RESIDUAL EMPIRICAL PROCESSES AND WEIGHTED SUMS FOR TIME-VARYING PROCESSES WITH APPLICATIONS TO TESTING FOR HOMOSCEDASTICITY

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1 JOURNAL OF TIME SERIES ANALYSIS J. Time. Ser. Aal. 38: 7 98 (7) Published olie 9 Jue 6 i Wiley Olie Library (wileyolielibrary.com) DOI:./jsa. ORIGINAL ARTICLE RESIDUAL EMPIRICAL PROCESSES AND WEIGHTED SUMS FOR TIME-VARYING PROCESSES WITH APPLICATIONS TO TESTING FOR HOMOSCEDASTICITY a b GABE CHANDLER a* AND WOLFGANG POLONI b Deparme of Mahemaics, Pomoa College, Claremo, CA, USA Deparme of Saisics, Uiversiy of Califoria, Davis, CA, USA I he coex of heeroscedasic ime-varyig auoregressive (AR)-process we sudy he esimaio of he error/iovaio disribuios. Our sudy reveals ha he o-parameric esimaio of he AR parameer fucios has a egligible asympoic effec o he esimaio of he empirical disribuio of he residuals eve hough he AR parameer fucios are esimaed o-paramerically. The derivaio of hese resuls ivolves he sudy of boh fucio-idexed sequeial residual empirical processes ad weighed sum processes. Expoeial iequaliies ad weak covergece resuls are derived. As a applicaio of our resuls we discuss esig for he cosacy of he variace fucio, which i special cases correspods o esig for saioariy. Received Augus 4; Acceped April 6 eywords: Cumulas; empirical process heory; expoeial iequaliy; locally saioary processes; o-saioary processes; ess of saioariy.. INTRODUCTION Cosider he followig ime-varyig auoregressive (AR) process ha saisfies he sysem of differece equaios p Y k Y k D ; D ;:::;; () kd where k are he AR parameer fucios, p is he order of he model, is a fucio corollig he volailiy ad.; / deoe he i.i.d. errors. Followig Dahlhaus (997), ime is rescaled o he ierval Œ; i order o make a large sample aalysis feasible. Observe ha his, i paricular, meas ha Y D Y ; saisfyig () i fac forms a riagular array. The cosideraio of o-saioary ime series models goes back o Priesley (965) who cosidered evoluioary specra, ha is, specra of ime series evolvig i ime. The ime-varyig AR process has always bee a impora special case, eiher i more mehodological ad heoreical cosideraios of o-saioary processes, or i applicaios such as sigal processig ad (fiacial) ecoomerics (e.g. Subba Rao, 97; Greier, 983; Hall e al., 983; Raja ad Rayer, 996; Giraul e al., 998; Eom, 999; Drees ad Sǎricǎ, ; Orbe e al., 5; Fryzlewicz e al., 6; Chadler ad Poloik, 6). Oe of our coribuios is he esimaio of he (average) disribuio fucio of he iovaios D. Suppose ha we have observed Y p ;Y p ;:::;Y,adleY D.Y ;:::;Y p / ;D ;:::;:Give a esimaor b of D. ;:::; p / ad correspodig residuals b D Y b Y, we cosider he sequeial empirical disribuio fucio of he residuals give by Correspodece o: Gabe Chadler, Deparme of Mahemaics, Pomoa College, Claremo, CA 97, USA. gabriel.chadler@pomoa.edu Copyrigh 6 Joh Wiley & Sos Ld

2 WEIGHTED SUMS AND RESIDUAL EMPIRICAL PROCESSES 73 bf. ; / D b c D ¹b º ; R; Œ; : The correspodig disribuio fucio of he rue iovaios is F. ; / D uder appropriae codiios (Theorem ), Œ;; R P b c D ¹ º: We show ha bf. ; / F. ; / D o P = ; () eve hough o-parameric esimaio of he parameer fucios k is ivolved. This meas ha by usig appropriae esimaors for he parameer fucios, he (average) disribuio fucio of he ca be esimaed jus as well as if he parameers were kow. I parameric siuaios, his pheomeo is o ew, ad i perhaps was firs observed i Boldi (98). We refer o oul (), Ch. 7, for more complex siuaios. No-parameric models usually do o allow for such a pheomeo. See, for isace, Akrias ad va eilegom (), oul (), Schick ad Wefelmeyer (), Cheg (5), Müller e al. (7, 9a, 9b, ) ad va eilegom e al. (8). I coras o his work, our model, eve hough o-parameric i aure, has he crucial srucure of beig liear i he lagged observaios, which also meas ha he Y have mea zero. For more o he role of a zero mea i his coex, see Wefelmeyer (994) ad Schick ad Wefelmeyer (). (Geeralized) Auoregressive Codiioal Heeroscedasic-ype processes are cosidered i Horváh e al. (), Sue (), oul (), oul ad Lig (6) ad Laïb e al. (8) (cf. Remark (a) give ex o Theorem 4 ). The proof of he resuls jus meioed ivolves he behaviour of wo ypes of sochasic processes, boh of which are ivesigaed i his aricle. Oe is he residual sequeial empirical process, ad he oher ype is a geeralized parial sum process or weighed sums process. Boh are of idepede ieres. The residual sequeial empirical process is defied as. ; ; g/ D p b c ² g Y C ³ F g Y C = ; (3) D where Œ; ; R; g W Œ;! R p ; g G p D ¹g D.g ;:::;g p / ;g i Gº wih G a appropriae fucio class such ha b G p wih probabiliy edig o (see i he succeedig exs) ad F. /deoes he disribuio fucio of he errors : Observe ha. ; ; / D p.f. ; / EF. ; //,where,here, D.;:::;/ deoes he p-vecor of ull fucios. The basic form of is sadard, ad residual empirical processes based o o-parameric models have bee cosidered i he lieraure as idicaed earlier. Our coribuio here is o sudy hese processes based o a o-saioary Y of he form (). The secod key ype of processes i his aricle is weighed sums of he form Z.h/ D p D h Y ; h H; where H is a appropriae fucio class. Such processes ca be cosidered as geeralizaios of parial sum processes. Expoeial iequaliies ad weak covergece resuls for such processes are derived laer. We use properies of Z.h/ for provig (), ad his moivaes is sudy here. We also sudy he empirical disribuio of he esimaed errors bf. ; / D P b c D.b /; where b D b b. / for a appropriae esimaor b of he variace fucio : We will see ha uder appropriae codiios, J. Time. Ser. Aal. 38: 7 98 (7) Copyrigh 6 Joh Wiley & Sos Ld wileyolielibrary.com/joural/jsa DOI:./jsa.

