Imputation Based on Local Linear Regression for Nonmonotone Nonrespondents in Longitudinal Surveys

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1 Ope Joural of Sascs, 6, 6, p:// SSN Ole: SSN Pr: 6-78X mpuao Based o Local Lear Regresso for Nomoooe Norespodes Logudal Surves Sara Pee, Carles K. Sego, Leo Odogo, George O. Orwa 3, Romaus O. Odambo 3 Pa Afrca Uvers sue for Basc Sceces, Tecolog ad ovao, Narob, Kea Deparme of Sascs ad Acuaral Scece, Keaa Uvers, Narob, Kea 3 Deparme of Sascs ad Acuaral Scece, Jomo Keaa Uvers of Agrculure ad Tecolog, Narob, Kea How o ce s paper: Pee, S., Sego, C.K., Odogo, L., Orwa, G.O. ad Odambo, R.O. (6) mpuao Based o Local Lear Regresso for Nomoooe Norespodes Logudal Surves. Ope Joural of Sascs, 6, p://dx.do.org/.436/ojs Receved: Ocober 3, 6 Acceped: December, 6 Publsed: December 7, 6 Coprg 6 b auors ad Scefc Researc Publsg c. Ts work s lcesed uder e Creave Commos Arbuo eraoal Lcese (CC BY 4.). p://creavecommos.org/lceses/b/4./ Ope Access Absrac Te sud focuses o e mpuao for e logudal surve daa wc ofe as ogorable orespodes. Local lear regresso s used o mpue e mssg values ad e e esmao of e me-depede fe populaos meas. Te asmpoc properes (ubasedess ad cossec) of e proposed esmaor are vesgaed. Comparsos bewee dffere paramerc ad oparamerc esmaors are performed based o e boosrap sadard devao, mea square error ad perceage relave bas. A smulao sud s carred ou o deerme e bes performg esmaor of e me-depede fe populao meas. Te smulao resuls sow a local lear regresso esmaor elds good properes. Kewords Logudal Surve, Nomoooe, Norespose, mpuao, Noparamerc Regresso. roduco Logudal surves refer o a pe of samplg surves doe repeaedl over me o e same sampled us. suc surves, daa wc are rc formao abou e specfc sampled u ca be obaed ad us suable for varous purposes. Wle logudal surves are regarded o be beer ad relable formg abou varous feaures of a sud u, e suffer from moooe ad erme paers of mssg daa. Ts s ofe as a resul of accessbl o or delberae refusal of respodes o provde formao afer avg parcpaed e surves us e occurrece of DO:.436/ojs December 7, 6

2 S. Pee e al. oresposes. Mssg daa are a problem because earl all sadard sascal meods presume complee formao for all e varables cluded e aalss. Usg daa w mssg values leads o reduco sample sze wc sgfcal affecs e precso of e cofdece erval, sascal power reduce ad based populao parameer esmaes. mpuao s oe of e approaces used o uvel fll ese mssg values. Over me, varous mpuao models ave bee developed ad e ave bee used o overcome que a umber of calleges caused b mssg daa. However, some sorcomgs sll exs suc as basedess ad effcec of esmaors. Ts s because mpuao models ave dffere assumpos bo paramerc ad oparamerc coexs. Paramerc meods lke maxmum lkelood esmao ave lmaos lke sesv o model msspecfcao wle oparamerc meods are more robus ad flexble []. Some of e meods used b [] are smple lear regresso mpuao ad Nadaraa-Waso ecque. From er smulao resuls, was foud a e smple lear regresso mpuao approac as e weakess of producg based esmaes eve we e resposes a a parcular me (cludg prevous values) are correcl specfed. O e oer ad, Nadaraa-Waso ecque of [3] ad [4] used e mpuao of mssg values e logudal daa as some weakesses of producg a large desg bas ad boudar effecs a gve urelable esmaes for ferece. As sow b [5] ad [6], a rval for Nadaraa-Waso ecque s e local lear regresso esmaor wc was foud o produce ubased esmaes wou boudar effecs. [7] suded e weged Nadaraa-Waso meod ad was cocered w e lmaos of e meod suc as cossec, asmpoc ormal ad e eror ad boudar po effecs. s sud, e foud a local lear regresso s muc beer a e weged Nadaraa-Waso meod as produces asmpocall ubased esmaes wou boudar effecs. Moreover, [8] also foud a e local lear regresso esmaor (roduced b [9]) as desrable properes. order o overcome e lmaos of Nadaraa-Waso esmaor, we derve a local lear regresso esmaor e mpuao of e orespodes a logudal daa se. Te asmpoc properes (ubasedess ad cossec) of e proposed esmaor are vesgaed. Comparsos bewee varous esmaors (paramerc ad oparamerc) are performed based o e boosrap sadard devao, mea square error ad perceage relave bas. A smulao sud s coduced o deerme e bes performg esmaor of e fe populao mea.. Assumpos ad Noaos ) All sampled us are observed o e frs me po ( ) ad rema e sample ll e fal me T. Te varable of eres, s e value of for e u a me po. ) Te predco process s pas las value depede ad e vecors 39

