On Comparison of Local Polynomial Regression Estimators for P = 0 and P = 1 in a Model Based Framework

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1 Internatonal Journal of Statstcs and Probablty; Vol. 7, No. 4; July 018 ISSN E-ISSN Publsed by Canadan Center of Scence and Educaton On Comparson of Local Polynomal Regresson Estmators for P = 0 and P = 1 n a Model Based Framework Conlet Bket Kkec 1 & Rcard Onyno Smwa 1 Statstcs and Operatons Researc Secton, Scool of Matematcs, College of Bologcal and Pyscal Scences, Unversty of Narob, Narob, Kenya Actuaral Scence and Fnancal Matematcs Secton, Scool of Matematcs, College of Bologcal and Pyscal Scences, Unversty of Narob, Narob, Kenya Correspondence: Conlet Bket Kkec, Statstcs and Operatons Researc Secton, Scool of Matematcs, College of Bologcal and Pyscal Scences, Unversty of Narob, Narob, Kenya. Emal: Kkecconlet@gmal.com Receved: May 16, 018 Accepted: May 31, 018 Onlne Publsed: June 8, 018 do: /jsp.v7n4p104 URL: ttps://do.org/ /jsp.v7n4p104 Abstract Ts artcle dscusses te local polynomal regresson estmator for P = 0 and te local polynomal regresson estmator for P = 1 n a fnte populaton. Te performance crteron exploted n ts study focuses on te effcency of te fnte populaton total estmators. Furter, te dscusson explores analytcal comparsons between te two estmators wt respect to asymptotc relatve effcency. In partcular, asymptotc propertes of te local polynomal regresson estmator of fnte populaton total for P = 0 are derved n a model based framework. Te results of te local polynomal regresson estmator for P = 0 are compared wt tose of te local polynomal regresson estmator for P = 1 studed by Kkec et al (018). Varance comparsons are made usng te local polynomal regresson estmator T 0 for P = 0 and te local polynomal regresson estmator T 1 for P = 1 wc ndcate tat te estmators are asymptotcally equvalently effcent. Smulaton experments carred out sow tat te local polynomal regresson estmator T 1 outperforms te local polynomal regresson estmator T 0 n te lnear, quadratc and bump populatons. Keywords: Asymptotc Propertes, Asymptotc Relatve Effcency, Fnte Populaton, Local Polynomal Regresson, Model Based Framework, Nonparametrc Regresson, Sample Surveys 1. Introducton Te teory of sample surveys nvolves prncples and metods of collectng and analyzng data from a fnte populaton of N unts and ten makng nferences about fnte populaton parameters on te bass of nformaton obtaned from te sample. For some early work on survey samplng teory, see Royall (1970a), Royall (1970b), Royall (1971), Smt (1976) and Pfeffermann (1993). In ts study, an estmator of te fnte populaton total s developed and ts propertes derved usng te local polynomal regresson procedure. Local polynomal regresson s a nonparametrc tecnque wc s a generalzaton of kernel regresson and s used for smootng scatter plots and modelng functons. Under normal condtons, wen p = 0, ts s referred to as local constant regresson, wen p = 1, ts s local lnear regresson and wen p, ts s local polynomal regresson. p s te order of te local polynomal beng ft. In local polynomal regresson, a low order wegted least squares regresson s ft at eac pont of nterest x, usng data from some negborood around x ( see Cleveland (1979) and Cleveland and Devln (1988)). Once a modelng approac s undertaken, tere s