oparametric aalysis of covariace Holger Dette Rur-Uiversitat Bocum Faultat fur Matemati 4478 Bocum Germay email: olger.dette@rur-ui-bocum.de FAX: +49 34 7 94 559 atalie eumeyer Rur-Uiversitat Bocum Faultat fur Matemati 4478 Bocum Germay email: atalie.eumeyer@rur-ui-bocum.de July 4, Abstract I te problem of testig te equality of regressio curves from idepedet samples we discuss tree metods usig oparametric estimatio teciques of te regressio fuctio. Te rst test is based o a liear combiatio of estimators for te itegrated variace fuctio i te idividual samples ad i te combied sample. Te secod approac trasfers te classical oe-way aalysis of variace to te situatio of comparig oparametric curves, wile te tird test compares te diereces betwee te estimates of te idividual regressio fuctios by meas of a L -distace. We prove asymptotic ormality of all cosidered statistics uder te ull ypotesis, local ad xed alteratives wit dieret rates correspodig to te various cases. Additioally cosistecy of a wild bootstrap versio of te tests is establised. I cotrast to most of te procedures proposed i te literature te metods itroduced i tis paper are also applicable i te case of dieret desig poits i eac sample ad eteroscedastic errors. A simulatio study is coducted to ivestigate te ite sample properties of te ew tests ad a compariso wit recetly proposed ad related procedures is performed. AMS Classicatio: Primary 6G5 Keywords ad Prases: oparametric aalysis of covariace, variace estimatio, compariso of regressio curves, goodess-of-t, wild bootstrap. Itroductio A importat problem i applied regressio aalysis is te compariso of a respose Y across several groups i te presece of a covariate eect. I geeral tis model ca be writte as (.) Y ij g i (t ij )+ i (t ij )" ij (i ;:::;; j ;:::; i ); were " ij are idepedetly idetically distributed errors, g i ; i are te regressio ad variace fuctio i te it group (i ;:::;) ad te covariate t ij varies i te
iterval [; ]: I tis paper we are iterested i te problem of testig te equality of te mea fuctios, i.e. (.) H : g g ::: g versus H : g i 6 g j (9 i; j f;:::;g): Muc eort as bee devoted to tis problem i te recet literature [see e.g. Hardle ad Marro (99), Kig, Hart ad Werly (99), Hall ad Hart (99), Delgado (993), Youg ad Bowma (995), Kulaseera (995), Kulaseera ad Wag (997), Hall, Huber ad Specma (997) or Dette ad Mu (998)]. Most autors cocetrate o te case of two samples ( ) ad a omoscedastic error i all groups. Hardle ad Marro (99) cosider a semi-parametric approac i te case of equal desigs (i.e. ::: ;t ij t j ;i;:::;): Kig, Hart ad Werly (99) ad Hall ad Hart (99) discuss te completely oparametric omoscedastic (i.e. i (t) ; i ;:::;) i model i te case of equal desigs poits. Wile te latter approac ca be geeralized to te case of uequal desigs poits [see Hall ad Hart (99)], Kulaseera (995) poits out some drawbacs of te test i tis case ad proposes several alteratives wic are applicable i te model (.) uder te additioal assumptio of omoscedasticity i all groups. Tis approac ca detect alteratives wic coverge to te ull at a rate of order p but te autor also metios some practical problems of tis procedure. O te oe ad te level of te test depeds sesitively o te smootig parameter, o te oter ad larger oises yield levels substatially dieret from te omial levels [see also Kulaseera ad Wag (997) for a detailed simulatio study ad data-drive guidelies for badwidt selectio]. Moreover, a geeralizatio to a eteroscedastic error or a multivariate predictor seems to be dicult. A rater dieret test was itroduced by Youg ad Bowma (995) wo geeralized te oe-way aalysis of variace to te model (.). Uder te assumptio of ormally distributed omoscedastic errors over all groups tese autors proposed a -approximatio for te distributio of te test statistic. Altoug te ite sample properties of te test uder tese assumptios loo promisig, a geeralizatio to te geeral eteroscedastic, oormal case does ot appear trivial ad te asymptotic properties of tis test ave ot bee ivestigated so far. To our owledge te problem of testig te equality of te regressio fuctios i te completely eteroscedastic model (.) wit a uivariate predictor ad uequal desig poits was rstly cosidered by Dette ad Mu (998) wo itroduced a simple estimator of te distace X i<j [g i (t) g j (t)] dt ad proved a asymptotic ormal law wit a p -rate for a correspodig test statistic. As a cosequece tis test is ot eciet from a asymptotic poit of view. I tis paper we discuss various tests for te ypotesis (.) wic are directly applicable i te geeral model (.), do ot require ay additioal assumptios (as omoscedasticity or equal desig poits) ad improve o te asymptotic eciecy of te test of Dette ad Mu (998). Moreover, te ew metods ca easily be exteded to te case of multivariate predictors. A rst metod for testig te ypotesis (.) is based o a dierece betwee a oparametric variace estimator i te combied sample fy ij jj ;:::; i ; i ;:::;g ad te correspodig estimators i te idividual samples fy ij jj ;:::; i g ad yields i fact a estimator of a alterative measure of equality. Our secod proposal is to use Youg ad Bowma's (995) test also i te situatio of a eteroscedatic error. Fially, we suggest a geeralizatio of Kig, Hart ad Werly's (997) test to te geeral setup (.), wic compares te estimates of te regressio fuctios i te idividual samples. Tis metod is closely related to a approac itroduced by Roseblatt (975) i te cotext of testig
