Asymptotic normality of the NadarayaWatson estimator for nonstationary functional data and applications to telecommunications.


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1 Asymptotic ormality of the NadarayaWatso estimator for ostatioary fuctioal data ad applicatios to telecommuicatios. L. ASPIROT, K. BERTIN, G. PERERA Departameto de Estadística, CIMFAV, Uiversidad de Valparaíso, Chile Istituto de Matemática y Estadística Rafael Laguardia, Facultad de Igeiería, Uruguay. July 6, 2007 Abstract We study a oparametric regressio model where the explaatory variable is a ostatioary depedet fuctioal data ad the respose variable is scalar. Supposig that the explaatory variable is a ostatioary mixture of statioary processes ad geeral coditios of depedece of the observatios implied i particular by weakly depedece, we obtai the asymptotic ormality of the NadarayaWatso estimator. Uder some additioal regularity assumptios o the regressio fuctio, it ca be obtaied asymptotic cofidece itervals for the regressio fuctio. We apply this result to the estimatio of the quality of service o the Iteret where the cross traffic is ostatioary. Keywords: Noparametric regressio, fuctioal data, asymptotic ormality, ostatioarity, depedece, quality of service. AMS subject classificatio: 62G05, 62G08, 62G20. Itroductio We study the problem of estimatig a oparametric regressio fuctio φ whe the explaatory variable is fuctioal ad may be ostatioary ad depedet. We observe {X, Y } = {X i, Y i : i = 0,..., } such that Y i = φx i + ε i, i = 0,...,, where φ is a fuctio φ : D R, X i D, Y i R, the ε i s are cetered ad idepedet of the X i s ad D is a semiormed liear space with semiorm. Istituto de Matemática y Estadística Rafael Laguardia, IMERL, Facultad de Igeiería, Julio Herrera y Reissig 565, Motevideo 200, Telfax: , s: Departameto de Estadística, CIMFAV, Uiversidad de Valparaíso, Gra Bretaña 09, 4to piso, Playa Acha, Valparaíso, Chile, Tel: ,
2 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data 2 Our aim is to fid the asymptotic distributio of the estimator i= φ Y x Xi ik h x = 2 i= K x Xi h for x D where the kerel K is a positive fuctio ad the sequece h > 0 is the badwidth. We use the covetio 0/0 = 0. The estimator 2 is a geeralizatio of the NadarayaWatso estimator cf. []. The asymptotic ormality of the NadarayaWatso estimator is proved i [2] whe the observatios X i, Y i are idepedet, ad for depedet observatios, i [3]. Ferraty, Goia ad Vieu [4], [5] proposed the estimator 2 to estimate the regressio fuctio whe the explaatory variable is fuctioal. For weakly depedet ad statioary observatios, they proved the complete covergece of 2 ad obtaied rates of covergece whe the fuctio φ satisfies some regularity coditios. Aspirot et al [6] have geeralized the result of complete covergece for the estimator 2 to a ostatioary case. Masry 2005 [7] has proved the asymptotic ormality of φ for statioary weakly depedet radom variables. Ferraty, Mas ad Vieu 2007 [8] cosider asymptotic ormality of the estimator φ i the case of idepedece from theoretical ad practical poit of view. Related work o desity estimatio with fuctioal data ca be foud i [9] ad [0] for mode estimatio ad i [] for asymptotic ormality of kerel estimators. Surveys o fuctioal data aalysis ca be foud i Ramsay ad Silverma 2005 [2] ad i Ferraty ad Vieu 2006 [3] for the oparametric case. Our goal i this article is to geeralize the result of [7] to the case whe X is a ostatioary mixture of statioary processes. More precisely, we suppose that there exist two radom processes ξ = {ξ : N}, Z = {Z : N} ad a fuctio ϕ : D R D such that ξ is statioary with values i D, ξ is idepedet of Z, Z takes values i a fiite set {z,...,z m } of R ad X satisfies X i = ϕξ i, Z i, i = 0,...,. 3 We suppose geeral coditios of depedece cf. assumptios i Subsectio 3., implied i particular by weakly depedece of the observatios ad of the process ξ, but the process Z may be ostatioary ad oweakly depedet. I Theorem, uder these coditios ad a set of techical assumptios, we obtai the asymptotic ormality of the estimator 2. This result is based o cetral limit theorems for fields idexed i N or Z d. I Propositio ad 2, we prove results of cetral limit theorems for triagular arrays of a R m valued cetered statioary radom field idexed i N or Z d. For these fields the existece of a cetral limit theorem depeds strogly o the geometry of the subset of Z d where the observatios are idexed. We apply these results to our model, i which we have a mixture of statioary fields X = ϕξ, Z where Z is ostatioary but X = ϕξ, Z coditioed to Z is statioary. We cosider here our observatios idexed i the level sets of Z. Some assumptios about the field Z should be made i order to esure properties of the level sets ad to have a cetral limit theorem i this case. I theorem 2, supposig additioally some regularity assumptios o the regressio fuctio φ ad o the badwidth h, we obtai the asymptotic ormality of φ x φx for x D, which allows us to costruct asymptotic cofidece iterval for φx.