a specal feature n fnte populaton estmaton problems tat te unknown quanttes are realzed values of random varables, so te basc problem as te feature of beng smlar to a predcton problem. In order to estmate m(x) at a gven pont x, te assocaton between te predctor varable and te response varable s explored. Ts metodology was ntroduced by Stone (1977). It as also been studed by Fan (1993), Fan and Gjbels (1996), Bredt and Opsomer (000) and Kkec et al (017). Lke n Stone (1977), te man am of ts procedure s to quantfy te contrbuton of te covarate X to te response Y per unt value of X n order to summarze te assocaton between te two varables, to predct te mean response for a gven value X and to extrapolate te results beyond te range of te observed covarate values. A wegt k. x x / s assgned to te pont 104

2 ttp://jsp.ccsenet.org Internatonal Journal of Statstcs and Probablty Vol. 7, No. 4; 018 (x, y ) were s te sze of te local negbourood and k(t) s te unmodal non-negatve functon. On te oter and, nferences may explore propertes of te process tat generate te populaton values (Montanar and Ranall (003)). An assumpton s made from te fact tat te fnte populaton as been generated by a super populaton model ξ = f(x, y, φ) and t s of nterest to estmate te populaton parameters φ, were φ = α + βx. Te super populaton model can be appled to predct te unobserved values y s after obtanng estmates of α and β usng te known auxlary nformaton x, = 1,, N (see Montanar and Ranall (005) and Rueda and Sancez-Borrego (009)). Te nonparametrc approac does not restrct te functonal form of te dstrbuton nor does t specfy te varous stocastc propertes suc as E ξ (. ), V ξ (. ) and MSE ξ (. ). Rater, t leaves tem to cover broad classes of models, tus allowng for more robust nference tan nference obtaned n parametrc approac. Usng te model ξ, te nonparametrc estmator of total, T as been derved by Nadaraya (1964), Watson (1964), Prestly and Cao (197), Gasser and Muller (1979), Dorfman (199) ), Cambers et al (1993) and Odambo and Mwall (000). In s study, Dorfman (199) as been able to prove te asymptotc unbasedness and MSE consstency of ts estmator. Te estmator, owever suffers from sparse sample problem, and more work needs to be done to come up wt anoter tecnque tat can overcome ts problem. Ts s were te local polynomal procedure comes n. See Kkec et al (017) and Kkec et al (018). Te local polynomal regresson s one of te most successfully appled desgn adaptve non parametrc regresson. Ts estmaton procedure s an attractve coce due to ts flexblty and asymptotc performance. Havng a local model (rater tan just a pont estmate) enables dervaton of response adaptve metods for bandwdt and polynomal order selecton n a stragtforward manner. Te procedure as also te advantage of elmnatng desgn bas and allevatng boundary bas. Furtermore, te metod adapts well to random, fxed, gly clustered and nearly unform desgns. Te wegted least squares prncple to be employed n te local polynomal approxmaton approac, opens te way to a wealt of statstcal knowledge and tus provdng easy computatons and generalzatons. See Fan (199), Fan (1993), Ruppert and Wand (1994) and Fan and Gjbels (1996) among oters. Kkec et al (018) employ a superpopulaton approac to estmate te fnte populaton total usng te procedure of local lnear regresson. Explctly, te autors derve