3 74 G. CHANDLER AND W. POLONI Œ;; R bf. ; / F. ; / f. / p d e D b p D o P = ; (4) where F. ; / D P b c D. / deoes he empirical disribuio of he rue errors ad f is heir desiy. The derivaio of his resul ivolves he sudy of a residual sequeial empirical process slighly more geeral ha (3) i order o accommodae for he esimaio of he variace. Noice ha, from (4), we ca see ha i coras o he esimaio of he AR parameer fucios, he esimaio of he variace fucio is o egligible, eve if he variace was o be assumed cosa. See Secio 3 for more o his. Dahlhaus (997) advaced he formal aalysis of ime-varyig processes by iroducig he oio of a locally saioary process. This is a ime-varyig process wih ime beig rescaled o Œ; ha saisfies cerai regulariy assumpios; see (4 6) preseed laer. We would like o poi ou, however, ha, i our aricle, local saioariy is oly used o calculae he asympoic covariace fucio i Theorem 5. All he oher resuls hold uder weaker assumpios. The oulie of he aricle is as follows. I Secios, 3 ad 4, we aalyze he large sample behaviour of he fucio-idexed residual empirical processes ad of fucio-idexed weighed sums, respecively, uder he ime-varyig model (), ad apply he obaied resuls o show () ad (4). As a applicaio of he heoreical resuls, we discuss i Secio 5 a mehod for esig for homoscedasiciy, which i special cases is equivale o esig for saioariy. Proofs are deferred o Secio 6. Remark o measurabiliy. Suprema of fucio-idexed processes will eer he heoreical resuls give i he succeedig exs. We assume hroughou he aricle ha such rema are measurable.. RESIDUAL EMPIRICAL PROCESSES UNDER TIME-VARYING AUTOREGRESSIVE MODELS I order o formulae oe of our mai resuls for he residual empirical process. ; ; g/ defied i (3) earlier, we firs iroduce some more oaios ad formulae he uderlyig assumpios. Le Hdeoe a class of fucios defied o Œ; ad le d deoe a meric o H.Foragiveı>;le N.ı;H;d/ deoe he miimal umber N of d-balls of radius ı ha are eeded o cover H. Thelog N.ı;H;d/is called he meric eropy of H wih respec o d: If he balls A k are replaced by brackes B k D¹h H W g k h g k º for pairs of fucios g k g k ;kd;:::;n wih d.g k ;g k / ı, he he miimal umber N D N B.ı; H;d/ of such brackes wih H S N kd B k is called a brackeig coverig umber, ad log N B.ı; H;d/ is called he meric eropy wih brackeig of H wih respec o d: For a fucio h W Œ;! R, we deoe khk WD uœ; jh.u/j ad khk WD P D h : We furher deoe by d he meric geeraed by kk ; ha is, d.h; g/ Dkhgk : Assumpios. (i) The process Y D Y ; has a Movig Average (MA)-ype represeaio Y ; D a ;.j / j ; (5) j D where i:i:d:.; /: The disribuio fucio F of has a sricly posiive Lipschiz coiuous Lebesgue desiy f: The fucio.u/ i () is of bouded variaio wih <m <.u/<m < for all u. (ii) The coefficies a ;./ i (5) saisfy ja ;.j /j ; j D ; ; ; : : : `.j / where, for j>, `.j / D j.log j/ C for some ; >. Forj D ; ; we le `.j / D : wileyolielibrary.com/joural/jsa Copyrigh 6 Joh Wiley & Sos Ld J. Time. Ser. Aal. 38: 7 98 (7) DOI:./jsa.

4 WEIGHTED SUMS AND RESIDUAL EMPIRICAL PROCESSES 75 (iii) We have max gg p g Y D OP./ as!: (6) (iv) xr jxj Cf.x/< for some >. Assumpios (i) ad (ii) have bee used i he lieraure o locally saioary processes before. I is show i Dahlhaus ad Poloik (5) (see also Dahlhaus ad Poloik, 9) by usig a proof similar o üsch (995) ha Assumpio (i) holds for ime-varyig AR processes () if he zeros of he correspodig AR polyomials are bouded away from he ui disk (uiformly i he rescaled ime u) ad he parameer fucios are of bouded j variaio. I case p D, he fucios a ;.j / are of he form a ;.j / D Q j `D ` ; where.u/ D./ for u<ad, similarly,.u/ D./ for u<. Assumpio (iii) is also o oo surprisig as i is similar o assumpios used i he aalysis of parameric residual empirical processes, [e.g. oul,, Thm..3, Ass (..8)]. Assumpio (iv) is eeded o corol he ails of f i he case whe he variable i he process. ; ; g/ is allowed o rage over he eire real lie. Recallig he defiiio of. ; ; g/ give i (3), we see ha he special case g D (he vecor of zero fucios) gives a process oly based o he iovaios, which are idepede bu o ecessarily ideically disribued. Our firs heorem says ha if he idex class G is o oo large, he. ; ; g/ ad. ; ;/ behave he same asympoically: Theorem. Suppose ha Assumpios (i) ad (ii) hold. Le G deoe a fucio class such ha, for some c>; Z c= The we have for ay <L< ad ı! as!ha q log N B u ; G;d du < 8 : (7) Œ;;.L;L/; gg p W P p kd kg k kı j. ; ; g/. ; ; /j Do P./: (8) If, i addiio, Assumpios (iii) ad (iv) hold, he (8) also holds wih L D. The proof of Theorem ress o he crucial echical Lemma, which is give i Secio 6. This lemma implies asympoic sochasic equicoiuiy of he residual empirical sequeial process. ; ; g/ of which he asserio of Theorem is a immediae cosequece. Saemes of ype (8) are of ypical aure for work o residual empirical processes [e.g. (8..3) i oul, ]. Observe, however, ha, here, we are dealig wih ime-varyig AR processes ad are cosiderig o-parameric idex classes. Also keep i mid ha we are cosiderig riagular arrays (recall ha Y D Y ; ). The fucio class G is geeric i he formulaio of Theorem. As idicaed i Secio, wha we have i mid is fucio classes G modellig he differeces b k k [cf. Assumpio (vii)]. A specific example of such a class is give i he discussio ex o he formulaio of Theorem. Noice ha Theorem is cosiderig he differece of wo processes where he righ cerig is used. I coras o ha, we ow cosider he differece bewee wo sequeial empirical disribuio fucios: oe of hem based o he residuals ad he oher o he iovaios. As already said earlier, uder appropriae assumpios, he differece of hese wo fucios is o P = p, so ha he o-parameric esimaio of he parameer fucios has a egligible asympoic effec o he esimaio of he (average) iovaio disribuio. J. Time. Ser. Aal. 38: 7 98 (7) Copyrigh 6 Joh Wiley & Sos Ld wileyolielibrary.com/joural/jsa DOI:./jsa.

5 76 G. CHANDLER AND W. POLONI Furher assumpios. (v) jcum k. /jkšc k for k D ;;:::for some C>. (vi) For k D ;:::;p;we have k b k k k D O P.m / wih m!as!: (vii) There exiss a class G wih b k./ k./ G; k D ;:::;p;wih probabiliy edig o as!such ha gg kgk < ; ad for some C;c > ; we have R c= log N B.u; G;d /du<c <8: Theorem. Assume Assumpios (i) pad (ii) ad ha G is such ha (7) holds. I addiio, assume ha Assumpios (v) (vii) hold, wih m saisfyig D o./ as m!,where is such ha max Y D O P. /. The we have for <L<ha bf. ; / F. ; / D o P.= p /: (9) Œ;;.L;L/ If furher Assumpios (iii) ad (iv) hold, he (9) also holds wih L D: Discussio of assumpios: Assumpio (v) holds, for isace, whe Ej j k C k for all k D ;;:::. This is obviously a srog assumpio. However, much of he lieraure o locally saioary processes is usig i, i paricular hose i which raes of covergece for he esimaors b ad/or b are derived. I our work, his assumpios eer he picure hrough Theorem 5, which is used i he proof of Theorem. I is worh poiig ou i his coex ha he assumpio o m is ied o he large-sample behaviour of he maximum of he Y ;D ;:::; T; ad he srog assumpio (v) eails weak codiios o he esimaors b: I fac, uder Assumpio (v), we ca choose D log, which follows as i he proof of Lem 5.9 of Dahlhaus ad Poloik (9). Weakeig codiio (v) would require a sroger codiio o ; he rae of covergece of he esimaor b : The assumpios o he eropy iegral [see Assumpios (vii) ad (7)] corol he complexiy of he class G. May classes G are kow o saisfy hese assumpios see below for a example. For more examples, R we refer o he lieraure o empirical process heory. A more sadard codiio o he coverig umbers is p c= log N.u;G;d /du < (or similarly wih brackeig). Compared wih ha, he eropy iegral i Assumpio p (iv) does o have a square roo, ad he laer is similar o codiio (7) where he iegrad is log NB.u ; G;d / (oice he u ). This makes boh our eropy codiios sroger ha he sadard assumpio. The reaso for his is ha he expoeial iequaliy uderlyig our derivaios is o of sub-gaussia ype (Lemma 3), which i ur is caused by he depedece srucure of our uderlyig ime-varyig process. A class of o-parameric esimaors saisfyig codiios (vi) ad (vii) is give by he wavele esimaors of Dahlhaus e al. (999). These esimaors lie i he Besov smoohess class Bp;q s.c / where he smoohess parameers saisfy he codiio s C >:The cosa C > is a uiform boud o he (Besov) max.;p/ orm of he fucios i he class. Dahlhaus e al. derive codiios uder which heir esimaors coverge a rae log s=.sc/ i he L -orm. For s ; he fucios i Bp;q s.c / have uiformly bouded oal variaio. Assumig ha he model parameer fucios also posses his propery, he rae of covergece i he d -disace is he same as he oe i L, because i his case, he error i approximaig he iegral by he average over equidisa pois is of order O. /: Cosequely, i his case, we have m D log s=.sc/. Iorder o verify he codiio o he brackeig coverig umbers from Assumpio (iii), we use Nickl ad Pöscher (7). Their Cor., applied wih s D ; p D q D ; implies ha he brackeig eropy wih respec o he L -orm ca be bouded by Cı =. (Whe applyig heir Corollary o our siuaio, choose, i heir oaio, D ; D U Œ; ; r D ad D, say). 3. ESTIMATING THE DISTRIBUTION FUNCTION OF THE ERRORS T Esimaig he error disribuio ca be reaed by usig similar echiques as used for he iovaio disribuio, alhough he esimaio of he variace is o egligible see Theorem 4 ad Remark (a) righ ex o he heorem. For oher work ivolvig he esimaio of he error disribuio ad he volailiy, see, for isace, Akrias ad va eilegom () ad Neumeyer ad va eilegom (). wileyolielibrary.com/joural/jsa Copyrigh 6 Joh Wiley & Sos Ld J. Time. Ser. Aal. 38: 7 98 (7) DOI:./jsa.