3 S. Pee e al. (,,, T,,,,, T, ) are depedel ad decall dsrbued (..d) from e superpopulao uder e model-asssed approac. For,, T ad,,, N ad e respose dcaor fuco, s 3) Te vecor ( ) ;, observed,,,, T () ;. uobserved,, T follows e Markov ca for logudal surve daa wou mssg values (,,,,,, ) (,,,, ) L L () 4) We assume a e populao P s dvded o a fxed umber of mpuao classes, wc are bascall uos of some small sraa. 3. Regular Codos Deoe f o be a probabl des fuco (pdf) of X ad g( x) p( x) f ( x) were p( x ) s defed b; ( ) (,, ) (, ) ad g ad f ave bouded secod dervaves p x P Y X P X (3) ) Te Kerel fuco K s a bouded ad wce couousl dffereable smmerc fuco o e erval [,], ad suc a k K( u) du, k uk ( u) du, k uk( u), k < ad { K( u) } du <. m s a leas wce couousl dffereable ever- ) Te regresso fuco ( ) were e egborood of x. ) Te sample surve varable of eres as a fe secod mome bouded o E <.,. Tus ( ) e erval ( ) v) Te codoal varace ( x ) Var ( X x ) 4. Meodolog 4.. mpuao Process σ s bouded ad couous. Cosderg e case of e las pas value, we do mpue for mssg value b e value obaed roug e predco procedure. Bu accordg o [], e jo dsrbuo of bvarae radom varables ( XY), s preserved we e mssg value, Y s mpued b e codoal dsrbuo of Y gve X. Terefore, cosderg e codoal mea mpuao approac for e sgle mpuao. Le ( ) E(,, ) ϕ,,,,,, (4) be e codoal expecao w respec o e superpopulao for uobserved value, w observed value, for. s erefore clear a we ϕ,, s kow, e e mpued value of uobserved, s gve b, ϕ,, ( ). cases were ϕ,, ( ) Equao (4) s ukow, for omoooe orespodes, we emplo e las value depede, 4