robustness propertes of te local lnear regresson estmator and carry out smulaton experments on te performances of ts estmator n comparson wt oter estmators tat exst n te lterature. Results ndcate tat te local lnear regresson estmator s more effcent and performng better tan te Horvtz-Tompson (195) and Dorfman (199) estmators, regardless of weter te model s specfed or mspecfed. In ts paper, te local polynomal regresson estmator of fnte populaton total for P = 0 s studed and asymptotc propertes derved. Analytcal comparsons are carred out between ts estmator and te local polynomal regresson estmator for P = 1 studed by Kkec et al (018) wc ndcate tat te estmators are asymptotcally equvalently effcent. Smulaton experments owever ndcate tat te local polynomal regresson estmator T 1 s superor and domnates te local polynomal regresson estmator T 0 n te lnear, quadratc and bump populatons.. Metod of Constructng te Local Polynomal Regresson Estmator T for P = 0 Te superpopulaton model consdered for estmatng te fnte populaton total s gven by, Y = m(x ) + (X ) (1) Specfcally, te followng assumptons old for te model consdered n te nonparametrc regresson estmaton of m(x ): E(Y X = x ) = m(x ) o (Y, Y X = x, X = x ) = { (x ), = 0, = 1,, 3,., N = 1,,3,, N. () Te propertes of te error are gven by, E( X = x ) = m(x ) o (, X = x, X = x ) = { (x ), = = 1,, 3,., N = 1,,3,, N. (3) 0, Te functons m(x ) and (x ) are assumed to be smoot and strctly postve. Consder te Taylor seres 105

3 ttp://jsp.ccsenet.org Internatonal Journal of Statstcs and Probablty Vol. 7, No. 4; 018 expanson of m(x ) expressed as, m(x ) = m(x + t) = m(x ) + tm (x ) + t m (x ) + 3 t 3 m (x ) + 3 = m(x ) + (x x )m (x ) + (x x ) m (x ) + (x x ) 3 m (x ) + (4) 3 Te Taylor seres expanson s wrtten n a general form expressed as, were x les n te nterval,x, x + - and y = α + (x x )β + (5) = (x x ) m (x ) + (x x ) 3 3 m (x ) + Te constants α and β are solved usng te least squares procedure by makng te subject of te formulae, squarng bot sdes, summng over all possble sample values and applyng te wegts to obtan a soluton to te wegted least squares problem of te form; Lettng, =.y α β(x x )/ φ =.y α β(x x )/ Dfferentatng φ wt respect to α and equatng to zero, gves Implyng tat Lettng φ α =.y α β(x x )/. x x Ten t follows from equaton (9) tat / y = α S, =. x x. x x. x x. x x. x x. x x / (6) / (7) / {{. x x /} 1 } = 0 (8) / + β (x x ). x x /. (9) / (x x ) Smlarly, dfferentatng φ wt respect to β and equatng to zero, gves Implyng tat and tus φ β =.y α β(x x )/ (x x ) (x x ). x x / y = α (x x ) (x x ) (10) / y = α(s,0) + β(s,1). (11). x x. x x. x x Multplyng equaton (11) and equaton (14) by (S,) and (S,1) respectvely, gves / {{. x x /} 1 } = 0 (1) / + β (x x ). x x /. (13) / y = α(s,1) + β(s,). (14) 106

4 ttp://jsp.ccsenet.org Internatonal Journal of Statstcs and Probablty Vol. 7, No. 4; 018 (S,). x x (S,1) (x x ) Subtractng equaton (16) from equaton (15), gves (S,). x x Makng α te subject of te formulae, gves / y = α(s,0)(s,) + β(s,1)(s,) (15). x x / y (S,1) (x x ) / y = α(s,1) + β(s,1)(s,) (16). x x / y = α(s,0)(s,) α(s,1) (17) α = {.S, S,1(x x )/ (S,0)(S,) (S,1).x x / y } (18) Smlarly, multplyng equaton (11) and equaton (14) by (S,1) and (S,0) respectvely, gves (S,1). x x (S,0) (x x ) Subtractng equaton (0) from equaton (19), gves (S,1). x x Makng