6 WEIGHTED SUMS AND RESIDUAL EMPIRICAL PROCESSES 77 The sequeial empirical disribuio fucio of heb ca be rewrie as bf. ; / D b c D ¹b º D b c Accordigly, we defie he modified residual sequeial empirical process as D. b / Y C b! μ C :. ; ; g;s/d b c p F D g ² ³ g Y C s C Y C s C = i ; () where we hik of s S wih he class S beig a model for he differece b S. The followig is he aalogue o Theorem. Theorem 3. Assume Assumpios (i) ad (ii) ad ha G ad S are classes of ses saisfyig he eropy codiio (7). The we have for ay <L<ad ı! as!ha Œ;;.L;L/ ¹.g;s/G p SW P p kd kgkkckskı;º j. ; ; g;s/. ; ; ;/jdo P./: () If furher Assumpios (iii) ad (iv) hold, he asserio () also holds wih L Dad ksk replaced by ksk : Remarks: (a) The fac ha, for L D; we have o replace ksk by ksk is agai remiisce of assumpios used i he aalysis of parameric error processes [e.g. oul ad Lig, 6, Ass (4.4)]. (b) The residual sequeial empirical process from (3) correspods o o esimaig he variace. Formally, his correspods o usig he esimae of he variace beig cosa equal o (i.e. o dividig he residuals by a esimae of he variace). Sice S models he differece b ; his resuls i he class S D¹º cosisig of oly oe fucio. Wih his choice, he process () reduces o he process (3). I fac, we have. ; ; g/ D. ; ; g; /: Furher assumpios. (viii) We have kb k D o P. = / as!: (ix) There exiss a fucio class S wih b S wih probabiliy edig o as!, ss ksk < ; ad for some C;c > ; we have R c= log N B.u; S;d /du<c < 8 : Theorem 4. Assume Assumpios (i), (ii), (v) (ix) ad ha G ad S are classes of ses saisfyig he eropy codiio (7). I addiio, assume p ha f is differeiable wih bouded derivaive ad ha he sequece ¹m º i Assumpio (vi) is saisfyig D o./ as m!,where is such ha max Y D O P. /.The we have for <L<ha Œ;;.L;L/ bf d e. ; / F. ; / f. / b p D o P.= p /: () If furher Assumpios (iii) ad (iv) hold ad if D kbk D O P./, he () also holds wih L D: J. Time. Ser. Aal. 38: 7 98 (7) Copyrigh 6 Joh Wiley & Sos Ld wileyolielibrary.com/joural/jsa DOI:./jsa.

7 78 G. CHANDLER AND W. POLONI Remarks. (a) The heorem shows ha, i coras o he esimaio of he AR parameers, he esimaio of he variace fucio is o egligible, ad his eve holds i he i.i.d. case whe he variace is cosa. Tha he esimaio of he AR parameers k is egligible is due o he echical fac ha, i he Taylor expasio of E F b F. ; /; he esimaes of he AR parameers appear i sums of he form P D.b k / Yk.Now,if. b k / P lies i a class of fucios G; he his sum ca be absoluely bouded by gg D g Yk, ad because he Y k have mea zero ad he fucios g have small orm, he laer sum is o P. = / uder appropriae assumpios. I coras o ha, he sum of he differeces.b / does o ivolve he zero mea Y s, so ha he same rick does o apply. P (b) The erm f. / d e b. p /. / D i () is he firs-order sochasic approximaio o he differece of he appropriae cerigs F.b /. / Y C b F. /[see (49) ad (5) show laer]. The facor f. / also appears i oher work ha ivolves he esimaio of variace fucios such as Horváh e al. () ad oul ad Lig (6). (c) The wavele-based esimaor of he variace fucio of Dahlhaus ad Neuma () saisfies he ecessary assumpios o b for <L<. (d) I order o be able o uilize he earlier resuls o develop saisical mehodology, fier kowledge abou he P asympoic behaviour of he erm p d e b. /. / D is eeded. I ca be expeced ha, uder appropriae assumpios, his erm becomes asympoically ormal for reasoable esimaors, bu o he bes of our. / kowledge, such a resul is o available i he lieraure for esimaors of he variace fucio uder local saioariy. While i migh be possible o derive such a resul for cerai esimaors uder appropriae assumpios, explorig his quesio goes beyod he scope of his work. I is also uclear, wheher he asympoic ormaliy, if i ca be derived, could be easily used for saisical purposes, for wha really is eeded is he kowledge abou he asympoic disribuio of he sum F. ; / C f. / P p d e b. /. / D : For his,. / P a sochasic expasio of he sum p d e b. /. / D. / would be ideal. Such a expasio will of course heavily deped o he specific esimaor cosidered. Moreover, oe he has o be able o esimae he resulig asympoic variace, which migh o be sraighforward eiher, because he asympoic variace migh have a complex form. We would like o poi ou, however, ha oe of he basic ideas uderlyig our esig approach discussed i Secio 5 is o avoid esimaig he variace fucio alogeher ad exrac iformaio of he shape of he variace fucios by oher meas. 4. WEIGHTED SUMS UNDER LOCAL STATIONARITY The secod ype of processes of imporace i our coex is weighed parial sums of locally saioary processes give by Z.h/ D p D h Y ; h H: (3) I he i.i.d. case, weighed sums have received some aeio i he lieraure. For fucioal ceral limi heorems ad expoeial iequaliies, see, for isace, Alexader ad Pyke (986) ad va de Geer () ad he refereces herei. We will show i he succeedig discussio ha, uder appropriae assumpios, Z.h/ coverges weakly o a Gaussia process. I order o calculae he covariace fucio of he limi, we assume ha he process Y is locally saioary as i Dahlhaus ad Poloik (9). Recallig Assumpio (i), we assume he exisece of fucios a.;j/w.;! R wih wileyolielibrary.com/joural/jsa Copyrigh 6 Joh Wiley & Sos Ld J. Time. Ser. Aal. 38: 7 98 (7) DOI:./jsa.