4 S. Pee e al. mecasm. Uder assumpo (), we ave ( ) E(,, ) ϕ,,,,,, (5) Usg Equao (4), we are lmed o do esmao b regressg e orespodes o e observed values based o e logudal surve daa, erefore, we appl e equvale Equao (5) wc allows esmao usg daa from all subjecs avg observed ad observed. Te, e mpuao of e orespodes s doe usg ϕ,, ( ) Equao (5) ad uder e las value depede assumpo, we are able o use auxlar surve daa regresso fg. Accordg o [], mpug oresposes usg (5) was doe for moooe case ad er approac s eas o appl f e codoal expecao sa, ϕ, ( x) (4) as a lear relaosp w x. Adopg e cocep of oparamerc meod [], ere, e local lear esmaor of ϕ, ( x) s ϕ, ( x). Le, be e varable of eres for e - u a me were,, N ad,, T. Assocaed w eac, are e kow x q,,, q,, Q, of q auxlar varables. To make e oaos ad wrgs smple, we relax e dex ad wre w a sgle subscrp, us, s wre as. Te regresso mpuao model η s gve b e relao ( ) m x + ε (6) suc a ε s are resduals wc are assumed o be depedel ormall dsr- bued w mea zero ad varace σ ( x ). s clear a ( ) ( ) E X x m x (7) ( ) σ x j Cov(, j X x, X xj) oerwse were m( x ) s a ukow regresso fuco wc s a smoo fuco of x. To oba e esmaor of m( x ) a local polomal fg b assumg a e regresso fuco w ( p + ) ad s dervaves, we use e weged dervaves a a po, sa x x, exss ad are couous. We ca rewre e mpuao model (6) as were approxmao of m ( x ) expaso. Te kerel weg gve as ( ) (8) m x + ε (9) abou s doe followg e Talor seres w ( x) K ( x x ) were s e badwd ad K s e kerel fuco wc sould be srcl posve ad K ( ) corols e wegs, x s e po of focus ad () x beg e covarae 4

5 S. Pee e al. w desg marx ceered a pas las value ad j s e order of e local polomal. Le Accordgl, for j, p S x x w x j j β j ( ) ( ) () m w ( x) ( x) ( x) Equao () s e Nadaraa-Waso esmaor. W esmaor e m ( x ), e codoal expecao gve b ( ) mpue e mssg values,.e. were ϕ ( ), ω s e surve weg ad Smlarl for j, w x x K ω x x K ω () ϕ s used o (3),,,,,,, for,, T (4), oerwse { β β( ) } ( ) S x x w x (5) Mmzg S w respec o β ad β Equao (5) ad solvg for β ad β, we ge ad w ( x) w( x)( x x) w ( x)( x x) w( x)( x x) β w( x)( x x) w( x) w( x)( x x) (6) w ( x)( x x) w( x) w ( x) w( x)( x x) β w( x)( x x) w( x) ( x x) w( x) (7) Defg: j Sj ( x) w ( )( ) x x x ad j T x w x x x, Tus: j ( ) ( )( ) Usg S ( ) j x, Equao (7), we oba ( x x) S( x) S( x) β w( x) (8) S( x) S( x) ( S( x) ) 4

6 S. Pee e al. ad w T ( ) j x, Equao (7), elds Smlarl, usg S ( ) j ( ) ( ) ( ) ( ) S ( x) S ( x) S ( x) S xt x S xt x β x, Equao (6) gves ( ) ( ) ( ) ( ) ( ) ( ) (9) S x x x S x β w ( x) () S x S x S x ad w Tj ( x ), Equao (6) becomes S( xt ) ( x) T( xs ) ( x) β S( x) S( x) S( x) Te local lear esmaor for e regresso fuco m ( ) m ( x) β ( x x ) β x s ow gve b: () + () Subsug for β (from Equao ()) ad β (from Equao (8)) Equao () gves, m ( x) w x + x x w x S( x) S( x)( x x) ( x x) S( x) S( x) ( ) ( ) ( ) (3) S( x) S( x) S( x) S( x) S( x) ( S( x) ) W esmaor, m ( x ), e codoal expecao gve b ( ) mpue e mssg values,.e. were ϕ S x S x x x ( ) ( )( ) ( ) ( ) ( ) ω, ( ) ( ) w x S x S x S x ω ( x x) S( x) S( x) ω + ( x x ) w x S ( x) S ( x) S ( x) ( ) ω ϕ s used o ω, s e weg accordg o e surve desg ad,, s as defed earler. 4.. Esmao of e Fe Populao Meas Usg e mpued Daa s sud, we cosder a fe populao from wc samples are draw. Before esmao of e populao parameers, mpuao process s doe. Suppose a e surve measuremes are,,, N o e varables B, B,, BN respecvel ad a smple radom sample wou replaceme, B, of sze s seleced from a fe populao, P of sze N. Te sample cosss of wo pars: B r ad B r, were B r s e se of all respodes e surve ad B r s e se of all o-respodes. Te mssg observaos of e sample u,, for are cosdered. mpuao of e mssg value, for B r ad s doe ad e a complee daa se s produced wc s e used e esmao of fe populao meas. Le Y be e fe populao mea a me po, for,,, T. Te value o be mpued for e o respode s deoed b, suc a e mpued daa s gve as (4) 43