β te subject of te formulae, gves Now t follows from equaton (5) tat / y = α(s,0)(s,1) + β(s,1) (19). x x / y (S,0) (x x ) / y = α(s,0)(s,1) + β(s,0)(s,) (0). x x β = { (S,0(x x ) S,1) (S,0)(S,) (S,1) / y = β(s,1) β(s,0)(s,) (1).x x / y } () y = α + (x x )β (3) If te value assgned s zero, assumng tat β s a pre-assgned constant, ten Terefore y = α (4) m (x ) = {.S, S,1(x x )/ (S,0)(S,) (S,1).x x / y } were = w (x )y w (x ) =.S, S,1(x x )/ (S,0)(S,) (S,1).x x Implyng tat te fnte populaton total estmator T for P = 0 can be estmated usng T = y + m (x ) / y = y + { {.S, S,1(x x )/ (S,0)(S,) (S,1).x x (5) / y } } (6) 107

5 ttp://jsp.ccsenet.org Internatonal Journal of Statstcs and Probablty Vol. 7, No. 4; Propertes of te Local Polynomal Regresson Estmator T for P = 0 In dervng te propertes of te local polynomal regresson estmator, te followng assumptons are made accordng to Ruppert and Wand (1994): () Te x varables le n te nterval (0, 1). () Te functon m (. ) s bounded and contnuous on (0, 1). () Te kernel (t) s symmetrc and supported on ( 1, 1). Also (t) s bounded and contnuous satsfyng te followng: (x) dx = 1, x (x) dx = 0, x (x) dx > 0, x dx <, d k = (t) dt (v) Te bandwdt s a sequence of values wc depend on te sample sze n and satsfyng 0 and n, as n. (v) Te pont x at wc te estmaton s takng place satsfes < x < 1. Fan (1993) mposed condtons on be relaxed. (. ) and are only used for convenence n terms of tecncal arguments and tus can 3.1 Te Expectaton of te Local Polynomal Regresson Estmator T for P = 0 Te expectaton of T for P = 0 s derved as, E(T ) = E(y ) + { {.S, S,1(x x )/ (S,0)(S,) (S,1) k.x x / E(y )}} Usng te Taylor seres expanson of te form, = m(x ) + { {.S, S,1(x x )/ S,0S, (S,1) k. x x / m(x )}} (7) m(x ) = m(x ) + tm (x ) + t m (x ) +, (8) Teorem 3 n Fan and Gjbels (1996) s suc tat under te condtons gven n ()-(v), allows E(T ) = m(x ) + { { S,k. x x / S,0S, (S,1) (m(x t ) + tm (x ) + m (x ) + )}} S,1(x x ) { { S,0S, (S,1) k.x x / (m(x ) + tm (x ) + t m (x ) + )}} = m(x ) + {( S,0S, (S,1) S,0S, (S,1) ) m(x )} + {( S,1S, S,1S, S,0S, (S,1) ) m (x )} + {( (S,) S,1S,3 m (x ) S,0S, (S,1) ) } = m(x ) + m(x ) + {( (S,) S,1S,3 m (x ) S,0S, (S,1) ) }. (9) 3. Te Bas of te Local Polynomal Regresson Estmator T for P = 0 Te bas of T s gven by s(t ) = {( (S,) S,1S,3 m (x ) S,0S, (S,1) ) }. (30) Terefore te asymptotc expresson of te bas of te local polynomal regresson estmator T s s (T ) = {.n k + o(n )/ m (x ) } (n 4 k + o(n )) = { 1 k m (x )} (31) 108

6 ttp://jsp.ccsenet.org Internatonal Journal of Statstcs and Probablty Vol. 7, No. 4; Te Varance of te Local Polynomal Regresson Estmator T for P = 0 Te varance of te local polynomal regresson estmator T s estmated usng te varance of te error, tus V r(t T) s derved as were, V r(t ) = V r { y + m (x ) y y } = V r { w (x )y y } = w (x ) w (x ) =.S, S,1(x x )/ (S,0)(S,) (S,1) (x ) + (x ) (3).x x Te asymptotc expresson for te varance of T s gven by te expresson usng te results of m (x ) tat ave been derved, tus V r (T ) = 1 n {. x x / (x ). x x 1 /} /. = d k n (x ). (33) 3.4 Te MSE of te Local Polynomal Regresson Estmator T for P = 0 Teorem I n Fan (1993) allows tat under condton () gves, MSE(T ) = * s(t )+ + V r(t ) = { {( (S,) S,1S,3 m (x ) S,0S, (S,1) ) }} + w (x ) (x ) + (x ) (34) Te asymptotc expresson for te MSE of te local polynomal regresson estmator T s gven by MSE (T ) = { { 1 k m (x )}} (35) Note tat results for te local polynomal regresson estmator of fnte populaton total T for P = 1 ave been derved by Kkec et al (018). 