idepedece ad furter developed by Hardle ad Mamme (993) ad Gozalez Mateiga ad Cao (993) for te problem of testig te parametric form of a regressio fuctio. We prove asymptotic ormality of all proposed test statistics uder te ull ypotesis ad xed alteratives wit dieret rates of covergece correspodig to bot cases. I Sectio we itroduce te dieret metods, state te mai asymptotic results ad discuss various lis betwee te dieret approaces. I Sectio 3 we ivestigate te ite sample properties of some of te proposed tests ad perform a compariso wit alterative procedures wic ave recetly bee suggested i te literature [see Hall ad Hart (99), Delgado (993), Kulaseera (995), Kusaleera ad Wag (997)]. It is demostrated tat a wild bootstrap versio of te test based o te dierece of variace estimators as excellet ite sample properties ad is very ofte remarably more powerful ta several oter tests proposed i te literature, wic ca detect alteratives covergig to te ull at a parametric rate. Fially, some of te proofs, wic are cumbersome, are give i te appedix of Sectio 4. Testig te equaltiy of regressio fuctios by erel based metods I order to motivate te dieret metods for testig te ypotesis of te form (.) ad to ivestigate te asymptotic distributio of te correspodig test statistics we eed a few regularity assumptios. Let P i i deote te total sample size ad assume (.) i i + O( ); i ;:::; for give costats ;::: (; ): Let r ;:::;r deote positive desities o te iterval [; ] suc tat te desig poits t ij satisfy (.) tij r i (t)dt j i ; j ;:::; i ;i;:::; [see Sacs ad Ylvisacer (97)]. Trougout tis paper we will assume te cotiuity of te variace fuctios i () (i ;:::;) ad additioally tat te desig desities i (.) ad te regressio fuctios are sucietly smoot, i.e. (.3) g j ;r j C (r) [; ]; j ;:::; were r ad C (r) [; ] deotes te space of r-times cotiuously dieretiable fuctios o te iterval [; ]: Let P i K t t ij j i Y ij (.4) ^g i (t) P i K t t ij j deote te adaraya-watso estimator of te it regressio fuctio g i ad i te correspodig badwidt [see adaraya (964) ad Watso (964)]. We assume tat te erel i (.4) is supported o a compact iterval, say [ ; ]; ad of order r [see Gasser, Muller ad Mammitzsc (985)], i.e. (.5) ( ) j j! K(u)u j du 8 >< >: i : j : j r r 6 : j r; 3
R were K (u)du <. If te ypotesis of equal regressio fuctios is valid, te total sample could be used to estimate te commo regressio, i.e. P P i K t Y i j ij (.6) ^g(t) P i P i j K were is a furter badwidt. For te sae of simplicity te asymptotic aalysis of te statistics proposed below is performed for te case of equal badwidts i (i ;:::;) i te estimators (.4) ad (.6) were te badwidt satises t ij t t ij ; (.7)!; O( (4r+) ) :. Comparig variace estimators Our rst measure of equality betwee te dieret regressio fuctios is obtaied by comparig te oparametric variace estimators of te idividual samples wit a oparametric variace estimator of te pooled sample. To be precise let (.8) ^ i i X i j (Y ij ^g i (t ij )) (i ;:::;) deote te estimator of te variace of te it sample itroduced by Hall ad Marro (99), were ^g i is te oparametric estimator of te regressio fuctio i te it sample deed i (.4). Altoug tese autors cosidered oly a omoscedastic model it will be sow i te appedix (see Lemma 4.) tat i te eteroscedastic model ^ R i cosistetly estimates te itegrated variace fuctio i (t)r i (t)dt of eac sample (i ;:::;). I te followig we will cosider te aalogue of (.8) for te total sample size (.9) ^ X i i j (Y ij ^g(t ij )) : It is proved i te appedix tat uder te ypotesis of equal regressio curves tis is essetially a estimator for a covex combiatio of te idividual itegrated variace fuctios, i.e. i i (t)r i i(t) dt: For tese reasos we propose as a test statistic T () ^ i i^ i : Te asymptotic properties of te statistic T () are listed i te followig teorem. 4
Teorem.. Assume tat (.), (.), (.3), (.5), (.7) are satised. (i) If te ypotesis of equal regressio fuctios is valid, te te statistic T () satises p (T () B () r D() ) D! (; ;) were (.) B () D () r ((g R) (r) K (u) du g R (r) ) (t) dt R(t) K() X j r j j j (t)r j(t) R(t) j ((g r j ) (r) dt j (t) dt ; g r (r) j ) (t) dt r j (t) te asymptotic variace is give by (.) ; + (K K K) (u) du j l l6j j r 4 j (t) j (t) j R(t) j (t) l (t) jr j (t) l r l (t) o dt ; R (t) dt K K is te covolutio of te erel wit itself ad R deotes te covex combiatio of te uderlyig desities, i.e. R(t) j j r j (t): (ii) Uder te alterative g i 6 g j (9 i; j f;:::;g) te statistic T () satises p (T () M ;) D! (; ;) were (.) M ; j l l<j ad te asymptotic variace is give by (.3) ; 4 j X. A AOVA-type statistic l l6j (g j g l ) (t) jr j (t) l r l (t) R(t) (g j (t) g l (t)) lr l (t) R(t) dt j (t) j r j (t) dt: Te followig metod for testig te equality of te regressio fuctios was itroduced by Youg ad Bowma (995) i te cotext of a omoscedastic ormal distributio for te error over all samples. Te correspodig statistic is closely related to te dierece of variace estimators itroduced i 5
Sectio.. It will be sow i tis sectio tat te metod proposed by tese autors is also applicable i te geeral situatio of oormal eteroscedastic errors. Te test statistic of Youg ad Bowma (995) is motivated by te classical oe-way aalysis of variace ad give by Y ^s T () ; were (.4) T () X i i j [^g(t ij ) ^g i (t ij )] ; ^g, ^g ;:::;^g are deed i (.6) ad (.4), respectively, ad ^s ( ) i X i j (Y i;j+ Y i;j ) is a pooled mea of te dierece based estimators for te variaces i te idividual samples [see e.g. Rice (984)]. Te statistic ^s is also a cosistet estimator of (.5) s j j j (t)r j (t)dt wic follows by a straigtforward calculatio [see also Kulaseera ad Wag (997) for a related result uder omoscedasticity]. ote tat tere is a strog li betwee te statistics T () () ad T. Wile te statistic T () is comparig te regressio fuctios troug residual sums of squares te statistic T () compares te curves troug te tted values. I te case of a xed desig, equal omoscedastic variaces i all groups ad a ormally distributed error Youg ad Bowma (995) proposed a -approximatio of te correspodig test statistic uder te ull ypotesis. Tese restrictios allow a rapid ad accurate calculatio of te p-value [see Youg ad Bowma (995) for more details]. It is also wortwile to metio tat te use of te same smootig parameters i te estimates of te idividual regressio fuctio yields a direct cacelatio of te bias. Obviously, te umerator of Y give i (.4) is a estimate for a appropriate measure of equality of te regressio curves ad we will also use T () as a test statistic for te ypotesis (.) i te geeral situatio of ot ecessarily omoscedastic ad ormally distributed errors. Te followig result maes tis euristic argumet more precise ad provides te asymptotic properties of te statistic (.4). As a by product it also proves cosistecy of te test proposed by Youg ad Bowma (995) if te required assumptios for te ite sample size approximatio used by tese autors are ot satised. Moreover critical values could be obtaied from a approximatio by a ormal distributio or a wild bootstrap procedure as proposed i Sectio 3. Teorem.. Assume tat (.), (.), (.3), (.5), (.7) are satised. (i) If te ypotesis of equal regressio curves is valid, te te statistic T () satises p (T () + D () B() r ) D! (; ;) deed i (.4) 6
