uder dfferet methods wth dfferet sample szes ad badwdths for model h= /3 Method = 5 = 50 = 00 MSE MAE MAPE MSE MAE MAPE MSE MAE MAPE CLR % % % LLR % % % LLB % % % LLJ % % % h= /5 CLR % % % LLR % % % LLB % % % LLJ % % % h= /7 CLR % % % LLR % % % LLB % % % LLJ % % % Table MSE, MAE, ad MAPE of the total estmato uder dfferet methods wth dfferet sample szes ad badwdths for model h= /3 Method = 5 = 50 = 00 MSE MAE MAPE MSE MAE MAPE MSE MAE MAPE CLR % % % LLR % % % LLB % % % LLJ % % % h= /5 CLR % % % LLR % % % LLB % % % LLJ % % % h= /7 CLR % % % LLR % % % LLB % % % LLJ % % % Model 4): m 4 x,x )=sx + x )+exp x ) δ 4 x,x )=3 x 0.04 ) I x >o.04) The populatos of X ad X are geerated as depedet ad detcally dstrbuted d) Uform -, ) radom varables. The smulato expermets wll be performed to compare the performace of the local lear regresso estmator wth the classc lear regresso estmator. Also, the effects of the bootstrap ad the jackkfe techques o those estmators wll be studed. The smulato wll be carred out as follows:. Frstly, we geerate populato of sze = 000 as above.. The smple radom samples wll be chose from the populato ad dfferet szes wll be cosdered, amely = 5, 50, ad 00 respectvely

7 J. Stat. Appl. Pro. 3, o., ) / 35 Table 3 MSE, MAE, ad MAPE of the total estmato uder dfferet methods wth dfferet sample szes ad badwdths for model 3 h= /3 Method = 5 = 50 = 00 MSE MAE MAPE MSE MAE MAPE MSE MAE MAPE CLR % % % LLR % % % LLB % % % LLJ % % % h= /5 CLR % % % LLR % % % LLB % % % LLJ % % % h= /7 CLR % % % LLR % % % LLB % % % LLJ % % % Table 4 MSE, MAE, ad MAPE of the total estmato uder dfferet methods wth dfferet sample szes ad badwdths for model 4 h= /3 Method = 5 = 50 = 00 MSE MAE MAPE MSE MAE MAPE MSE MAE MAPE CLR % % % LLR % % % LLB % % % LLJ % % % h= /5 CLR % % % LLR % % % LLB % % % LLJ % % % h= /7 CLR % % % LLR % % % LLB % % % LLJ % % % Secodly, for each sample, we estmate the totalt = Y + j mx j). The lear regresso ad the local lear regresso wll be used to estmate mx). Also, the bootstrap ad the jackkfe techques wll be combed wth those regresso methods to estmate mx). We cosder the ormal kerel fucto wth dfferet badwdth values h = /3, /5 ad /7 for the local lear regresso, each smulato settg s appled to all four models ad repeated M = 000 tmes. Thrdly, the mea square error MSE) of the total T) uder the two types of the regresso methods wll be calculated. Also, the mea absolute error MAE) ad the mea absolute percetage error MAPE) wll be calculated. Fally, the effects of the bootstrap ad the jackkfe techques o the estmato of total T) wll be studed, these effects based o the bas, MSE, MAE, MAPE. Tables,, 3 ad 4 reveals the values of the mea squared error MSE), mea absolute error MAE) ad the mea absolute percetage error MAPE) of the estmators for the four models, whe the sample sze ) has dfferet values =5,50,ad00 ad the badwdth has values h= /3, /5 ad /7.

8 36 E. A. Rady ad D. Zeda: Estmato of Populato Total usg Local... 8 Results of the Smulato Study Tables,, 3 ad 4 summarze the followg coclusos about our smulato study:.for the four models the local lear regresso estmator domates the classcal lear regresso estmator whe the regresso model s correctly specfed..the local lear regresso estmator wth bootstrap s overall the best choce for all models ad badwdths uder study. 3.The effect of the bootstrap o the estmator s better tha the jackkfe at the most. 4.The badwdth h= /5 s the best choce at the most for all models. 5. For all estmators as the sample sze creases the mea squared error MSE), the mea absolute error MAE) ad the ma absolute percetage error MAPE) decrease, for the three badwdths h) cosdered ad for the four models. Abbrevato: CLR: classcal lear regresso, LLR: local lear regresso, LLB: local lear regresso wth bootstrap ad LLJ: local lear regresso wth jackkfe. Refereces [] Bredt, F.J. ad Opsomer, J.D., Local polyomal regresso estmators survey samplg. Aals of Stat, 000). [] Casella, G. Feberg, S., ad Olk, I, All of oparametrc Statstcs, Sprger, ew York, 006). [3] Chambers RL, Drofma AH, Wehrly TE. Bas robust estmato fte populatos usg oparametrc calbrato. J Am Stat Assoc., 88, ). [4] Drofma AH, Hall P. Estmators of the fte populato dstrbuto fucto usg oparametrc regresso. A Stat., 6, ). [5] Drofma AH. A comparso of desg-based ad model-based estmators of the fte populato dstrbuto fucto. Aust J Stat., 35, ). [6] Fredl, H. ad Stampfer, E., Jackkfe Resamplg, Ecyclopeda of Evrometrcs,, , 00). [7] Km, J Y, Bredt, FJ, ad Opsomer, JD. oparametrc Regresso Estmato of Fte Populato Totals uder Two-Stage Samplg. Techcal Report, Departmet of Statstcs, Colorado State Uversty, 009). [8] Kuo, L. Classcal ad predcto approaches to estmatg dstrbuto fuctos from survey data, I Proceedgs of the secto o survey research methods,, Amer. Statst. Assoc., Alexadra, VA, ). [9] Rueda, M., Arcos, A. O estmatg the meda from survey data usg multple auxlary formato. Metrka, 54, ). [0] Sahler, S. ad Topuz, D., Bootstrap ad Jackkfe Resamplg Algorthms for Estmato of Regresso Parameters, Joural of Appled Quattatve methods,, ). [] Ste, R., Bootstrap predcto tervals for regresso, J. Amer. Statst. Assoc, 80, ). [] Ste, R., Moder Methods of Data Aalyss; Edt: by Joh Fox, Scotlad, ) [3] Wu, C.F.J. Jackkfe, Bootstrap ad Other Resamplg Methods Regresso Aalyss, Aals of Statstcs, 4, ). [4] Ye, A., Hydma, R, J., ad L, Z., Local lear multvarate regresso wth varable badwdth the presece of heteroscedastcty, Workg Paper 8/06, Departmet of Ecoometrcs ad Busess Statstcs, Moash Uversty.