3.5 Te Asymptotc Relatve Effcency Te relatve effcency of two procedures s te rato of ter effcences, but t s often possble to use te asymptotc relatve effcency, defned as te lmt of te relatve effcences as te sample sze grows, as te prncpal measure of comparson. Let T 0 be te local polynomal regresson estmator of fnte populaton total for P = 0 and T 1 be te local polynomal regresson estmator of fnte populaton total for P = 1 as studed by Kkec et al (018). If T 0 and T 1 are bot unbased estmators of T, ten te relatve effcency of T 0 to T 1 s gven by, Eff(T 0, T 1) = V r(t 1) V r(t 0). (36) If T 0 and T 1 are bot asymptotcally unbased estmators of T, ten te asymptotc relatve effcency of T 0 to T 1 s gven by, E(T 0, T 1) = Eff(T 0, T 1) = V r(t 1) V r(t 0). (37) 109

7 ttp://jsp.ccsenet.org Internatonal Journal of Statstcs and Probablty Vol. 7, No. 4; 018 Terefore, te estmators of fnte populaton totals for T 0 and T 1 are respectvely gven by, T 0 = y + { {.S, S,1(x x )/ (S,0)(S,) (S,1).x x T 1 = + { {.S, S,1(x x )/ (S,0)(S,) (S,1) k.x x / y }} / y } }. (38) x x + {( S,0S, (S,1) ) {(S,0(x x ) S,1)k. x x / y }}. (39) Te varance of te local polynomal regresson estmator T 0 s gven by, V r(t 0) = w (x ) (x ) + (x ) (40) Te asymptotc expresson for te varance of te local polynomal regresson estmator T 0 s estmated by, V r (T 0) = d k n Te varance of te local polynomal regresson estmator T 1 s gven by, (x ) (41) V r(t 1 ) = w (x ) (x ) + (x x ) w (x ) (x ) + (x ) (4) Te asymptotc expresson for te varance of te local polynomal regresson estmator T 1 s estmated by, Note tat n Kkec e tal (017), V r V r (T 1) = d k n.m LL (x )/ = d k (x ). (43) (x ) and V r.m NW (x )/ = d k Tus te asymptotc relatve effcency of te local polynomal regresson estmator T 0 to te local polynomal regresson estmator T 1 derved by Kkec et al (018) s gven by, V r (T 1) E(T 0, T 1) = Eff(T 0, T 1) = { V r 4. Smulaton Study 4.1 Descrpton of te Data Sets (T 0) } = { d k (x ) n (x ) } = 1. (44) d k n (x ) In ts secton, smulaton experments are carred out to evaluate te performance of te estmators. Te data are generated from te regresson model of te form, Y = m(x ) + (X ) = 1,,, n (45) Te data sets are obtaned by smulaton usng specfc models avng relatons of te form, y = 1 + (x 0.5) + (46) y = 1 + (x 0.5) + (47) y = 1 + (x 0.5) + ( 00(x 0.5) + (48) for te lnear, quadratc and bump populatons respectvely. Te x s are generated as ndependent and dentcally dstrbuted (d) unform (0, 1) random varables. Te errors are assumed to be ndependent and dentcally dstrbuted (d) random varables wt mean 0 and constant varance. Te analyss and comparson n terms of performance s based on te local polynomal regresson estmator T 0 and te local polynomal regresson estmator T 1. Te Epanecncov kernel gven s used for kernel smootng on eac of te populatons due to ts smplcty and easy computatons usng well desgned computer programs and s defned as, 110