were B () B () is deed i (.), (.6) D () K (u)du ad te asymptotic variace is give by (.7) ; j (K K) (u)du + j l l6j j r j (t) j R(t) j (t)dt; j r 4 j (t) j (t) R(t) j (t) l (t) jr j (t) l r l (t) R (t) (ii) Uder te alterative g i 6 g j (9 i; j f;:::;g) te statistic T () p (T () M ;) D! (; ;) o dt dt were M M ad ; ; ; ; are deed i (.) ad (.3), respectively..3 Pairwise compariso of regressio curves deed i (.4) satises Followig Roseblatt (975), Hardle, Mamme (993) ad Gozalez Mateiga, Cao (993) a obvious alterative test of te ypotesis (.) could be obtaied from a pairwise compariso of te estimators of te regressio fuctios. To tis ed we cosider te statistic (.8) i X T (3) i j [^g i (t) ^g j (t)] w ij (t)dt were w ij () are positive weigt fuctios satisfyig w ij w ji ( j<i): A similar statistic was cosidered by Kig, Hart ad Werly (99) i te case of two samples wit equal desig poits (ere te itegral was approximated by a sum ad a costat weigt was used). A calculatio similar as give i te proof of Lemma 4. of te appedix sows tat (.9) E[T (3) ] ( D(3) were te costats D (3) ;M ;3 are deed by (.) (.) D (3) M ;3 j + O( ) uder H D(3) + M + O( ) uder H ;3 K (u) du l l<j j j (t) j r j (t) l l6j (g j g l ) (t)w jl (t) dt: w jl (t) dt; ote tat i cotrast to Teorem. ad. tere does ot appear a term of order r i (.9), wic is i fact a result of te applicatio of equal badwidts i te estimates of te regressio fuctios i te idividual samples. Te followig result ca be proved usig similar argumets as give for te proof of Teorem. i te appedix [see Sectio 4]. 7
Teorem.3. Assume tat (.), (.), (.3), (.5), (.7) are satised. (i) If te ypotesis of equal regressio curves is valid, te te statistic T (3) satises p (T (3) were te asymptotic variace is deed by (.) ;3 (K K) (u) du + D(3) j j ) D! (; ;3) l l6j 4 j (t) j r j (t) w jl (t) dt l l6j j (t) l (t) j r j (t) l r l (t) w jl(t) dt o : deed i (.8) (ii) Uder te alterative g i 6 g j (9 i; j f;:::;g) te statistic T (3) deed i (.8) satises p (T (3) M ;3) D! (; ;3) were M ;3 is deed i (.) ad te asymptotic variace is give by ;3 4 i j j6i l l6i (g i (t) g l (t))(g i (t) g j (t))w ji (t)w li (t) i (t) i r i (t) dt: Remar.4. It is wortwile to metio tat tere is a strog li betwee te tree statistics T (), T (), T (3), wic ca icely be explaied by looig at te classical oe-way aalysis of variace model, were X ij ( i ; ) ; j ;:::; i ; i ;:::;: Here te deomiator of te correspodig F -test correspods to te statistic T () Bowma (995) ad ca be decomposed as (.3) Xi X i i X i i j X ij X X i i j X ij Xi ; of Youg ad were te rst term o te rigt ad side is a estimator of te variace from te pooled sample (assumig equal meas i all samples) ad te secod term is a combiatio of te variace estimators i te idividual samples. Cosequetly te rigt ad side of (.3) correspods to te statistic T () itroduced i Sectio.. Terefore i liear models bot statistics are equivalet, wile for oparametric models tere appear diereces because te cross product terms ivolve a ovaisig bias. Similary, we ave te represetatio i i Xi X i i Xi j Xj j wic establises a aalogy betwee te statistics T () ad T (3). 8
.4 Some asymptotic power comparisos As a cosequece of Teorem. -.3 we obtai cosistet, asymptotic level tests by rejectig te ypotesis of equal regressio curves weever (.4) p (T (i) D (i) B (i) r ) > ;iu ; i ;; 3 were B (3),B (i) ;D(i), ;i are deed i Teorem. -.3 ad ave to be replaced by cosistet estimators. I te followig sectio we will illustrate te performace of a wild bootstrap versio of te tests give by (.4), because te speed of covergece uder te ull ypotesis is usually rater slow [see also Azzalii ad Bowma (993), Hjellvi ad Tjsteim (995) or Alcala, Cristobal ad Gozalez Mateiga (999) for similar observatios]. Moreover, te secod part of Teorem. -.3 provides a importat advatage i te applicatio of tese tests (compared to most of te procedures proposed i te literature). It is well ow tat i te problem of testig goodess-of-t te essetial error is te type II error ad a large observed p-value does ot give ay empirical evidece for te ull ypotesis [see e.g. Berger ad Delampady (987), Staudte ad Seater (99)]. Te secod part of Teorem. -.3 ow provides a approximatio for te type II error of te test by p o (.5) P ("rejectio") ;i M ;i u ;i p p M ;i ; i ;; 3: ;i We remar tat te approximatio by a ormal distributio uder xed alteratives is more reliable ta uder te ull ypotesis, because it is similar to te approximatio by a ormal distributio i te classical cetral limit teorem (see te proof i Sectio 4.3). Moreover, te secod part of Teorem. -.3 ca also be used for testig te precise ypoteses [see Berger ad Delampady (987)] (.6) H : M > versus H ;i : M ;i were is a sucietly small costat suc tat weever M ;i te experimeter agrees to aalyze te data uder te assumptio of equal regressio curves. ote tat te rejectio of H i (.6) allows us to sow tat te regressio fuctios are \close" at a cotrolled error rate. Remar.5. Te possibility of coosig te weigt fuctios i (.8) leaves some freedom for te statistic T (3) ad usig w ij ir i j r j (.7) R gives a statistic wit a asymptotically similar beaviour as described i Teorem. ad. for te tests based o T () () ad T. Tis weigt fuctio is very atural because uder te additioal assumptio of omoscedasticity it maximizes te asymptotic power for comparig te curves g i ad g j wit respect to te coice of te weigt fuctio. To be precise assume tat ;te a straigtforward calculatio [see also te derivatio of (.5)] sows tat te probability of rejectio is a icreasig fuctio of (.8) (M ;3) ;3 R R ( (g g ) (t)w (t)dt) (g g ) (t)w(t)( + )dt r(t) r(t) (g g ) ( r (t) + r (t) ) dt (M ;) 9 ;