8 WEIGHTED SUMS AND RESIDUAL EMPIRICAL PROCESSES 79 j D ja.u; j /j u `.j / ; (4) a;.j / a ;j ; (5) TV.a.;j// `.j / ; (6) where for a fucio g W Œ;! R; we deoe by TV.g/he oal variaio of g o Œ; : Codiios (4) (6) hold if he zeros of he correspodig AR polyomials are bouded away from he ui disk (uiformly i he rescaled ime u) ad he parameer fucios are of bouded variaio (see Dahlhaus ad Poloik, 6). Furher, we defie he ime-varyig specral desiy as he fucio wih A.u; / WD f.u;/wd ja.u; /j j D a.u; j / exp.ij/; ad c.u;k/ WD Z f.u;/exp.ik/d D j D a.u; k C j/a.u;j/ is he ime-varyig covariace of lag k a rescaled ime u Œ; : We also deoe by cum k./, hekh-order cumula of a radom variable. Theorem 5. Le H deoe a class of uiformly bouded, real-valued fucios of bouded variaio defied o Œ;. Assume furher ha for some C;c > ; Z c= log N.u;H;d /du<c < 8 : (7) The we have uder Assumpios (i), (ii) ad (v) ha, as!, he process Z.h/; h H; coverges weakly o a igh, mea zero Gaussia process ¹G.h/;h Hº: If, i addiio, (4) (6) hold, he he variace covariace fucio of G.h/ ca be calculaed as C.h ;h / D R h.u/ h.u/ S.u/ du; where S.u/ D P kd c.u;k/: Remarks. (a) Here, weak covergece is mea i he sese of Hoffma Jørgese see va der Vaar ad Weller (996) for more deails. (b) Weighed parial sums of he form Z. ; h/ D p b c g Y J. Time. Ser. Aal. 38: 7 98 (7) Copyrigh 6 Joh Wiley & Sos Ld wileyolielibrary.com/joural/jsa DOI:./jsa. D

9 8 G. CHANDLER AND W. POLONI are i fac a special case of processes cosidered i he heorem. Here, h.u/ D h g;.u/ D Œ;.u/ g.u/. Noe ha if g G ad G saisfies he assumpios o he coverig iegral from he aforemeioed heorem, he so does he class ¹h g;.u/ W g G; Œ; º. I his case, he limi covariace ca he be wrie as C.h g; ; h g; / D R ^ g.u/ g.u/ S.u/ du: (c) Assumpios (4) (6) are oly used for calculaig he covariace fucio of he limi process. The mai igredies o he proof of Theorem 5 are preseed i he followig resuls. These resuls are of idepede ieres. Theorem 6. Le ¹Y ; D ;:::;º saisfy Assumpios (i), (ii) ad (v), ad le H D¹h W Œ;! Rº be oally bouded wih respec o d : Furher, le A D P D Y M,whereM>. There exis cosas c ;c ;c >such ha, for all >saisfyig <6M p (8) ad >c Z log N.u;H;d /du _ 8M p! ; (9) we have P " # ² jz.h/j >;A c exp ³ : hh; khk c 5. APPLICATIONS We cosider a applicaio ha moivaes he aforemeioed sudy of he wo ypes of processes. The applicaio cosiss of esig for he cosacy of he variace fucio, ha is, esig for homoscedasiciy. We oe ha, for our es saisics, his paricular applicaio does o require esimaio of he variace fucio.u/;raher, he quesio is abou he cosacy of his fucio. This allows us o use he resuls of Theorem, avoidig he eed o hadle he bias erm arisig i Theorem 4 [see Remark (i) followig his heorem]. As we are primarily cocered wih quesios of homoscedasiciy, as well as lackig mehodology dealig wih disribuioal ess regardig he error erms i his seig o which o compare, we do o explore quesios regardig F here. We oe ha such applicaios would be more difficul because of his addiioal erm. However, whe explorig quesios of homoscedasiciy ad relaed quesios abou saioariy, he curre mehodology is sigificaly easier o impleme ha he mehods o which we compare; see succeedig discussio. Addiioally, more geeral quesios of homoscedasiciy are o able o be hadled by hose mehodologies. I he case of a ime-varyig AR model, he homoscedasic ad heeroscedasic model boh live i he aleraive space for he ess of saioariy. A secod esig problem, closely relaed alhough more ivolved ha he firs, is a es for deermiig he modaliy of he variace fucio, deails of which ca be foud i Chadler ad Poloik (). 5.. A es of homoscedasiciy We are ieresed i esig he ull hypohesis H W.u/ D for all u Œ;. (Noe ha if we addiioally assume ha he AR parameers do o deped o ime, he uder mild codiios o he k, his is also a es for wileyolielibrary.com/joural/jsa Copyrigh 6 Joh Wiley & Sos Ld J. Time. Ser. Aal. 38: 7 98 (7) DOI:./jsa.

10 WEIGHTED SUMS AND RESIDUAL EMPIRICAL PROCESSES 8 weak saioariy.) To his ed, cosider he process bg ;. / D b c D b bq ; Œ; ; () where bq is he empirical quaile of he squared residuals. Noice ha bg ;. / cous he umber of large (squared) residuals wihi he firs. /% of he observaios Y ;:::;Y, where large is deermied by he choice of. If he variace fucio is cosa, he, sice oe ca expec o have a oal of bc large residuals, he expeced value of bg ;. / approximaely equals : This moivaes he form of our es saisic, T D r Œ;. / bg ;. / : Followig he discussio of our applicaio, we argue ha he large-sample behaviour of his es saisic uder he ull is ha of he remum of a Browia bridge (Secio 5.). While here seems o be a obvious robus compoe o his mehodology, i is o obvious ha i should be very powerful, ad perhaps, oo much iformaio is beig los i cosiderig he daa i his way. As we are uaware of ay compeig ess for cosacy of he variace fucio i a oherwise o-saioary seig, we aemp o allay fears abou a lack of power by cosiderig he followig es of saioariy via our mehodology. We cosider he uivariae locally saioary model used i Puchsei ad Preuß (6), ; D C Z ; () where he Z N.; / i.i.d. Thus, we ru our es o simply he squared observaios. As compared wih our es, mos ess for saioariy i he lieraure (e.g. vo Sachs ad Neuma, ; Paparodiis, 9, ; Preuß e al., 3; Puchsei ad Preuß, 6) work i he specral domai, comparig a esimae of he ime-varyig specrum wih he saioary esimae. We compare he power behaviour of our es based o T o hree of hese es. To his ed, we use resuls preseed i Puchsei ad Preuß (6), which compare he mehods of Paparodiis () ad Preuß e al. (3) wih heir proposed mehod. This laes mehod has he beefi of o requirig a widow size o be chose, while he oher wo mehods were ru a wo differe widow sizes o opimize he procedures. For our proposed ime domai mehod, we use a D :8, alhough his does o prove o be oo criical. For example, wih D 56 ad uder Model (), he power was.87,.885 ad.86 for D.:7; :8; :9/. The proposed model is compeiive (Table I) ad much simpler boh cocepually ad implemeaio-wise. Accordig o our heory, he esimaio of he parameer fucios should o have a effec o he es, asympoically. To see how far his also holds for fiie samples, we simulaed from a var() of he form ; D :8 si ; C Z Table I. Simulaio resuls for model () T Paparodiis () Preuß e al. (3) Puchsei ad Preuß (6) J. Time. Ser. Aal. 38: 7 98 (7) Copyrigh 6 Joh Wiley & Sos Ld wileyolielibrary.com/joural/jsa DOI:./jsa.