8 ttp://jsp.ccsenet.org Internatonal Journal of Statstcs and Probablty Vol. 7, No. 4; (1 1 5 t ) t < 5 (49) Te bandwdts are data drven and are determned by te least squares cross valdaton metod. For eac of te tree artfcal populatons of sze 00, samples are generated by smple random samplng wtout replacement usng sample sze n = 60. For eac combnaton of mean functon, standard devaton and bandwdt, 500 replcate samples are selected and te estmators calculated. Table 1. Computatonal Formulae for te Local Polynomal Regresson Estmators T 0 and T 1 Estmator P E, T 0 Formulae T 0 = Y + m 0 (x ) P E, T 1 T 1 = Y + m 1 (x ) LINEAR RELATIONSHIP Y X Fgure 1. Scatter Dagram for te Lnear Populaton QUADRATIC RELATIONSHIP Y X Fgure. Scatter Dagram for te Quadratc Populaton 111

9 ttp://jsp.ccsenet.org Internatonal Journal of Statstcs and Probablty Vol. 7, No. 4; 018 BUMP RELATIONSHIP Y X Fgure 3. Scatter Dagram for te Bump Populaton 4. Results Te results of te bas and mean squared error (MSE) for te local polynomal regresson estmator T 0 for P = 0 and te local polynomal regresson estmator T 1 for P = 1 n te lnear, quadratc and bump populatons are provded n te table below. Table. Te Bas and MSE for T 0 and T 1 n te Tree Artfcal Populatons Lnear Quadratc Bump T 0 T T 0 T T 0 T BIAS MSE Dscusson In estmatng m (x ) for te local polynomal regresson estmator T 0, β as been assumed to be a pre-assgned constant and n partcular te value assgned s zero. It as terefore been sown n secton tat te estmator m (x ) s based leadng to a based estmaton of te fnte populaton total. On te oter and, wen estmatng m (x ) for te local polynomal regresson estmator T 1, te value of β s not pre-assgned but rater determned by te set of data provded and tus mnmzng te bas. Wt regard to asymptotc relatve effcency, tere s no dfference n te performance of te local polynomal regresson estmator T 0 studed n ts paper and te local polynomal regresson estmator T 1 studed by Kkec et al (018). Te reason for ts beng tat ter rato converges to 1 as n becomes large, see equaton (44). Ts terefore mples tat te estmators are asymptotcally equvalently effcent. However, t s observed from smulaton experments conducted tat te bases and MSEs computed n table for te local polynomal regresson estmator T 1 are small n all te tree populatons. Te results terefore ndcate tat te local polynomal regresson estmator T 1 s superor and domnates te local polynomal regresson estmator T 0 for te lnear, quadratc and bump populatons. 6. Concluson In ts artcle te local polynomal regresson estmators T 0 and T 1 of fnte populaton totals ave been studed n a model based framework. Analytcally, varance comparsons are explored usng te local polynomal regresson estmator T 0 for P = 0 and te local polynomal regresson estmator T 1 for P = 1 n wc results ndcate tat te estmators are asymptotcally equvalently effcent. Smulaton experments carred out n terms of te bases and MSEs sow tat te local polynomal regresson estmator T 1 outperforms te local polynomal regresson estmator T 0 n all te tree artfcal populatons and terefore, T 1 s te most effcent estmator. 11