were M ; M ; ad ; ; are deed i (.) ad (.3), respectively, ad te secod lie follows from Caucy's iequality. ow discussig te equality i (.8) sows tat te maximal power (wit respect to te coice of te weigt fuctio w ) is obtaied by te weigt (.7). For tese reasos te tests based o T () () [Youg ad Bowma (995)] ad T [proposed i Sectio ] sould be prefered because tey automatically adapt to te best possible (but uow) weigt fuctio for te maximizatio of te power at ay xed alterative. Remar.6. I te remaiig part of tis sectio we will cocetrate o te asymptotic beaviour of te dieret tests wit respect to local alteratives. For te sae of trasparecy we will cocetrate o te case of samples. Tere is o dierece i te discussio of te geeral situatio of 3 regressio fuctios. We will adopt a approac of Roseblatt (975), wo proposed to cosider alteratives of te form c (.9) g () g ()+ s were s is a cotiuously dieretiable fuctio of order r ad c [; ] a give costat. are sequeces covergig to suc tat ad (.3) p ; o() ; r o( r ) (a typical example i te case r is 9, 336 ad 6 ). For alteratives of te form (.9) satisfyig (.3) it follows by similar argumets as give i te appedix for te proof of Teorem. tat (.3) p T (i) D (i) B (i) r D! ( (i) ; ;i) were te costats B (i), D(i) ad ;i are deed i Teorem. -.3 (ote tat B(3) ) ad (.3) (i) (i) 8 >< >: 8 >< >: s (x)dx r (c)r (c) r (c)+ r (c) if i ; s (x)dx w (c) if i 3 A similar result is obtaied for local alteratives of te form (.9) wit c ad, i.e. g g + s ( p ). I tis case (.3) is still valid, wit a dieret expectatio i te limit distributio, i.e. s (x) r (x)r (x) dx if i ; r (x)+ r (x) (.33) s (x)w (x)dx if i 3: For a asymptotic aalysis of te tree testig procedures wit respect to tese local alteratives we use te optimal (but uow) weigt fuctio (.7) for w i te deitio of te statistic
. Te compariso ca ow easily be performed by looig at te dieret variaces i (.3) ad observig te relatio (K K) (x)dx K (x)dx (K K K) (x)dx T (3) wic as bee proved by Biederma ad Dette (). From tis iequality it follows tat ; ;3 ; ad cosequetly te procedures based o T (), T (3) are asymptotically more eciet as te test based o T (). However, some care is ecessary wit tis iterpretatio, because te speed of covergece i (.3) is rater slow ad te asymptotic aalysis usually requires a rater large sample size [see Azzalii ad Bowma (993), Hjellvi ad Tjsteim (995) or Alcala, Cristobal ad Gozalez Mateiga (999) for similar observatios]. For realistic sample sizes te approximatio (.5) idicates a similar beaviour of all tree metods. Moreover, for moderate sample sizes te bias as always to be tae ito accout ad a superiority of oe of te tree metods ca ot be establised i geeral. I te examples preseted i Sectio 3 we observed a muc better performace of te test based o T ()..5 Geeralizatios: Dieret badwidts, smootig teciques ad radom desig Remar.6. ote tat we ave assumed te equality of all badwidts i Sectio. -.3, wic substatially simplies te presetatio of te asymptotic results ad teir proofs. everteless i practice it is strictly recommeded to coose te badwidt i (i ;:::;) ad of eac estimator ^g i ad ^g accordig to te size of te correspodig sample. If tere exist costats b ;:::;b (; ) suc tat tese badwidts satisfy (.7) ad (.34) i b i + O( ) ; i ;:::; as!, similar results as give i Teorem. -.3 ca be establised, were te costatsb (i) ad D (i) additioally deped o te proportios b ;:::;b (ote tat B (3) does ot vais i tis case). For more details we refer to eumeyer (999) ad Dette ad eumeyer (999). Remar.7. It sould also be poited out tat te asymptotic results give i Teorem. -.3 do ot deped o te special structures of te smootig procedures used i te costructio of te variace estimators. We used te adaraya-watso estimator for te calculatio of residuals because for tis coice te proofs give i te appedix are more trasparet. For example, a local polyomial estimator [see Fa (99) or Fa ad Gijbels (996)] ca be treated i te same way wit greater matematical complexity but witout cagig te structure of te asymptotic results. Altoug local polyomial estimators ave various advatages for te estimatio of te regressio fuctio, particulary at te boudaries, our simulatio results sowed tat tis superiority is ot reected i te problem of testig te equality of regressio fuctios. A euristical explaatio of tis observatio is tat te metods preseted i Sectio. -.3 essetially avoid te direct estimatio of te regressio fuctio ad oly use estimates for quatities smooted by liear itegral operators. everteless tere are still teoretical advatages of usig local smootig i te deitio of te statistics T (i) (i ;; 3). O te oe ad te use of tese estimators allows weaer assumptios o te desig desities, because oly te cotiuity of te desig desity is required (for local
polyomials of of odd order). O te oter ad te bias of local polyomials of odd order is te same for all curves irrespective of te desig patter. More precisely, if equal badwidts are used for te local polyomial estimatio of te idividual regressio fuctios [see (.4) ad (.6)] te terms B () ad B () ad i asymptotic variace ;i i Teorem. ad. vais, wile te erel K i te asymptotic bias D (i) (i ;; 3) of Teorem. -.3 as to be replaced by te equivalet iger order erel correspodig to te local polyomial estimator [see Wad ad Joes (995)]. A similar observatio was made by Alcala, Cristobal ad Gozalez Mateiga (999) i te cotext of testig for a parametric form of te regressio fuctio. Remar.8. Te test statistics T (), T () (3) ad T ca be directly used for a multivariate predictor ad a radom desig. Uder te assumptio of a radom desig t i ;:::;t ii are realizatios of i.i.d. radom variables T i ;:::;T ii wit positive desity r i o te iterval [; ] (i ;:::;). I tis case te rst parts of te statemets of Teorem. -.3 regardig te asymptotic beaviour of te statistics ud te ypotesis of equal regressio fuctios remai valid ad cosistet tests are obtaied exactly as i te case of a xed desig. However, it is wortwile to metio tat uder te alterative a dieret asymptotic variace is obtaied i all tree cases. Cosider for example te situatio of Teorem. i te case of idepedet radom samples. Uder a xed alterative te asymptotic variace of te statistic T () is give by Var(T () )4 + Var + Var 3 Simulatio results r (g g ) (t)( r (t)(t)+ r (t)(t)) (t)r (t) ( r + r ) (t) dt (g g ) (T ) r(t )+ r (T )r (T ) ) ( r + r ) (T ) (g g ) (T ) r(t )+ r (T )r (T ) ) ( r + r ) (T ) + o( ): I similar problems it was observed by several autors [see e.g. Azzalii ad Bowma (993), Hjellvi ad Tjsteim (995) or Alcala, Cristobal ad Gozalez Mateiga (999)] tat te asymptotic ormal distributio uder te ull ypotesis does ot provide a satisfactory approximatio of te distributio of te statistics T (i) for reasoable sample sizes. For tese reasos may autors propose te applicatio of bootstrap procedures i tese problems [see e.g. Hall ad Hart (99) or Hardle ad Mamme (993)]. I tis sectio we study te ite sample performace of a wild bootstrap versio of te test (.4) ad compare its power properties wit several oter procedures suggested i te literature. Some remars regardig te cosistecy of tis procedure are give i Sectio 4.4 of te appedix. Because all simulatio results publised so far cosider te two sample case wit equal omoscedastic variace (i.e. (t) (t) ); ad we are iterested i a compariso, we maily restrict our study to tis case. Moreover we will cocetrate o te statistic T () variace estimators, because it performed better ta T () te specicatio of a weigt fuctio (i cotrast to te statistic T (3) a asymptotic equivalet test statistic give by based o te dierece of (see Sectio 3.) ad it does ot require ). I our study we used i fact ~T ^ ^ ^