7 S. Pee e al., Br observed value, B r mpued value #,,, (5) Te mea of e fe populao s gve b Y N Y N Now, usg e mpued daa, e esmaor of e fe populao oal s e sample oal of e mpued daa deoed b ad s gve b ( ) + (6),, B Tus, usg e mpued daa, e esmaor of e fe populao mea s e sample mea of e mpued daa deoed b #, B, gve b ω (7) Assumg a for eac B for eac P. N E ω (8) s B Te mpued values are reaed as f e were observed suc a bo observed ad e mpued are used e esmao of e populao mea: Sample mea for e mpued daa becomes w + w,, Br B r (9) Noe a e same weg due o samplg desg s used Equao (9) for all us e sample. for,, T. +,, Br B r (3) Sce s used as a cosa varable, Equao (3) s re-wre as + Br B r (3) As for [], e local cosa esmao for e orespodes Equao (3) s obaed as: ( ) ϕ, S x K x K s,, ad e local lear esmao for e orespodes, b: ω ω,,,,, (3) Equao (3) s gve 44

8 S. Pee e al. S ( x) S ( x)( x x ) ω w x ( ) ( ) ( ) ( x x) S( x) S ( x) ω,, ( x x + ) w ( ) x S( x) S( x) S ( x) ω,,,, ϕ, ( ) ( ) S x S x S x ω,, Clearl, Equao (3) s subsued b Equao (3) ad Equao (33) for use of local cosa esmaor ad local lear regresso esmaor respecvel. 5. Asmpoc Properes of e Esmaor e dervao of e asmpoc properes, we use e se of regular codos. Accordg o [], e asmpoc eor developme s provded b e cocep of a sequece of fe populaos { P ν } w ν sraa P ν ν. s assumed a ere s a sequece of fe populaos ad e correspodg sequece of samples. Te fe populao P dexed b ν s assumed o be a member of e sequece of e populaos. Te sample sze deoed b ν ad e populao sze deoed b N ν approac f as ν. Te uform respose ad e sze m ν of e orespomν des se B r sasf e codo α <. All lmg processes wll be uderν sood as ν suc a e regular codos are sasfed. For eas oao, e subscrp ν wll be gored e subseque work. Teorem. Assumg e regular codos ()-(v) ad also e assumpos seco old. Te uder e regresso mpuao model η, (6), e esmaor, Equao (3), s asmpocall ubased ad cosse for e populao mea Y. Proof. ) Bas of Y. Te geeral formula for e fe populao oal s gve b: Y + B BN (33) (34) were B ad BN are e sampled ad e o sampled ses respecvel. Equao (34) ca be decomposed as Y + + (35) Br Br BN For smplc, deoe B r, B r ad N r N respecvel rougou e remag work. From Equao (3), e esmaor for e fe populao oal s gve b Now cosder e dfferece, B b r, ( ) ad ( ) ( ) (36) + r r Y m x Y Y m ( x ) + + r r r r N 45