11 8 G. CHANDLER AND W. POLONI q Figure. T D q.q/ jj OG Gjj (horizoal) vs p jjfo F jj (verical) ad he esimaed he parameer fucio.u/ D :8 si.u/ usig local leas squares followed by a kerel smooh. Noe ha his paricular form of he esimaor does o saisfy he assumpios uderlyig our heory, bu he discussio laer demosraes he robusess of he resul o hese assumpios. We geeraed daa ses accordig o his model wih D 56, ad he es saisic regardig he residuals was compued for each. We he geeraed a sample from a uiform disribuio ad compued he olmogorov Smirov saisic, properly ormalized for D as a sample from he limiig ull disribuio. A QQ plo of he wo es saisics is provided i Figure. We he simulaed from he model ; D :8 si ; C C Z ; which has he same volailiy fucio as he modulaed whie oise process (). Wih D 56, 884 of ime series from his model resuled i a rejecio of he ull of cosa variace usig a D :5. We oe ha his is very close o he esimaed power (.878) foud for Model (), as expeced. Based o he resuls comparig his idea wih muliple mehods i he beer sudied quesio abou saioariy, we are led o believe ha hese resuls are promisig, despie lackig a secod mehod o which o compare. 5.. Asympoic disribuio of a es saisic T Noice ha bg ;. / is closely relaed o he sequeial residual empirical process, ad as ca be see from he proof of Theorem 7, weighed empirical processes eer he aalysis of bg ;. / hrough hadlig he esimaio of q. Theorem 7 give i he succeedig exs provides a approximaio of he es saisic T by idepede (bu o ecessarily ideically disribued) radom variables. This resul crucially eers he proofs i Chadler ad Poloik (), where a similar es saisic is used o es for he modaliy of he variace fucio. I paricular, i implies ha he large-sample behaviour of he es saisic T uder he ull hypohesis is o iflueced by he, i geeral, o-parameric esimaio of he parameer fucios, as log as he rae of covergece of hese esimaors is sufficiely fas. Firs, we iroduce some addiioal oaio. Le f u deoe he pdf of.u/,hais,f u. / D f,.u/.u/ ad G ;. / D b c q ; where q is defied by meas of. / D Z F D du.u/ Z F du; ; ().u/ wileyolielibrary.com/joural/jsa Copyrigh 6 Joh Wiley & Sos Ld J. Time. Ser. Aal. 38: 7 98 (7) DOI:./jsa.

12 WEIGHTED SUMS AND RESIDUAL EMPIRICAL PROCESSES 83 as he soluio o he equaio.q / D : (3) Noice ha his soluio is uique sice we have assumed F o be sricly moooic, ad if.u/ D is cosa for all u Œa; b, heq equals he upper -quaile of he squared iovaios D : The approximaio resul ha follows does o assume ha he variace is cosa, however. Theorem 7. Le Œ; ad pose ha a<bare o-radom. The, uder Assumpios (i) (v), wih = log D o./; we have as!ha m where p bg ;. / G ;. / C c. /.G ;./ EG ;.// D op./; (4) Œ; c. / D R Œ f u.q / C f u.q / du R Œf u.q / C f u.q / du : Uder he ull hypohesis.u/ >for u Œ; ; we have c. / D : Moreover, i case he AR parameer i Model () is cosa ad p -cosise esimaors are used, he he mome assumpios o he iovaios ca be sigificaly relaxed o E < : Uder he ull hypohesis,.u/ D for all u Œ;, he iovaios are i.i.d. Usig ha i his case c. / D ad EG ;. / D ; we see ha he aforemeioed resul implies ha, i his case,.. // =p.bg ;. / / coverges weakly o a sadard Browia Bridge (cf. Chadler ad Poloik, ). 6. PROOFS Throughou he proofs, we use he oaio F u. / D P..u/ /. 6.. A crucial maximal iequaliy The proofs of Theorems ad 3 res o he followig lemma, which is of idepede ieres. I is modelled afer a similar resul for empirical processes (va de Geer,, Thm 5.). Le H L D Œ;.L; L/ Gp S deoe he idex space of he process defied i (), where <L. Defie a meric o H L as d ;W.h ;h / Dj jcjw. / W. /jc p kg ;k g ;k k Cks s k ; (5) where W.x/ D R x w.y/dy wih w.y/ > iegrable such ha, wih >from Assumpio (iv), we have lim jyj! y Cw.y/ D. Observe ha W is a sricly icreasig, posiive, bouded fucio o R ad he ails of f are o heavier ha he oes of w, assumig ha Assumpio (iv) holds. Lemma. Assume Assumpios (i) ad (ii). Furher assume ha boh G ad S are oally bouded wih respec o d.lea D P sdpc Y s C \ gg jg ± Y j <C : Defie D ss ksk C m C L C p pc : Suppose ha C ;; >are such ha kf k m J. Time. Ser. Aal. 38: 7 98 (7) Copyrigh 6 Joh Wiley & Sos Ld wileyolielibrary.com/joural/jsa DOI:./jsa. kd

13 84 G. CHANDLER AND W. POLONI 6 p ; (6) p. ^ /; (7) C Z = 8 p The for <L< ad C 6 p, we have wih C D q log N B u ; H L ;d ;W du _ : (8) 6. 6 C/ C C ha " # P.h /.h /j ; A C exp h ;h H L W d;w.h;h/ 6. 6 C / : If, i addiio, Assumpio (iv) holds ad, for some c >;we have if ss if uœ; Œ.s C /.u/ >c ; he he asserio also holds for L Dwih a modified (see Proof). Remarks: (a) The asserio also holds (wih a slighly modified ) o he simplified se A D P sdpc Y s C if we, i addiio, assume ha gg kgk < : (b) This lemma of course also applies o H L D Œ;.L; L/ G p replacig H L [wih he appropriaely modified (simplified) d ;W ]. Sice his case formally correspods o puig S D¹º, all he assumpios ivolvig S are rivially saisfied. Moreover, i his case, he assumpios gg kgk < ad xr jxj Cf.x/< for some >ca boh be dropped (see also Proof). (c) The assumpio if ss if uœ; Œ.s C /.u/ >c for some c >is fulfilled if ss ksk is arbirarily small (recall ha we assumed o be bouded away from zero). I our applicaio, we hik of s modellig he differece b, so ha (uiform) cosisecy of he esimaor b will esure ha, wih high probabiliy, b will lie i S; saisfyig he assumpio. Proof We oly prese a oulie of he proof. Le h D. ; ; g;s/ H L. Firs oice ha.h/ D. ; ; g;s/ is a sum of bouded marigale differeces. To see his, le ;g;s D g Y C s C ad ;g;s Q D ;g;s E ;g;s jf,wheref D. ; ;:::/ deoes he -algebra geeraed by ¹ ; ;:::º,he. ; ; g;s/d p D Q ;g;s : Obviously, also. ; ; g ;s /. ; ; g ;s / are sums of marigale differeces. The proof of he lemma is based o he basic chaiig device ha is well kow i empirical process heory, uilizig he followig expoeial iequaliy for sums of bouded marigale differeces from Freedma (975). Lemma (Freedma 975). Le Z ;:::;Z deoe marigale differeces wih P respec o a filraio ¹F ; D ;:::; º wih jz j C for all D ;:::;: Le furher S D p D Z ad V D V.S / D wileyolielibrary.com/joural/jsa Copyrigh 6 Joh Wiley & Sos Ld J. Time. Ser. Aal. 38: 7 98 (7) DOI:./jsa.