10 ttp://jsp.ccsenet.org Internatonal Journal of Statstcs and Probablty Vol. 7, No. 4; 018 References Bredt, F. J., & Opsomer, J. D. (000). Local Polynomal Regresson Estmaton n Survey Samplng. Annals of statstcs, 8, Cambers, R. L., Dorfman, A. H., & Werly, T. E. (1993). Bas robust estmaton n fnte populatons usng nonparametrc calbraton. J. Amer Statst Assoc., 88, Cleveland, W. S. (1979). Robust Locally Wegted Regresson and Smootng Scatter Plots. J. Amer. Statst. Assoc. 74, Cleveland, W. S., & Devln, S. (1988). Locally Wegted Regresson: An Approac to Regresson Analyss by Local Fttng. J. Amer. Statst. Assoc. 83, Dorfman, A. (199). Nonparametrc Regresson for Estmatng Totals n Fnte Populatons, Proceedngs of te Secton on Survey Researc Metods. Amercan Statstcal Assocaton, Fan, J. (199). Desgn Adaptve Nonparametrc Regresson. Journal of Amercan Statstcal Assocaton, 87, Fan, J. (1993). Local Lnear Regresson Smooters and Ter Mnmax Effcences. Annals of Statstcs, 1, ttps://do.org/10.114/aos/ Fan, J., & Gjbels, I. (1996). Local Polynomal Modelng and ts Applcatons. London: Capman and Hall. Gasser, T., & Muller, H. G. (1979). Kernel Estmaton n Regresson Functons. Smootng Tecnques for Curve Estmaton, Horvtz, D. G., & Tompson, D. J. (195). A Generalzaton of Samplng wtout Replacement from a Fnte Unverse. Journal of Amercan Statstcal Assocaton, 47, ttps://do.org/ / Kkec, C. B., Smwa, R. O., & Pokaryal, G. P. (017). On Local Lnear Regresson Estmaton n Samplng Surveys. Far East Journal of Teoretcal Statstcs, 53(5), ttps://do.org/ /ts Kkec, C. B., Smwa, R. O., & Pokaryal, G. P. (018). On Local Lnear Regresson Estmaton of Fnte Populaton Totals n Model Based Surveys. Amercan Journal of Teoretcal and Appled Statstcs, 7(3), ttps://do.org/ /j.ajtas Montanar, G. E., & Ranall, M. G. (003). Nonparametrc Metods n Survey Samplng. In: Vnc, M., Monar, P., Mgnan, S. and Montanar, A., Eds., New Developments n Classfcaton and Data Analyss, Sprnger, Berln, Montanar, G. E., & Ranall, M. G. (005). Nonparametrc Model Calbraton Estmaton n Survey Samplng. Journal of te Amercan Statstcal Assocaton, 100, ttps://do.org/ / Nadaraya, E. A. (1964). On Estmatng Regresson. Teory of Probablty Applcatons, 10, Odambo, R. O., & Mwall, T. (000). Nonparametrc Regresson for Fnte Populaton Estmaton. East Afrcan Journal of Scence, II(), Pfeffermann, D. (1993). Te Role of Samplng Wegts Wen Modelng Survey Data. Internatonal Statstcal Revew, 61(), ttps://do.org/10.307/ Prestley, M. B., & Cao, M. T. (197). Nonparametrc Functon Fttng. Journal of te Royal Statstcal Socety, B34, Royall, R. M. (1970a). On Fnte Populaton Samplng under certan Lnear Regresson Models. Bometrka, 57, Royall, R. M. (1970b). Fnte Populaton Samplng-On Labels n Estmaton. Journal of te Annals of Matematcal Statstcs, 41, Royall, R. M. (1971). Lnear Regresson Models n Fnte Populaton Samplng Teory Holt, Rnart and Wnston, Toronto, Canada, 54, Rueda, M. & Sancez-Borrego, I. (009). A Predctve Estmator of Fnte Populaton Mean Usng Nonparametrc Regresson. Computatonal Statstcs 4, ttps://do.org/ /s x Ruppert, D., & Wand, M. P. (1994). Multvarate Locally Wegted Least Squares Regresson. Annals of Statstcs,, ttps://do.org/10.114/aos/ Smt, T. M. (1976). Te Foundatons of Survey Samplng. Journal of Royal Statstcal Socety Assocaton, 139, Part

11 ttp://jsp.ccsenet.org Internatonal Journal of Statstcs and Probablty Vol. 7, No. 4; 018 Stone, C. (1977). Consstent Nonparametrc Regresson. Annals of Statstcs, 5, Watson, G. (1964). Smoot Regresson Analyss. Sankya Seres A, 6, Copyrgts Copyrgt for ts artcle s retaned by te autor(s), wt frst publcaton rgts granted to te journal. Ts s an open-access artcle dstrbuted under te terms and condtons of te Creatve Commons Attrbuton lcense (ttp://creatvecommons.org/lcenses/by/4.0/). 114