9 S. Pee e al. ( ( ) ) (37) r N Y Y m x ( ( ) ( ) ( ) ) (38) + r N Y Y m x m x m x Takg expecao o bo sdes of Equao (38), we ave ( ) ( ) ( ) ( ) ( ( ) ) ( ) (39) + r r N E Y Y E m x m x E m x E Clearl, ( ( ) ) sce E( ) m( x) Now, Assumg, (4) ad ece, E m x r. ( ) ( ) ( ) ( ) ( ) (4) E Y Y E m x m x E r N ( ) ( ) ( ) ( ) ( ) (4) E Y Y E m x m x m x r N Equao N N suc a ( N ), e m( x ) Bu from Lemma (see Appedx), ( ) ( ) ( ) r ( ) E Y Y E m x m x (4) E m x m x + um x + k + u k m x (43) were u x x. Tus e bas of Y becomes ( ) ( ) ( ( )) ( ) ( ) ( ) 3 k ( ) ( ) ( k + x x k 3 m x Bas Y) ( x x) m ( x) + r k (44) ) Varace of Y. Te varace of Y s gve b e varace of e error erm Y ( ) ( ) Y. Ta s, Var Y Var Y Y (45) Var m x r ε N ( ( ) ) (46) Tus, w ( x) σ ( x) σ ( x) σ ( x) (47) r r N ( ) σ ( ) ( ) σ ( ) ( ) σ ( ) Varas Y Y dk x r x N x (48) ( ) σ ( ) Var Y Y d x (49) as k r for suffcel large suc a ( N ) ad ( r) d K ( u) du. k 3) Mea square error (MSE) of Y. ; were 46

10 S. Pee e al. Fall, we ave ( ) ( ) MSE Y Bas Y + Var Y (5) k + ( x x ) k 3 m ( x) ( ) ( ) ( ) (5) MSE Y x x m x + + x d σ wc s e asmpoc expresso of e MSE for Y. MSE Y, ad us Y s cosse. k r K r Cosequel, s asmpocall ubased ad cosse. 6. Smulao Sud 6.. Descrpo of Logudal Daa as ad s seco, a sud of e fe populao mea esmaors based o four measures of performace (perceage relave bas (%RB), MSE ad boosrap sadard devao (SD boosrap)) s carred ou. Smulaos ad compuaos of e fe populao mea esmaors were doe usg R (R verso 3..3 (5--)) based o rus. For e e local lear ad local cosa esmaors, e Gaussa kerel w a fxed badwd of.75 was used. To f e oparamerc regresso, e loess fuco R was used. For comparso purposes, we used complee daa as our ma referece e evaluao of e performace of e esmaors (Proposed local lear esmaor, local cosa esmaor ad e smple lear regresso esmaor). s smulao sud, a sample of sze 5 was cosdered. Te logudal daa for eac of e sampled us s of sze T 4 a s,,,3,4. Ts wll eld 3 dffere paers of e logudal daa w eac of respode ad orespode values beg deoed b ad respecvel a dffere me pos. Logudal daa was geeraed accordg o wo models: ) model, smulao of (,,,3,4) s doe from a mulvarae ormal dsrbuo w e meas for e 4 me pos as.33,.94,.73, 3.67 respecvel ad e covarace marx followg e AR ( ) model w sadard error ad correlao coeffce.9. ) model, smulao of ( log ( ),,,3,4) s doe from a mulvarae ormal dsrbuo w e meas for e 4 me pos as.33,.94,.73, 3.67 respecvel ad e covarace marx followg e AR ( ) model w sadard error ad correlao coeffce.9. order o oba e omoooe paer e smulaed daa, we used e predeermed ucodoal probables of [3] sow Table. 6.. Boosrap Varace Esmao Te followg seps were used o oba e boosrap varace. ) We cosruced a pseudo populao b replcag e sample of sze 5 mes roug smulao rus. 47