were ; are ormalizig costats covergig to, suc tat ^ i is ubiased for costat regressio g i [see Hall ad Marro (99)]. More precisely, tese costats are deed by were ` ` w (`) i X` i w (`) ii + X` i; K( t`i t` ) ` P ` s K( t`i t`s ` (w (`) i ) ; ` ;; ) ; ` ; : We used te commo wild bootstrap of residuals based o a oparametric t [see Hardle ad Mamme (993)] (3.) ^" ij Y ij ^g(t ij ) ; j ;:::; i ; i ; were ^g is te estimator of te regressio curve from te total sample deed i (.6). Let V ij (i ;, j ;:::; i ) deote i.i.d. radom variables wit masses ( p 5 +) p 5 ad ( p 5 ) p p p 5 at te poits ( 5) ad ( + 5) (ote tat tis distributio satises E [Vij] ;E [V ij ] E [V 3 ij ] ). Fially dee " ij V ij ^" ij ad te bootstrap sample by (3.) Y ij ^g(t ij )+" ij ; j ;:::; i; i ; : For te test at level te ull ypotesis is rejected if ~ T is bigger ta te correspodig quatile of te bootstrap distributio of ~ T ; i.e. (3.3) ~T > ~ T (b(b( )c) ; were ~ T (i) deotes te it order statistic of te bootstrap sample T ~ ;:::; ~ ; T ;B : I our study we resampled B times ad used simulatios for te calculatio of te level ad power i eac sceario. Moreover, we used te same badwidt for te geeratio of te bootstrap sample (3.) ad te deitio of te test statistic ~ T. Te cosistecy of tis procedure is idicated i Sectio 4.4. We cosidered two samples at te desig poits (3.4) t i i ; i ;:::; t j j ; j ;:::; ad ormally distributed errors i bot samples uless it is stated oterwise, i.e. (3.5) " `; " ` (; ): Te erel was cose as K(x) 3 4 ( x )I [ ;] (x) (wic yields r ) ad te badwidts are (3.6) (3.7) i R i (t)dt i! 3 i i 3 R R! 3 (t)dt + (t)dt ( + ) ; i ; + 3 3
were te last equalities follow i te case of omoscedasticity ad : ote tat we use dieret badwidts for te estimators ^g, ^g ad ^g i our study. ( ; ) (, ) (,) (,3) (, 5) (,) %.99.96.99.5. 5%.6.5.5.54.54.5%.3.3.6.3.5 ( ; ) (,3) (,5) (3,3) (3,5) (5,5) %.98.8.99.9.8 5%.54.5.48.47.48.5%.9.8.5.6.5 Table 3.. Simulated level of te test (3.3) for various sample sizes ad stadard ormal errors. Te desigs are uiform [accordig to (3.4)] ad g (t) g (t) t : 3. Simulatio of te level Our rst example ivestigates te approximatio of te level by te wild bootstrap versio of te test (.4). Firstly we cosidered quadratic regressio fuctios g (t) g (t) t ; stadard ormal distributed errors ad dieret sample sizes ; ;; 3; 5: Te results are summarized i Table 3. wic sows te simulated rejectio probabilities of te wild bootstrap test wit level %, 5% ad.5%. Our secod Table 3. sows te correspodig results for te regressio fuctios g (t) g (t) cos(t):we observe a reasoable approximatio of te level by te wild bootstrap procedure i all cases, eve i te case of very small samples [see also Hall ad Hart (99), wo obtaied a similar coclusio for teir resamplig procedure]. ote tat for te more oscillatig regressio fuctios g i (t) cos(t) te approximatio is sligtly worse compared to te more smoot case g (t) g (t) t ; wic ca be partially explaied by a larger bias i te variace estimators ^ ; ^ ad ^ : ( ; ) (, ) (,) (,3) (, 5) (,) %.98.4.7.9.97 5%.54.56.55.5.53.5%.3.3.8.8.3 ( ; ) (,3) (,5) (3,3) (3,5) (5,5) %..96.98.95. 5%.53.48.5.5.5.5%.3.6.3.8.7 Table 3.. Simulated level of te test (3.3) for various sample sizes ad stadard ormal errors. Te desigs are uiform [accordig to (3.4)] ad g (t) g (t) cos(t): 4
As poited out by a referee it migt be of iterest to ivestigate te approximatio of te level uder a eteroscedastic error distributio. To tis ed we cosidered te quadratic regressio fuctios g (t) g (t) t ad te variace fuctios (3.8) (3.9) (t) (t) (t) e t e t R ex dx ; R ex dx ; (t) e t R ex dx ; were te rst ad secod sceario correspod R to te case of equal ad uequal variace fuctios, respectively, ad we ormalized suc tat i (t)dt ( i ;). Te results are listed i Table 3.3 ad 3.4 ad demostrate a excellet performace of te wild bootstrap procedure uder eteroscedasticity. ( ; ) (, ) (,) (,3) (, 5) (,) %..88.94.9. 5%.57.46.48.59.49.5%.3.6...4 ( ; ) (,3) (,5) (3,3) (3,5) (5,5) %.88.93.95.9.6 5%.46.47.55.47.48.5%.3..3..8 Table 3.3. Simulated level of te test (3.3) for various sample sizes ad stadard ormal but eteroscedastic errors. Te desigs are uiform [accordig to (3.4)], g (t) g (t) t, ad te variace fuctios give by (3.8). ( ; ) (, ) (,) (,3) (, 5) (,) %.97.84.87.84.5 5%.46.5.43.4.5.5%.35.8.9.7.9 ( ; ) (,3) (,5) (3,3) (3,5) (5,5) %.89.86.95.9.3 5%.5.44.5.47.44.5%.6..33..3 Table 3.4. Simulated level of te test (3.3) for various sample sizes ad stadard ormal but eteroscedastic errors. Te desigs are uiform [accordig to (3.4)], g (t) g (t) t, ad te variace fuctios give by (3.9). 5
3. Te test of Kulaseera ad Wag (997) Recetly Kulaseera (995) proposed a ew testig procedure for te ypotesis (.) i te case of two samples wit omoscedastic errors. Because tis test is applicable for dieret desigs i bot groups ad ca detect alteratives covergig to te ull at a rate p ; we will discuss it i a little more detail. Te test is based o te quasi residuals ad te correspodig partial sums e i Y i ^g (t i ) ; i ;:::; e j Y j ^g (t j ) ; j ;:::; i (t) b i tc X j e ij p i ; <t<; i ;: Te test statistic proposed by Kulaseera (995) is deed as suitable fuctio of or K (i) i S i K (i) S 3 i X i i ( i ) ; i ; i (t )d i (t) ; i ; were S i is a cosistet estimators of i (i ;): ote tat tis test does ot require equal desigs i bot groups. Kusaleera ad Wag (997) ivestigated te fuctios W mifk () ;K() g, W mifjk () j; jk () jg ad proposed a metod for coosig te badwidt, wic rougly speaig, maximizes te power at a specic alterative. As poited out i te latter paper te data-based smootig parameters iate te size of te test ad te discrepacy from te actual size depeds largely o te variability of te reposes ad te sample size. For tese reasos Kulaseera ad Wag (997) used simulatio (for g g ) for dig te critical poits. I Table 3.5 we compare te test (3.3) wit te procedure proposed by Kulaseera ad Wag (997). For te sae of compariso we cosed te setup cosidered i Table 3 of te latter paper, tat is ormally distributed errors wit variace :5 ad te followig regressio fuctios (3.) (a) g (x) g (x) :5 cos(x) (b) g (x) g (x) :5 si(x) (c) g (x) g (x) x cos(x) (d) g (x) g (x) cos(x) (e) g (x) g (x) x cos(x) (f) g (x) g (x) cos(x): Comparig te results of Table 3.5 wit te correspodig results of Kulaseera ad Wag's (997) i Table 3 of teir paper we observe tat te test proposed i tis paper yields a substatial improvemet wit respect to te power i all cosidered cases. ote tat Kulaseera ad Wag (997) cosed te badwidts suc tat te power is maximized (at te cost of a simulated level) ad we could obtai a furter improvemet i power for te test (3.3) by applyig a similar tecique. Altoug 6