14 WEIGHTED SUMS AND RESIDUAL EMPIRICAL PROCESSES 85 P D E Z j F : The we have for all ; >ha P S ; V exp C C p! : (9) I order o be able o apply (9) o our problem, i is crucial o corol he quadraic variaio V.Weow idicae how o do his. Le ; ;g;s D ;g;s Q : We have for h D. ; ; g ;s /; h D. ; ; g ;s / H wih d ;W.h ;h / ha V DV..h /.h // " D E e ;s ; ;g D D j jc C D E ;g ;s For he las sum, we obai by elescopig C C D ;g E ;s F D F D F D e ; ;g;s E ;g;s F Q ;g ;s F ;g;s F : C F# D hq ;g E ;s ;g;s Q F i g Y C.s C / F g Y C.s C / (3) g Y C.s C / F g Y C.s C / (3) g Y C.s C / F g Y C.s C / (3) We esimae he hree erms o he righ separaely. To corol (3), le V.y/ D F Y C.s C / W.y/,whereW is iroduced i (5) earlier. The g.3/ D D V.W. // V.W. // D f g Y C.s C / W./ w W./ W. / W. /.s C / D (33) wih bewee W. / ad W. /, ad hus, by moooiciy of W; we have W./.L; L/ (sice ;.L; L/). I fac, for L<; we ca choose w.x/ D for x.l; L/ ad w.x/ D else, ad he erm (33) J. Time. Ser. Aal. 38: 7 98 (7) Copyrigh 6 Joh Wiley & Sos Ld wileyolielibrary.com/joural/jsa DOI:./jsa.

15 86 G. CHANDLER AND W. POLONI (i.e. (3)) ca herefore be bouded by kf k. ss ksk C m / m : For L D ; we also eed o cosider cases i which W./ is large. We fix L large eough so ha, for W./ > L,wehaveoA ha g Y C.s C / W./ > jw./=j ad ha jxj>l jxj Cw.x/ > =. The laer ca be achieved by our assumpio o w (give righ before he formulaio of he lemma). To see he former, assume ha (a) s C >c ad (b) gg jg Y j <C. The, if j j C c, he i is easy o see ha j. /j j j ; which is he asserio. (a) holds by assumpio ad (b) holds o A. Usig hese properies, we obai f g Y C.s C / W./ M w W./ j g Y C.s C / W./ /Cw W./ C.m / W./ C C.m / M: w.w.// Thus, for L D; we have he boud kf k.3/ max m ; C.m / xcf.x/ ksk C m : xr ss Nex, we cosider (3). By a applicaio of a oe-erm Taylor expasio, here exiss lyig bewee s ad s such ha, wih he shorhad oaio. / D g Y C C,wehaveha F g Y C.s C/ F g Y C.s C/ D f. /.s s / : Cosequely, for L<,.3/ D D f.s. / s / kf k L: For L D, we argue as follows. Sice lies bewee s ad s, we have from our assumpio ha C >c : As meioed earlier i he esimaio of (33), i ow follows ha if j j C c,he j. /j j j ; adwehaveforj j C c ha.3/ D D D f.s. / s / f m xr jxf.x/j m m xr jxf.x/j m :. /.s s. / /.s s / D m. /. / j j j. /j wileyolielibrary.com/joural/jsa Copyrigh 6 Joh Wiley & Sos Ld J. Time. Ser. Aal. 38: 7 98 (7) DOI:./jsa. (34)

16 WEIGHTED SUMS AND RESIDUAL EMPIRICAL PROCESSES 87 Cosequely, we have uiformly over R.3/ m C max kf k c ; m xr jxf.x/j : As for (3), we have o A (ad oe ha we do o eed he eve gg jg ± Y j <C o hold here).3/ u;x f u.x/ kf k m.g g / D v p u kg k g k k p kd Y Dp Y p pc kf k m : (35) Puig everyhig ogeher, we obai ha, for h ;h H L wih d ;W.h ;h / ad,wehaveo A for <L< ha V..h /.h // ; (36) wih D kf k m ss ksk C m C L C p pc ; as defied i he formulaio of he lemma. If L D; he he formula jus show shows ha (36) holds where he correspodig ca immediaely be obaied from he aforemeioed esimaes. Proofs of saemes i Remarks give afer he lemma (a) If we modify A by droppig he eve gg jg ± Y j <C gg kgk <, he we ca esimae he las average i (34) as follows: D ad isead assume ha v.s s / u g Y.s s / g Y / D D v u g Y / D v u kgk p Y gg DpC C = kgk p! : gg This he leads o a slighly modified cosa, which ca easily be deermied. (b) If H L is replaced by H L he, as has bee discussed earlier, his formally correspods o seig S D¹ º, which is a class cosiss of jus oe fucio, implyig ha, i he aforemeioed proof, s s D.Ioher words, all he assumpios used o boud (3) become obsolee. J. Time. Ser. Aal. 38: 7 98 (7) Copyrigh 6 Joh Wiley & Sos Ld wileyolielibrary.com/joural/jsa DOI:./jsa.

17 88 G. CHANDLER AND W. POLONI The jus show corol of he quadraic variaio i cojucio wih Freedma s expoeial boud for marigales ow eables us o apply he (resriced) chaiig argume i a way similar o he proof of Thm 5. i va de Geer (). Deails are omied. 6.. Proofs of Theorems ad 3 We oly prese he proof of Theorem 3. The proof of Theorem is exacly he same. Recall ha he meric d ;W.h ;h / o H L is defied i (5), ad observe ha, for h D. ; ; g;s/ ad h D. ; ; ;/; we have d ;W.h ;h / D P p kd kg k k Cksk. We hus obai for >ad <L<ad C>ha B C j. ; ; g;s/. ; ; ;/ja Œ;;.L;L/;gG Pp p kd kg k kckskı P A { C P d ;W.h ;h /ı h ;h H L C j.h /.h /j; A A ; where as i Lemma, A D P sdpc Y s C \ gg jg ± Y j <C for some C ;C >: A applicaio of Lemma gives he asserio oce we have show ha P A { ca be made arbirarily small q log N B u ; H L ;d ;W du <. (for sufficiely large ) adha R c To see his, oice ha, by our assumpios, we have boh R c p log NB.u ; G;d /du < ad R q R q c log N B u ; S;d du < : This implies c log N B u ; H L ;d ;W du <, because for hi D. i; i; g i ;s i /; i D ; wih j j < ; jw. / W. /j < ; kg 4 4 k g k k < ;k D ;:::;p ad 4p ksk <, we obviously have d 4 ;W.h ;h /<;ad hus, by usig sadard argumes, i is o difficul o see ha log N B ; H L ;d ;W C log 4 C p log N B ; G;d 4p C log N B ; S;d 4 (37) for some C >. I remais o show ha P.A { / ca be made arbirarily P small (for large) by choosig C ;C sufficiely large. Because of Assumpio (iii), his follows from D E Y < ; whichiurfollows easily from EY D E 4 j D 3 a ;. j/ j 5 D E for some C>. Here, we are usig Assumpio (ii) Proof of Theorem 5 D j D kd j D a ;. j/a ;. k/ j k a ;. j/ j D <C <; `.j / Firs, we formulae ad prove a lemma ha is eeded i he proof of Theorem 5. For radom variables ;:::; k, we deoe by cum. ;:::; k / heir joi cumula, ad if i D for all i D ;:::;k, he cum. ;:::; k / D cum.;:::;/d cum k./,hekh-order cumula of. wileyolielibrary.com/joural/jsa Copyrigh 6 Joh Wiley & Sos Ld J. Time. Ser. Aal. 38: 7 98 (7) DOI:./jsa.