11 S. Pee e al. Table. Probables of orespose paers for 4. Paer pe Norespose paer Normal/Log-ormal daa Toal Probabl Moooe Nomoooe Complee daa ) A smple radom sample of sze was draw w replaceme from e pseudo populao. 3.) We appled e smple lear regresso, local cosa ad local lear regresso mpuao models o mpue e mssg s of e sample. 4) Repeag e seps ad 3 for a large umber of mes ( B ) o oba ( ) ( B) ( ) Y,, Y were Y b s e aalog of Y, for e b- boosrap sample. 5) Oba e boosrap varace of Y b e formula B b. (.) Vboo Y Y Y were Y s e mea boosrap aalog of Y, gve b ( ) ( ) ( ) ( ) b Y B. (.) B ( b) Y b B 6.3. Resuls ad Dscusso Te resuls of s smulao sud are summarzed Table ad Table 3. erms of e perceage relave bas (%RB), a me po, ca be see a e local lear esmaor as e leas value followed b e Nadaraa-Waso esmaor ad e e smple lear regresso esmaor, wc was e larges value of %RB. A me po 3, observe a e e smple lear regresso esmaor as e leas %RB value compared o a of e local lear esmaor ad e Nadaraa- Waso esmaor performed wors w e larges %RB. Te %RB values of e local lear esmaor ad e smple lear regresso esmaor are ver muc closer o zero a ose for e oer esmaors. A me po 4, observe a e local lear esmaor as e leas %RB value followed b e smple lear regresso esmaor ad e Nadaraa-Waso esmaor performed wors. Troug comparsos based o %RB w referece o e complee daa, e local lear esmaor as s %RB values approacg zero. erms of MSE, a me po, Nadaraa-Waso esmaor as e leas values followed b e local lear esmaor ad lasl e smple lear regresso esmaor wc as e larges values. A me po 3, e local lear esmaor as e leas values of MSE followed b e smple lear regresso esmaor ad lasl 48

12 S. Pee e al. Table. Smulaed resuls for mea esmao (ormal case). Meod Qua 3 4 Complee daa Local Lear Regresso Nadaraa-Waso Smple lear regresso Mea Sadard devao %RB.... MSE SD boosrap Mea Sadard devao %RB MSE SD boosrap Mea Sadard devao %RB MSE SD boosrap Mea Sadard devao %RB MSE SD boosrap Table 3. Smulaed resuls for mea esmao (log-ormal case). Meod Qua 3 4 Complee daa Local Lear Regresso Nadaraa-Waso Smple lear regresso Mea Sadard devao %RB.... MSE SD boosrap Mea Sadard devao %RB MSE SD boosrap Mea Sadard devao %RB MSE SD boosrap Mea Sadard devao %RB MSE SD boosrap

13 S. Pee e al. e Nadaraa-Waso esmaor wc as e larges MSE value. A me pos 4, Nadaraa-Waso esmaor as e leas values of MSE followed b e smple lear regresso esmaor ad lasl e local lear esmaor wc as e larges MSE value. erms of e boosrap sadard devao, ca be see a e local lear esmaor performs e bes a all e ree me pos, 3, ad 4 wc s values are eve lower a ose of e complee daa mplg a e resuls go w e local lear esmaor are e bes. Te smple lear regresso ad Nadaraa-Waso esmaors are compeg ercageabl erms of performace for e boosrap samples. erms of e perceage relave bas (%RB), a me pos ad 4, observe a e smple lear regresso esmaor as e leas %RB values followed b e local lear esmaor ad e Nadaraa-Waso esmaor as e bgges %RB values. Based o ese aforemeoed resuls, s vable o coose e bes esmaor as e local lear esmaor wc adles bo lear ad olear models. A me pos 3, observe a e local lear esmaor as e leas %RB value followed b smple lear regresso esmaor ad lasl e Nadaraa-Waso. Ts mples a, for 5, e local lear esmaor as e smalles bas close o zero as for e complee daa, ece e bes esmaor compared o oers. erms of e MSE, a me pos ad 4, Nadaraa-Waso esmaor as e leas values of MSE, followed b e smple lear regresso esmaor ad lasl e local lear esmaor wc as e larges values of MSE. A me po 3, e e local lear esmaor as e leas values mplg a performed well a me po 3. erms of e boosrap sadard devao, observe from Table 3 a e local lear esmaor performs e bes a all e ree me pos sce as e leas boosrap sadard devaos ad ese values are eve smaller a ose of e complee daa order of creasg me. From Table 3 of resuls, s ca be see a e boosrap sadard devaos of e local lear esmaor are more close o ose of e Nadaraa-Waso esmaor a e smple lear regresso esmaor. 7. Cocluso Geerall, orespodes a surve daa as a sgfca mpac o e bas ad e varace of e esmaors ad erefore, before usg suc daa sascal ferece, mpuao w a approprae ecque oug o be doe. s sud, e ma objecve was o oba a mpuao meod based o local lear regresso for omoooe orespodes logudal surves ad deerme s asmpoc properes. Comparg e paramerc ad oparamerc meods, oparamerc meods performed beer a e paramerc meods. Ts was evde from e MSE ad %RB values bo e ormal ad log-ormal daa. Amog e oparamerc meods, e local lear esmaor was e bes esmaor as beaved beer 5