tis would ave teoretical advatages, we do ot recommed tis approac i practice, because tis data based coice of te smootig parameter usually yields a large discrepacy betwee te size of te test ad te actual level. model 5 5 % 5% :5% % 5% :5% (a).84.736.6.985.97.95 (b).84.738.653.979.964.936 (c).988.975.955... (d).987.973.95... (e).979.96.946... (f).983.973.943...998 Table 3.5. Simulated rejectio probabilities of te test (3.3) for various alteratives give i (3.). Te desigs are uiform [accordig to (3.4)] ad te errors are ormal wit variace :5: We remar tat te test (3.3) ca oly detect alteratives covergig to te ull at a rate ( p ) [wic gives 74 for te coice (3.7)], wile Kulaseera ad Wag's (997) test acieves te parametric rate : O a rst glace tis is a cotradictio to te results obtaied i our simulatio. However, tese observatios ca be explaied by te fact tat te metod of te lastamed autors implicitly uses a sample splittig. Oe sample is used for estimatig te regressio wile te oter sample is used for te calculatio of te residuals. For te sae of compariso we also studied te performace of te test of Youg ad Bowma (995) i tis situatio. Te results are listed i Table 3.6. We observe a larger power of te test (3.3) based o te diereces of variace estimators i most cases. model 5 5 % 5% :5% % 5% :5% (a).388.36.36.897.8.66 (b).77.65.547.99.975.956 (c).789.683.59.973.947.9 (d).998.996.99... (e).833.7.6.997.996.989 (f).983.969.94...998 Table 3.6. Simulated rejectio probabilities for te test of Youg ad Bowma (995) for various alteratives give i (3.). Te desigs are uiform [accordig to (3.4)] ad te errors ormal wit variace :5: 7
3.3 Te tests of Delgado (993) ad Dette ad Mu (998) Te test recetly proposed by Dette ad Mu (998) was te rst procedure wic was applicable i te geeral model (.). Tis test is based o a simple estimate of a L -distace betwee te regressio fuctios wic does ot deped o a smootig parameter. Altoug tis procedure ca oly detect alteratives wic coverge to te ull at a rate of 4 ; te test as promisig ite sample properties wit respect to te quality of approximatio of te level [see Dette ad Mu (998)]. Moreover, a compariso wit Delgado's (993) test, wic ca detect alteratives covergig to te ull at a rate ; idicates tat for realistic sample sizes tis test is comparable wit procedures wic are eciet from a asymptotic poit of view. Delgado's (993) test requires equal desig poits ad is based o te sup-orm of a mared empirical process of te pairwise diereces from bot samples. I order to compare te ew test (3.3) wit tese procedures we cosidered te setup give i Sectio 4. of Dette ad Mu (998), i.e. 5;3; (g g )(t) ; (g g )(t) si(t) ad tree types of error distributios [see also Hall ad Hart (99)] (i) ( ; ) p p (3.) (ii) ( j ; j j p p ) (iii) ( jj ; j j): Te results are listed i Table 3.7 ad a compariso of te power at te 5% level sows te followig. Wile Delgado's (993) test performs better for te smoot alterative g g ; Dette ad Mu's (998) test is more eciet for te oscillatig alterative. Te ew test (3.3) as a reasoable performace i bot cases. O te oe ad it is substatially more powerful as Delgado's test for te oscillatig alterative ad as Dette ad Mu's test for te smoot alterative. O te oter ad it is comparable wit tese procedures i te remaiig cases. (g g )(t) si(t) (" i ;" i ) (i) (ii) (iii) (i) (ii) (iii).5%.55.936.875.76.57.5 i 5 5%.648.963.93.55.65.64 %.734.977.94.347.73.73.5%.87.999.99.386.863.77 i 3 5%.95.999.998.49.9.97 %.954...68.95.95 Table 3.7. Simulated rejectio probabilities of te test (3.3) i te sceario cosidered by Dette ad Mu (998) Sectio 4.. Te desig is uiform [accordig to (3.4)] ad te error distributios are give by (3.). Our al example compares te ew test wit te bootstrap test itroduced by Hall ad Hart (99). Tese autors maily cosidered te case of equal desig poits ad briey metioed a geeralizatio of teir approac to te geeral case. However, Kulaseera (995) observed tat tis geeralizatio is ot reliable ad recommeds te applicatio of Hall ad Hart's test oly i te case of equal desigs. ote tat tis test ca detect alteratives covergig to te ull at a rate : 8
For a compariso wit our test we cosed te setup of Table 3 i Hall ad Hart (99). Te test proposed by tese autors depeds o a smootig parameter p ad Table 3 i Hall ad Hart (99) lists results for tree coices of p: More precisely te errors are give by (3.) ad te alteratives by g g ad(g g )(x) x were g :Te results are give i Table 3.8 ad sow tat te ew test is a serious competitor. I most cases we observe a better power for te ew test (3.3), eve if we compare it wit te best coice of p i te procedure of Hall ad Hart. (g g )(t) t (" i ;" i ) (i) (ii) (iii) (i) (ii) (iii).5%.55.98.874.97.493.493 i 5 5%.643.957.93.76.6.59 %.738.98.96.384.76.7.5%.666.989.966.37.64.59 i 5%.75.99.98.335.74.685 %.844.999.99.447.87.779.5%.848..995.369.76.78 i 3 5%.898..998.487.847.83 %.939..999.584.96.899.5%.99...54.94.94 i 5 5%.995...649.969.96 %.999...74.983.98 Table 3.8. Simulated rejectio probabilities of te test (3.3) i te sceario cosidered by Hall ad Hart (99), Table 3. Te desig is uiform [accordig to (3.4)] ad te error distributios are give by (3.). 4 Appedix: Proofs 4. Prelimiaries We will restrict ourselves to a proof of Teorem. i te case of regressio fuctios. Te geeral case 3 ad te asymptotic results give i Teorem. ad.3 for T () (3) ad T follow by exactly te same argumets wit a additioal amout of algebra ad otatio. For te sae of a trasparet otatio we will omit all idices referrig to te umber of samples ad to te specic statistic discussed i Sectio. I oter words we write B istead of B (), T istead of T () etc. Recallig te deitio of te weigts (4.) K( tij ti j P i K( t ij t il ) ; i ; l w (i) ) te adaraya-watso estimators (.4) of te idividual regressio fuctios evaluated at te poits t ij ca be rewritte as ^g i (t ij ) X i w (i) j Y i; i ;: 9