18 WEIGHTED SUMS AND RESIDUAL EMPIRICAL PROCESSES 89 Lemma. Le ¹Y ;D;:::;º saisfy Assumpios (i) ad (ii). For j D ;;:::,leh j be fucios defied o Œa; b wih kh j k < : The here exiss a cosa < such ha for all k such ha cum.z.h /;:::;Z.h k // k cum k. / If, i addiio, kh j k M<; jd ;:::;k, he for k 3; ky kh j k : j D cum.z.h /;:::;Z.h k //. / k M k cum k. / k : Proof P We have cum.z.h /;:::;Z.h k // D k h k ; ;:::; kd :::hk cum.y ;:::;Y k / by uilizig mulilieariy of cumulas. I order o esimae cum.y ;:::;Y k /; we uilize he special srucure of he Y - variables. Sice he j are idepede ad have mea zero, we have ha cum. j ;:::; jk / D uless all he j`; `D ;:::;k are equal, ad by agai usig mulilieariy of he cumulas, we obai cum.y ;:::;Y k / D cum k. / mi¹ ;:::; kº j D a ;. j/:::a k;. k j/: (38) Thus, cum.y ;:::;Y k / cum k. / P cum.z.h /;:::;Z.h k // k D k j D Q k id cum k. / cum k. / `.j i ; ad cosequely, j j/ ky 4 hi j D id i D Y 4 hi j D id i D ky 4 i hi id3 i D i i `.j i j j/ 3 5 `.j i j j/ 3 5 `.j i j j/ 3 5 : Uilizig he Cauchy Schwarz iequaliy, we have for he las produc 3 v ky 4 i hi ky u 5 i h i `.j i j j/ id3 i D id3 i D v k Y u k kh i k id3 k k v uu `.j i j j/ D ky kh i k ; id3 i D `.jj/ (39) J. Time. Ser. Aal. 38: 7 98 (7) Copyrigh 6 Joh Wiley & Sos Ld wileyolielibrary.com/joural/jsa DOI:./jsa.

19 9 G. CHANDLER AND W. POLONI r P P where we used he fac ha D `.jj/ D boud (39) does o deped o he idex j aymore, so ha cum.z.h /;:::;Z.h k // k By usig P j D `.j j j/ `.jj/ ky kh i k cum k. / id3 for some < : Noice ha he j D id 3 Y 4 i hi 5 : `.j i j j/ i D D P `.j j j/ j D P `.j j j/ `.j. /C j j/ `.j j/ j D for some `.j j/ >ad Cauchy Schwarz iequaliy, we obai 3 Y 4 i hi 5 D h `.j i j j/ j D id i D D D h h `.j j/ D v D v u u h h `.j j/ D kh k kh k : D D h D j D `.j j/ `.j j j/ `.j j j/ `.j j j/ This complees he proof of he firs par of he lemma. The secod par follows similar o he aforemeioed by observig ha if kh i k <Mfor all i D ;:::;k, he, isead of he esimae (39), we have wih D P D ha `.jj/ 3 ky 4 i hi Y k 5 M k `.j i j j/ id3 i D id3 i D `.j i j j/.m / k : Now, we coiue wih he proof of Theorem 5. Showig weak covergece of Z.h/ meas provig asympoic ighess ad covergece of he fiie dimesioal disribuio (e.g. va der Vaar ad Weller, 996). Tighess follows from Theorem 6. I remais o show he covergece of he fiie dimesioal disribuios. To his ed, we will uilize he Cramér Wold device i cojucio wih he mehod of cumulas. I follows from Lemma ha all he cumulas of Z.h/ of order k 3 coverge o zero as!: Usig he lieariy of he cumulas, he same holds for ay liear combiaio of Z.h i /; i D ;:::;:The mea of all he Z.h/ equals zero. I remais o show ha covergece of he covariaces cov.z.h /; Z.h //: The rage of he summaio idices show ex are such ha he idices of he Y -variables are bewee ad. For ease of oaio, we achieve his by formally seig h i.u/ D for u ad u>;id ; : We have cov.z.h /; Z.h // D D h D jkj p D sd h h s cov.ys ;Y / cov.y ;Y k / C R ; h k (4) wileyolielibrary.com/joural/jsa Copyrigh 6 Joh Wiley & Sos Ld J. Time. Ser. Aal. 38: 7 98 (7) DOI:./jsa.

20 WEIGHTED SUMS AND RESIDUAL EMPIRICAL PROCESSES 9 where for sufficiely large, jr j h D jkj> p k h cov.y ;Y k /: From Proposiio 5.4 of Dahlhaus ad Poloik (9), we obai ha jcov.y ;Y k /j for some cosa : Sice boh h ad h are bouded ad P kd < ; we ca coclude ha R `.k/ D o./: The mai erm i (4) ca be approximaed as where jr ; j D jkj p h D jkj p h `.k/ k h c ;k C R ; (4) k h cov.y ;Y k / c : ;k Proposiio 5.4 of Dahlhaus ad Poloik (9) also gives us ha, for jkj p,wehave P c ;k C jkj for some >. Usig his, we obai jr ; j p h D kd p p kd p D h k cov.y ;Y k / c cov.y ;Y k / c ;k as! : Nex, we replace h. k / i he mai erm of (4) by h bouded by h D jkj p h k h ;k p kd p D cov.y ;Y k / C jkj D o./ `.jkj/. The approximaio error ca be `.jkj/ D o./: Here, we are usig he fac ha u jc.u;k/j (see Prop 5.4 i Dahlhaus ad Poloik, 9) ogeher `.jkj/ wih he assumed (uiform) coiuiy of h, he boudedess of h ad he boudedess of P kd.we `.jkj/ have see ha cov.z.h /; Z.h //D h h c k p ;k C o./: D Sice S.u/ D P kd c ;k < ; we also have cov.z.h /; Z.h //D kd h h c ;k C o./: D J. Time. Ser. Aal. 38: 7 98 (7) Copyrigh 6 Joh Wiley & Sos Ld wileyolielibrary.com/joural/jsa DOI:./jsa.

21 9 G. CHANDLER AND W. POLONI Fially, we uilize he fac ha TV.c.;k//, which is aoher resul from Prop 5.4 of Dahlhaus ad `.jkj/ Poloik (9). This resul, ogeher wih he assumed bouded variaio of boh h ad h ; allows us o replace he average over by he iegral A crucial expoeial iequaliy The followig expoeial iequaliy is a crucial igredie o he proof of Theorem 6. I relies o he corol of he cumulas, which is provided i Lemma show earlier. Lemma 3. Le ¹Y ;D;:::;º saisfy Assumpios (i) ad (ii). Le h be a fucio wih khk <. Assume ha here exiss a cosa C>such ha, for all k D ;;:::;we have jcum k. /jkšc k : The here exis cosas c ;c >such ha, for ay >;we have ² P Œ jz.h/j > c exp c khk Proof Usig Lemma, we have he assumpios o he cumulas implyig ha Z.h/./ D log Ee Z.h/.C khk / k ; kd assumig ha >is such ha he ifiie sum exiss ad is fiie. We obai Choosig D P ŒjZ.h/j >e E e Z.h/ exp C ³ : (4) μ.c khk / k : C khk gives he asserio wih c D e = ad c D C : The fac ha cum j. / jšc j ; j D ;;::: (43) kd holds if Ej j k C k ;kd;;:::;ca be see by iducio. Deails are omied Proof of Theorem 6 Usig Lemma 3, we ca mimic he proof of Lem 3. from va de Geer (). As compared wih ha of va de ± Geer, our expoeial boud is of he form c exp c khk raher ha c exp ²c khk ³.Iiswellkow ha his ype of iequaliy leads o he coverig iegral beig he iegral of he meric eropy raher ha he square roo of he meric eropy. (See, for isace, Thm..4 i va der Vaar ad Weller, 996.) This idicaes he ecessary modificaios o he proof i va de Geer. Deails are omied Proofs of Theorems ad 4 Proof of Theorem 4 Recall ha bf. ; / D P b c D.b / Y C.b / ± C ad ha, by wileyolielibrary.com/joural/jsa Copyrigh 6 Joh Wiley & Sos Ld J. Time. Ser. Aal. 38: 7 98 (7) DOI:./jsa.