14 S. Pee e al. a e Nadaraa-Waso esmaor erms of %RB. erms of e boosrap sadard devao, e local lear esmaor performs e bes a all e ree me pos sce as e leas boosrap sadard devaos for e wo daa ses. Geerall, e local lear esmaor performs relavel well ad parcular e ormal daa. We coclude a use of e oparamerc esmaors seem plausble bo eorecal ad praccal scearos. Ackowledgemes We ws o ak e Afrca Uo Commsso for full fudg s researc. Refereces [] Dorfma, A.H. (99) Noparamerc Regresso for Esmag Toals Fe Populao. Proceedg Seco of Surve Meodolog. Amerca Sascal Assocao Alexadra, VA, [] Xu, J., Sao, J., Pala, M. ad Wag. L. (8) mpuao for Nomoooe Las-Value- Depede Norespodes Logudal Surves. Surve Meodolog, 34, [3] Nadaraa, E.A. (964) O Esmag Regresso. Teor of Probabl ad s Applcaos, 9, 4-4. [4] Waso, G.S. (964) Smoo Regresso Aalss. Sak: Te da Joural of Sascs, 6, [5] Hase, T.J. ad Loader, C. (993) Local Regresso: Auomac Kerel Carper (w Dscusso). Sascal Scece, 8, -43. [6] Wad, M.P. ad Joes, M.C. (995) Kerel Smoog. Capma & Hall, Lodo. [7] Ca, Z. () Weged Nadaraa-Waso Regresso Esmao. Sascs & Probabl Leers, 5, [8] Fa, J. ad Gjbels,. (996) Local Polomal Modellg ad s Applcaos. Capma ad Hall, Lodo. [9] Soe, C.J. (977) Cosse Noparamerc Regresso. Te Aals of Sascs, 3, [] Rub, D.B. (987) Mulple mpuao for Norespose Surves. Jo Wle & Sos, c., New York. ps://do.org/./ [] Pak, M.C. (997) Te Geeralzed Esmag Equao Approac We Daa Are No Mssg Compleel a Radom. Joural of Amerca Sascal Assocao, 9, [] Ceg, P.E. (994) Noparamerc Esmao of Mea Fucoals w Daa Mssg a Radom. Joural of e Amerca Sascal Assocao, 89, [3] Sao, J., Kle, M. ad Xu, J. () mpuao for Nomoooe Norespose e Surve of dusral Researc ad Developme. Surve Meodolog, 38, [4] Masr, E. (996) Mulvarae Local Polomal Regresso for Tme Seres. Uform Srog Cossece ad Raes. Joural of Tme Seres Aalss, 7,