I order to obtai a similar represetatio for te estimator i te combied sample dee te weigts (4.) w l;ij K( t l t ij ) P l P l K( t l t ij ) : Te adaraya Watso estimator (.6) evaluated at te poits t ij usig te total sample ca ow be writte as (4.3) ^g(t ij ) X X l l We ally itroduce te otatio, ad w l;ij Y l : (4.4) (t) r (t)+ r (t) [ r (t)+ r (t)] wic will be used frequetly trougout tis sectio. Our rst result establises te asymptotic expasio for te bias of te estimator T (). Lemma 4.. Assume tat (.), (.), (.3), (.5), (.7) are satised, te (4.5) E[^ i ] i (t)r i (t) dt + d i r + o( r )+O( i ) + i K (u) du K() i (t) dt were te costat d i is deed by d i r [(g i r i ) (r) (t) g i r (r) i (t)] r i (t) dt (i ;:::;): Moreover, if te ull ypotesis of equal regressio fuctios is valid we ave for te estimator (.9) i te case (4.6) E[^ ] (t)r (t) dt + (t)r (t) dt + K (u) du K() (t) r (t)+ (t) r (t) ( r + r )(t) dt were te costat C is deed by + C r + o( r )+O( ); C r [(g ( r + r )) (r) dt g ( r + r ) (r) ] (t) ( r + r )(t) : Uder te alterative we obtai for te estimator (.9) i te case (4.7) E[^ ] (t)r (t) dt + (t)r (t) dt + M + O( r )+O( )
were te costat M is deed by (4.8) r M (g (t) g (t)) (t)r (t) r (t)+ r (t) dt: Proof of Lemma 4.. Te rst part (4.5) of te Lemma is obtaied from te represetatio (4.6) by cosiderig equal variace fuctios ad desig desities. Te proof of (4.6) ad (4.7) essetially follows te argumets of Hall ad Marro (99) ad we will oly metio te mai modicatios ere, wic tae ito accout te mixture of two desig desities. Dee (4.9) ij g i (t ij ) X X l l w l;ij g l (t l ) te te expectatio of te variace estimator (.9) from te total sample splits ito two parts, i.e. E[^ ] X X i ij + X X i X X l (4.) E i (t ij )" ij w l;ij l (t l )" l : i j i j A straigtforward but tedious calculatio sows l (4.) (4.) ij (t ij ) + R(t ij) r 3 i (t ij )(g i g 3 i )(t ij ) i + K( t ij t ) (g r )(t ij ) (g r )(t)+g i (t ij )f(t) (t ij )g +(g r )(t ij ) (g r )(t) dt + O( )+O(r ) i r 3 i (t ij )(g i g 3 i )(t ij ) + O( r )+O( (t ij ) ); uiformly i j ;:::; i ;i;. Here te fuctio R is deed by (4.3) R(x) (x) R K x t (x) (t) dt O( r ) ; were te estimate o te rigt ad side follows from te dieretiability of te desig desities ad te momet assumptios o te erel. ow te evaluatio of te rst term i (4.) gives for g 6 g X X i i j ( ij r ) (x)r (x) (g g ) (x) dx + (x) (g g )(x) + r (x)r (x) K( x t (x) +g (x)((t) (x)) + (g r )(x) (g r )(t) r (x)r(x) (g g ) (x)dx (x) o ) (g r )(x) (g r )(t) dt dx
(4.4) R(x) + (x) ( r (x)(g + r (x)r (x) (g g )(x) (x) g )(x)) r (x) dx K( x t + g (x)((t) (x))+(g r )(x) (g r )(t) + ) (g r )(x) (g r )(t) R(x) (x) (r (x)(g g )(x)) r (x) dx + O( r )+O( ) r (x)r (x) (g g ) (x) dx (x) + r (x)r (x) (g g )(x) K( x t (x) ) (g r )(x) (g r )(t) + g (x)((t) (x))+(g r )(x) (g r )(t) (g r )(x)+(g r )(t) g (x)((t) (x)) (g r )(x)+(g r )(t) + R(x)(g g ) (x)r (x) r (x) dx + O( r )+O( ) r (x)r (x) (g g ) (x) dx + r (x)r (x) (g (x) g )(x) (x) K( x t )(g (x) g (x))((t) (x)) dt dx + R(x)(g g ) (x)r (x) r (x) dx + O( r )+O( ) r (x)r (x) (x) M + O( r )+O( ); o o dt dt (g g ) (x) dx + O( r )+O( ) were we used te deitio of i te rst equality, te deitio of R(x) i (4.3) ad of M i (4.8) for te last step. Uder te assumptio of equal regressio curves g g (4.) simplies ad we obtai observig (.7) ad (4.4) X X i i j K( x t ) (g )(x) (g )(t) ij (x) + R(x) o + g (x)((t) (x)) dt + O( ) r (x) dx + (x) + R(x) K( x t o + g (x)((t) (x)) dt + O( dx dx ) (g )(x) (g )(t) ) r (x) dx + O( )
(4.5) + O( ) r r i (x) (x) + R(x) r r g (r) (g ) (r) (x)+o( r )+O( r C + o( r )+O( ) (x) [g (r) (g ) (r) ] (x) dx + o( r )+O( ) ) dx For te secod term i (4.) we obtai by a straigtforward but cumbersome calculatio U X X i X X l (4.6) E i (t ij )" ij w l;ij l (t l )" l X i + i X X i i j j i (t ij ) i i (t)r i (t) dt X i " K() X X i i j +O( r )+O( )+O( () ) l i (t ij )w ij;ij + i i (t) r i(t) dt + (t) (t)r (t) dt + (t)r (t) dt + K (u) du K() i (x)r (x) (x) X X i i K ( t dx + j X X l l l (t l )w l;ij # x i (x) i r i(x) ) dt dx (t) (x) r (x) (x) o dx + O( ) ad te assertios (4.6) ad (4.7) follow from (4.), (4.4), (4.5) ad (4.6). 4. Proof of Teorem.: te ull ypotesis of equal regressio fuctios I a rst step we itroduce te otatio [observig (4.)] (4.7) ij g i (t ij ) X i w (i) j g i(t i ); j ;:::; i ; i ; ad decompose te cetered versio of T as follows (4.8) T E[T ]^ ^ ^ E[T ]R ; + R ; 3
were (4.9) (4.) R ; R ; X X i i j X X i i j ^T () + ^T () ; + 7X j3 X X i i j X X i i j X X i i j X X i i j ^T (j) : ij i (t ij )" ij ij i (t ij )" ij i (t ij )" ij E i (t ij )" ij i (t ij )" ij E i (t ij )" ij X X l l X i X X l l l X i X i w l;ij l (t l )" l w (i) j i(t i )" i w l;ij l (t l )" l X X l w l;ij l (t l )" l w (i) j i(t i )" i w (i) j i(t i )" i Here te radom variables ^T (j) (4.) (4.) (4.3) (4.4) ad te coeciets ij are give by (4.5) ij are deed by ^T (i) ^T (+i) ^T (4+s) ij ^T (7) ij X i j X i j X s s i i ij " ij ; i ; ij (" ij ) ; i ; X l l6i X X j X X l l r (s) il " si " sl ; s ; t ij " i " j l w ij;l + X i i w (i) i i (t ij ) (j ;:::; i ; i ;) were ij ad ij are deed i (4.9) ad (4.7), respectively. Te coeciets ij ;r (s) il ; tij i te represetatio of R ; are deed as follows: X l X X i (4.6) ij w (i) jj w ij;ij + w ij;l (w (i) i ) i (t ij ) ; i ;; l 4
(4.7) (4.8) r (s) il t ij w (s) il w si;sl + X X j j w i;j w j;i + w j;si w j;sl X X l l X s w (s) i w(s) l s (t si ) s (t sl ) ; s ;; w l;i w l;j (t i ) (t j ) : (j) Te ext Lemmata specify te asymptotic beaviour of te terms ^T o te rigt ad side of (4.9) (j) ad (4.). ote tat all terms i tese represetatios are cetered, i.e. E[ ^T ] (j ;:::;7): Lemma 4.. If te assumptios of Teorem. are satised we ave uder te ypotesis of equal regressio curves ^T (j) H o p ( p ); j ; ad uder te alterative (i) Var( ^T ) H 4 ( ) r (g g ) (t) i (t) (t)r (t) i r i (t)( r (t)+ r (t)) dt + o( ) ; i ;: () Proof: We oly prove te assertio for te statistic ^T ; te remaiig case follows by exactly te same argumets. From (4.) it follows (4.9) were, by (4.5) ad (4.) (4.3) i r (t i ) (t i ) (g g )(t i ) X X l l + O( r )+O( ) Var( ^T () ) l r 3 l (t l ) (t l ) X i i (g g )(t l )K tl t i o (t i ) (t l ) r (t i ) (t i ) (g g )(t i )+O( r )+O( (t i ) ) uiformly wit respect to i ;:::; : Te last equality i (4.3) uses te fact tat te itegral approximatios of te two sums ave te same absolute value wit opposite sigs. ow (4.9) implies uder te ypotesis of equal regressio curves () Var( ^T )o( ) ad a applicatio of Cebysev's iequality proves te rst part of te Lemma. For te secod part we obtai from (4.9) ad (4.3) Var( ^T () ) 4 X i r (t i ) (t i ) (t i ) (g g ) (t) (t) r (t)( r ) (t) (t) (g g )(t i ) + (O(r )+O( )) 5 dt + o( );