22 WEIGHTED SUMS AND RESIDUAL EMPIRICAL PROCESSES 93 assumpio, P b ; b G p S! as!.le F. ; ; g;s/d b c D ² g Y C.s C / ³ : (44) The we have wih F. ; ; ;/ D F. ; ;b ;b /. Furher, le P b c D ¹ º ha F. ; / D F. ; ; ;/ ad bf. ; / D E. ; ; g;s/d b c D F g Y C.s C / (45) deoe he codiioal expecaio of F. ; ; g;s/ give F, wheref D.Y ;Y ;:::/. The purpose of iroducig E is o make F E a marigale differece (see i he followig exs). We ow have bf. ; / F. ; / D F. ; ; b ;b / F. ; ; ;/ i D h.f E /. ; ; b ;b /.F E i /. ; ; ;/ C he. ; ; b ;b / E. ; ; ;/ (46) DW T. ; ;b ;b /C T. ; ;b ;b /: Usig his decomposiio, we will show he asserio by provig he followig wo properies: p T. ; ;b ;b / D op./ (47) Œ;;.L;L/ " p T. ; ;b ;b / f. / Œ;;.L;L/ b # D o P./: (48) D Verificaio of (47). Le. ; ; g;s/ D p F E. ; ; g;s/deoe he error sequeial empirical process. By akig io accou ha, by assumpio, P p kd kb k k k D O P.m / ad kb k D O P. =4 /; we have ha, for ay >;we ca fid a C>such ha, wih probabiliy a leas for large eough ; p T. ; ;b ;b / Œ;;.L;L/ Œ;;.L;L/; gg p ; P p kd kg k kcm ss; kskc =4 j p T. ; ; g;s/j: Tha he righ-had side is o P./ is a immediae applicaio of Theorem 3, ad (47) is verified. Verificaio of (48). We have by a simple oe-erm expasio ha J. Time. Ser. Aal. 38: 7 98 (7) Copyrigh 6 Joh Wiley & Sos Ld wileyolielibrary.com/joural/jsa DOI:./jsa.

23 94 G. CHANDLER AND W. POLONI " p T. ; ;b ;b / D p d e F D f. / D d e p C p d e D. b / Y C b! # F. / ". b / Y C b # (49) ". b / f.. // Y C b # (5) D wih. / bewee ad.b/. / Y C b. /. The erm (5) is a remaider erm ha will be reaed as show. /. / laer. The sum i (49) ca be wrie as a sum of wo erms f. / d e p D.b / Y C f. / d e p D b : (5) The firs erms are (fiie sum of) weighed sums of he Y s, ad we will ow use ± Theorem 6 o show ha his erm is o P./. By recallig ha, by assumpio, he probabiliy of b G p eds o as!, we ca assume ha.b / k G for all k D ;:::;p.forh H D¹h ;g D.u/ g.u/ I Œ; ; g Gº wih he.u/ shorhad oaio.u/ D.u /, le Z k;.h/ D p D The we ca wrie he firs erm i (5) as f. / P p ± eve b G p )ha h Y k ; k D ;:::;p: (5) kd Z k;.b/ k : Wih his oaio, we have (o he d e f. / p D.b / Y f.x/ x p kd hh jz k;.h/j: I hus remais o show ha hh jz k;.h/j D o P./ for all k D ;:::;p: To his ed, we will apply Theorem 6. Clearly, hh khk m gg kgk m m D o./: Sice EZ k;.h/ D for all h H, a applicaio of Theorem 6 ow gives he asserio oce we have verified ha he class H saisfies he codiio o he coverig umbers. This, however, follows easily because by assumpio, G has a fiie coverig iegral wih respec o d ; is jus a sigle fucio ha is bouded above ad below ad he class of fucios.u/ ¹.u/; Œ; º is a Vapik-Chervoekis (VC)-subgraph class (ad hus has a fiie coverig iegral as well). Now, we show ha he erm i (5) is o P./. Wehave.5/ p C p D D ". b / # f.. // Y (53) " b # f.. // ; (54) wileyolielibrary.com/joural/jsa Copyrigh 6 Joh Wiley & Sos Ld J. Time. Ser. Aal. 38: 7 98 (7) DOI:./jsa.

24 WEIGHTED SUMS AND RESIDUAL EMPIRICAL PROCESSES 95 ad uder our assumpios, boh (53) ad (54) are o P./ uiformly over. I fac, for (53), we have p D ". b / f.. // x f.x/ p m D # Y.b / Y ; where for he las sum we have (recall ha is such ha max Y D O P / p D.b / Y p D b / k kd p max Y p OP p m p Y j j D p D p m A O P./ D o P./; where he las equaliy uses our assumpios. The fac ha (54) i o P./ follows immediaely oce we have show ha f.. // D O P./: (55) R To see ha (55) holds, firs observe ha, by assumpio, we have lim jxj! jxf.x/j D; ad a applicaio of l Hospial s Rule gives lim jxj! jx f.x/j D: Sice f is coiuous, we coclude ha f is bouded, ad hus, i is clear ha (55) holds for j j <L<: For L D, he argume is as follows. The previous argumes show ha xr jf.x/x j < : Sice max R see ha i is sufficie o show ha max R jf.. // j D R max jf.. //. /jj. /j, we. / D O P./: (56) This follows by a argume alog he lies used o boud (33) i he proof of Lemma 3. Furher deails are omied. Treaig residual empirical process, i.e. provig Theorem, is very similar. We omi he deails. Proof of Theorem 7 Firs recall ha bf. ; / D we ca wrie P b c D.b / ad F. ; / D p b G ;. / G ;. / P b c D : Wih his oaio, D p h i bf. ;bq / C bf. ; bq /. F. ; q / C F. ; q // h p D F. ;bq / bf. ;bq / p i F. ; bq / bf. ; bq / p. F. ;bq / F. ; q / / p. F. ; bq / F. ; q / / DW I. / II. / (57) J. Time. Ser. Aal. 38: 7 98 (7) Copyrigh 6 Joh Wiley & Sos Ld wileyolielibrary.com/joural/jsa DOI:./jsa.

25 96 G. CHANDLER AND W. POLONI wih I. / ad II. /; respecively, deoig he wo quaiies iside he wo Œ-brackes. The asserio of Theorem 7 follows from ji. /j Do P./ ad (58) Œ; jii. / c. /.G ;./ EG ;.// jdo P./: (59) Œ; Clearly, Œ; ji. /j Œ;; R jbf. ; / F. ; /j; ad hus, propery (58) follows from Theorem. Propery (59) will ow be show by usig empirical process heory based o idepede, bu o ideically disribued, radom variables. Proof of (59) Defie We ca wrie F. ; / WD EF. ; / D b c D F! : II. / D p.f F /. ;bq /.F F /. ; q / (6) C p.f F /. ; bq /.F F /. ; q / (6) C p.f. ;bq / F. ; q // p.f. ; bq / F. ; q //: (6) The process. ; ; / D p.f F /. ; / is a sequeial empirical process, or a iefer Müller process, based o idepede, bu o ecessarily ideically disribued, radom variables. This process is asympoically sochasically equicoiuous, uiformly i wih respec o.v; w/ D F.; v/ F ;.; w/. Tha is, for every >;here exiss a >wih lim P! " # j. ; ; /. ; ; /j > Œ;;. ; / I fac, wih. ; /;. ; / / Dj jc. ; /; we have Œ; j. ; ; /. ; ; /j ; ;R;. ; / ; Œ;; ; R.. ; /;. ; // D : (63) j. ; ; /. ; ; /j: Thus, (63) follows from asympoic sochasic d -equicoiuiy of. ; ; /. This i ur follows from a proof similar o, bu simpler ha, he proof of Lemma. I fac, i ca be see from (35) ha, for g D g D ; we simply ca use he meric.. ; /;. ; / / Dj j C. ; / i he esimaio of he quadraic variaio, which i he simple case of g D g D amous o he esimaio of he variace, because he radomess oly comes i hrough he. Wih his modificaio, he proof of he -equicoiuiy of. ; ; / follows he proof of Lemma. Thus, if bq is cosise for q wih respec o, he i follows ha boh (6) ad (6) are o P./. The proof of he cosisecy is omied here. (I is available from he auhors.) This complees he proof. wileyolielibrary.com/joural/jsa Copyrigh 6 Joh Wiley & Sos Ld J. Time. Ser. Aal. 38: 7 98 (7) DOI:./jsa.