15 S. Pee e al. Appedx LEMMA. Te bas of m ( ) x s gve b Bas m x um x + k + u k m x (5) ( ) ( ) ( ( )) ( ) ( ) 3 k Uder e regular codos seco 3, ( ( )). PROOF OF LEMMA. Proof. From Equao (3), were ( ) Bas m x as ad ( ) ( ) + ( ) ( ) (53) m x w x x x w x ( ) ( ) ( ) ( ) ( ) ( ) s x x x s x s x s x s x ( ) w x w x w x w x ( x x) s( x) s( x) ( ) s( x) s( x) ( s( x) ) Te expecao of m ( ) x s gve b, ( ), were w ( x) K x x. ( ) ( ) [ ] + ( ) ( ) [ ] (54) E m x w x E x x w x E ( ) { ( ) + ( ) ( )} E m x m x x x m x ( ) ( ) ( ) + ( )( ( ) ( ) ( ) ( )) S ( x) S ( x) S ( x) S x S x S3 x x x S x S3 x S x S x + m Te bas of m ( ) Bas ( m ( x) ) ( x x ) m ( x) x s erefore gve b ( ) ( ) ( ) + ( )( ( ) ( ) ( ) ( )) S ( x) S ( x) S ( x) S x S x S3 x x x S x S3 x S x S x + m For fxed desg pos of,, e expresso j j+ j+ 3 S j ( x) ( x ) ( ) x w x k j + o( ) almos everwere, see [4]. Now, ) ) 3) x s o e erval ( ) ( ) ( ) ( ) + ( ) ( ) + ( ) 6 8 k+ o ( ) S x S x S3 x k o o k3 o ( ) ( ) ( ) ( ) S x S x S x S x o( ) k3 o( ) o( ) k o( ) o( ) ( ) ( ) ( ) + ( ) + ( ) ( ) 4 6 k+ o ( ) k + S x S x S x k o o o ( x ) ( x ) (55) (56) 5

16 4) S ( x) S ( x) S ( x) S ( x) o( 3 ) o( 4 ) o( 4 ) o( 3 ) S. Pee e al. + + Equao (56) becomes Leg { } ( ( )) ( ) ( ) + + ( ) Bas m x x x m x k x x k x m ( x ) 3 (57) k x u u x. x Hece, e bas of m ( ) ad ece e resul. x ca be re-wre as Bas m x um x + k + u k m x (58) ( ) ( ) ( ( )) ( ) ( ) 3 k LEMMA. Te asmpoc expresso of e varace of m ( ) ad ; were ( ) as PROOF OF LEMMA. Proof. Usg Equao (3), sce ( j) x s gve b dk Var ( m ( x) ) ( x) σ (59) d K u du. k ( ( )) ( ) ( ) ( ) + ( ) ( ) (6) Var m x w x Var x x w x Var Cov,. follows a ( ( )) ( ) ( ) ( ) ( ) σ + σ ( ) (6) Var m x w x x x x w x x were ad w ( x) w ( x) (6) ( o( ) + o( ) o( ) o( ) ) 4 6 ( + ( )) w ( x) w( x) k o as. Tus, Var m x w x x (63) ( ( )) ( ) σ ( ) Te asmpoc expresso of e varace of m ( ) d K u du. Hece e resul. were ( ) k x becomes dk Var ( m ( x) ) ( x) σ (64) 53

17 S. Pee e al. MSE of m ( x ) From LEMMA ad, e MSE of m ( ) x becomes d MSE m ( x) ( x x ) m ( x ) + k + ( x x ) k m x + x σ k ( ) ( ) ( ) 3 k (65) Subm or recommed ex mauscrp o SCRP ad we wll provde bes servce for ou: Accepg pre-submsso qures roug Emal, Facebook, Lked, Twer, ec. A wde seleco of jourals (clusve of 9 subjecs, more a jourals) Provdg 4-our g-qual servce User-fredl ole submsso ssem Far ad swf peer-revew ssem Effce peseg ad proofreadg procedure Dspla of e resul of dowloads ad vss, as well as e umber of ced arcles Maxmum dssemao of our researc work Subm our mauscrp a: p://papersubmsso.scrp.org/ Or coac ojs@scrp.org 54