wic completes te proof, by te deitio of ad : Lemma 4.. Uder te assumptios of Teorem. we ave o p( p ) ; j 3;4: ^T (j) (j) Proof. ote tat E[ ^T ] for j 3;4: Recallig te deitio of te coeciets i i (4.6) we obtai K() (4.3) i r (t i ) K ( t K() (t i ) + K ( s t i o ) r (t) dt t i ) (s) ds (t i )+o( ) O( ) uiformly i i ;:::; : Tis implies for te variace of ^T (3) (3) Var( ^T ) X i ivar(" i) o( ) ad proves Lemma 4. for te case j 3 :Te remaiig case is obtaied by exactly te same argumets ad terefore omitted. Lemma 4.3. Uder te assumptios of Teorem. we ave Var( ^T (4+i) ) 4 i (x) i r i (x) (x) i dx (K K K) (u) du + o( ); i ;: Proof. We oly setc a proof of te rst part i of te assertio, te remaiig case i follows by exactly te same argumets. Recallig te deitio of te weigts r (s) il i (4.7) we obtai by straigtforward algebra r () il ti t l K r (t i ) t ti K K t tl ti t l K (t i ) + s ti K r (t) dt (t i) (t l) + o( ) uiformly for i; l ;:::; ad straigtforward but tedious algebra sows s tl K (s) ds 6
(5) Var( ^T ) X 4 i 4 X l l6i [(r () il ) + r () il r () K (u) du (K K)(u)K(u) du (x) 4 r (x) (x) li ] (x) 4 i 4 (x) X i r (x) (x) X l il ) + o( ) (r () i dx + r (x) (x) i dx (K K) (u) du + o( ) (x) 4 r (x) i dx (x) (K K K) (u) du + o( ): Lemma 4.4. Uder te assumptios of Teorem. we ave (7) Var( ^T ) 4 (K K K) (u) du r (x) r (x) (x) (x)(x) dx + o( ): Proof. A straigtforward calculatio sows for te coeciets tij i (4.8) (4.3) t ij K ti + t j K s ti (t i ) + (t j ) K s tj (s) ds (t i ) (t j )+o( ) (uiformly for i ;:::; ;j ;:::; ) wic implies for te variace of ^T (7) (7) Var( ^T ) 6 X X t ij i j r (x) r (x) (x) K (u) du 6 (x) (x) dx 4 (K K K) (u) du (K K)(u)K(u) du +4 (K K) (u) du + o( ) r (x) r (x) (x) (x)(x) dx + o( ): Lemma 4.5. Uder te assumptios of Teorem. we ave for te covariaces of te statistics deed i (4.) { (4.4) (i) Cov( ^T ; (j) ^T ); if fi; jg 6f; 3g; f; 4g: 7
Uder te ypotesis of equal regressio fuctios it follows wile uder te alterative g 6 g Cov( ^T () ; ^T (3) )o( ); Cov( ^T () ; ^T (4) )o( ) Cov( ^T () ; ^T (3) )o( ); Cov( ^T () ; ^T (4) )o( ): Proof. Te rst part of Lemma 4.5 is obvious. From (4.) ad (4.) we obtai Cov( ^T () ; ^T (3) ) X i i E[" 3 i] were i ; i are deed i (4.5) ad (4.6), respectively. ow (4.3) gives i 8 < i : O(r )+O( ) : if g g O() : if g 6 g (uiformly for i ;:::; ): Similary, we ave from (4.3) (uiformly for i ;:::; ) wic implies uder te ull-ypotesis, ad i O( ); Cov( ^T () ; ^T (3) )o( ) Cov( ^T () ; ^T (3) )o( ) uder te alterative g 6 g : Tis proves te secod part of te assertio for te statistics ^T () ^T (3) : Te remaiig case follows by exactly te same argumets ad is terefore omitted. ad Proof of Teorem. (i): Observig Lemma 4. ad (.7) we obtai p T B r D p T E[T ] + o() p (5) ( ^T + (6) ^T + (7) ^T )+o p(); X s XX s p i X (s) r il " si " sl + s i l l6i i X j p tij " i " j + o p () were te secod equality follows from (4.8), (4.9), (4.), Lemma 4. { 4.5 ad te costats B ad D are deed i (.) for. Deig r (s) il : r(s) il + r (s) li ; s ;; 8
te rigt ad side of tis equatio ca be writte as a symmetric quadratic form wit vaisig diagoal elemets, i.e. W p (5) ^T + (6) ^T + (7) ^T X T AX were X (X ;:::;X ) T (" () ;" () ) T ;" (i) (" i ;:::;" ii )(i ;); te matrix A (a ij ) i;j;:::; is give by a ii (i ;:::;) (4.33) a ij : 8 >< >: p r () ij : i; j ;:::; ;i6 j p t i;j : i ;:::; ; j +;:::; + p t j;i : i +;:::; + ;j;:::; pr () i ;j : i; j +;:::; + ;i6 j ad r (s) ; ij tij are deed i (4.7) ad (4.8), respectively. I order to sow asymptotic ormality of te statistic W uder te ypotesis of equal regressio curves we apply Teorem 5. i de Jog (987). For te asymptotic variace of W we obtai from Lemma 4.3 { 4.5 ad te deitio of i (4.4) (4.34) p (5) Var( ( ^T + (6) ^T + (7) ^T )) + o() O(); were ; is deed i (.) for. Observig (4.3) we ave X t ij 4 ti t K (t i ) + (t) (t) r (t) dt j + 4 K s ti s t K 3 (s) ds r (t) (t) dt + 8 ti t K K s ti s t K r (t) (t) (t i ) + (t) ad a similar argumet implies From tese estimates it follows (4.35) X j a ij X j (r () X j (r () (s) ds dt o (t i )+o( ) O( ) ij ) O( ): ij ) + X t ij O( 4 ) ; i ;:::; j ad a aalogous argumet sows tat (4.35) is also valid for i +;:::;:Terefore coditio ) ad ) i de Jog's (987) teorem are satised wit K log : I order to establis te remaiig coditio 3) i te latter teorem we ote tat by Gerscgori's teorem te eigevalues ;:::; of te matrix A ca be estimated as follows max i j i j max i X l ja il j p X l 9 jr () il j + p X l j til j O( p )
were we used te deitio of r (s) il ad (4.7), (4.3) i te last estimate. Similarly, we obtai max i j i j O( p ) wic implies max i i o(): Te assertio of Teorem.(i) ow follows from de Jog's (987) teorem ad (4.34), i.e. W D! (; ): 4.3 Proof of Teorem.: xed alteratives If g 6 g we ave from Lemma 4. { 4.4 tat p T M p (4.36) T E[T ] p X X i i j + o() p () ( ^T + () ^T )+o p() ij " ij + o p () W + o p () were te last equality dees W. Te assertio ow follows from te stadard cetral limit teorem usig Ljapuo's coditio. To tis ed we ote tat uder a xed alterative Var(W ) + o(); were we used Lemma 4., 4.5 ad ; is deed i (.3) for. For te coeciets ij i (4.36) we ave from (4.3) for te case i ad a similar argumet i te case i 4 ij 6 ( i r 3 i ) 4 (t ij ) 4 4 i (t ij )(g g ) 4 (t ij )+o() (i ;) (t ij ) wic implies Ljapuo's coditio, i.e. 4 X i j 4 p E ij" ij 6 ( r ) 4 (t)r (t) 4 4 4 (t) (t)(g g ) 4 (t) dt r(t) 4 + r (t) 4 4 (t) (t)(g g ) 4 (t) dt + o() o(); ad completes te proof of Teorem. (ii). 4.4 Some commets o te cosistecy of te wild bootstrap I tis subsectio we briey idicate te cosistecy of te wild bootstrap procedure used i te simulatio study of Sectio 3. For te sae of brevity we restrict ourselves to te statistic T () based o a dierece of variace estimators for samples. Correspodig results for T () ;T(3) ad 3 regressio fuctios ca be proved followig a similar patter. Recall tat we agai omit all 3