3 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data 3 The ostatioary assumptio for the data appears aturally whe modellig the Iteret traffic cf. [4], [5], [6]. I this paper, we are iterested i estimatig the quality of service for voice ad video applicatios. We call quality of service parameters some etwork characteristics like delay, packet loss, etc.. The iterest i voice or video applicatios is based i the fact that these etwork traffic the same as may real time applicatios have high quality of service requiremets very low delay ad losses, etc.. I Sectio 4, we use the model 3 to estimate quality of service parameters for traffic over the Iteret. We assume that these parameters are a fuctio of the Iteret traffic, represeted by X = ϕξ, Z. I the literature, it is usual to assume that the traffic is piecewise statioary, i this model Z is a ostatioary radom process that selects betwee differet statioary behaviors. The process Z idicates for example if the etwork is very much loaded or ot ad it could describe seasoal or periodic behaviors. The process ξ is related with obtaied measuremets variatios. We give simulatios of the estimator 2 ad cofidece iterval, for simulated ad real traffic o the Iteret. The orgaizatio of the paper is as follows. I Sectio 2, we give some prelimiaries o the otios of asymptotically measurable sets ad fields ad state a cetral limit theorem for triagular arrays of R m valued cetered statioary radom fields. I Sectio 3, we give our result o the asymptotic ormality of the estimator φ ad we make the assumptios uder which we obtai this result. Sectio 4 describes the applicatio of our result to the problem of estimatig the quality of service o the Iteret. Sectio 5 is devoted to the proofs. 2 Prelimiaries I this sectio we preset some prelimiary results. I subsectio 2. we defie the otio of asymptotically measurable subsets that allows, i subsectio 2.2, to give a cetral limit theorem for triagular arrays of R m valued cetered statioary radom fields. I subsectio 2.3 we preset the otio of asymptotically measurable field that allows to state a cetral limit theorem for a field described by equatio 3. I this sectio, we cosider both the case whe the field is idexed i N ad i Z d ad i the followig, L stads for N or Z d. Whe we have L = N, we deote l = ad for a set A N, A = A {0,,..., } ad A c = A {0,,..., }. Whe we have L = Z d, we deote l = 2 + d ad for a set A Z d, A = A {,...,} d ad A c = A c {,...,} d. 2. Notio of asymptotically measurable subsets I what follows carda idicates the cardial of subset A. Defiitio. A subset A L is said to be a asymptotically measurable subset if, for all k L, the followig limit exists Fk, A = lim card{a c A k}. l The fuctio F., A is called the border fuctio of the subset A. Defiitio 2. The subset family { A i : i =,...,m } i L is a asymptotically measurable family if, for all i {,...,m}, the subsets A i are asymptotically measurable ad, for all k L ad
4 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data 4 i, j {,...,m}, the followig limit exists F k, A i, A j = lim card{a i A j k}. l 2.2 Cetral limit theorem for triagular arrays of R m valued cetered statioary radom field I the followig, the otatio = w meas covergece i law ad N d 0, Σ represets a zero mea ormal distributio i dimesio d with covariace matrix Σ. Moreover, for X = Xk k L a R m valued cetered statioary radom field, we deote the compoets of X by X = X,,...,X m, ad X k = X, k,...,xm, k for k L. Defiitio 3. We defie the class BL as the class of R m  valued cetered statioary radom fields X = X k k L that satisfies the followig coditios: H For all i, j {,...,m} ad N, we have { } E X i, 0 Xj, k <. k L H 2 Let X,J = X,,J,...,X m,,j be the trucatio by J of X, defied for k L ad J > 0 by [ ] X,J k = Xk { Xk J} E Xk { Xk J}, where represets the euclidia orm o R m. i There exists a sequece γk 0 such that k L γk < ad such that for all k L, N, i, j {,...,m} ad J > 0 we have E { X i,,j 0 X j,,j k } γk. ii There exists a sequece bj such that lim J bj = 0 ad for all set B L, i {,...,m}, N ad J > 0 where E [ S B, X i, S B, X i,,j ] 2 bjcardb, l S B, X i, = l /2 X i, k. k B H 3 There exists a sequece of reals umbers CJ such that for all B L, N, J > 0 ad i {,...,m} we have [ E S B, X i,,j ] 4 cardb 2 CJ. l
5 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data 5 H 4 There exist a fuctio h : R + R +, such that lim x + hx = 0 ad a fuctio g : R + R m R +, with gj, t < for all fixed J > 0 ad sup t R m gj, t = g J <, such that [ ] [ ] [ ] E e isb C, t,x,j E e isb, t,x,j E e isc, t,x,j gj, thdb, C, for all disjoit sets B, C L, all N, J > 0 ad t R m. Here, represets the scalar product o R m. H 5 There exist sequeces γ J i, j, k ad γi, j, k, i, j {,...,m}, k L such that for all J > 0 we have { } lim E X i,,j 0 X j,,j k = γ J i, j, k, ad for i, j {,...,m} ad k L. We have the two followig propositios proved i Sectio 5. lim J γj i, j, k = γi, j, k 4 Propositio. If X = X k k N belogs to BN, the for ay asymptotically measurable family { A i : i =,...,m } i N, we have as teds to where for i, j {,...,m} S A, X,,...,S A m, X m, w = N m 0, Σ, Σi, j = γi, j,0fi, j,0 + k {γi, j, kfi, j, k + γj, i, kfj, i, k} ad Fi, j, k = lim card{a i A j k}. Propositio 2. If X = X k k Z d belogs to BZd, the for ay asymptotically measurable family { A i : i =,...,m } i Z d, we have as teds to S A, X,,...,S A m, X m, w = N m 0, Σ, where for i, j {,...,m} ad Σi, j = k L γi, j, kfi, j, k card{a i A j k} Fi, j, k = lim 2 + d.
6 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data Notio of asymptotically measurable field I what follows a.s. meas almost sure covergece as teds to ad A is the idicator fuctio of a set A. I the case L = N, we set D = {0,..., } ad if L = Z d, we set D = {,...,} d. Defiitio 4. The R m valued radom field Z = Z L is a asymptotically measurable field i L if there exists a radom probability measure R 0 i B, where B is the Borel σalgebra i R m, such that a.s. R 0 B l m D {Zm B} ad for all k L {0} there exists a radom measure R k i B 2, where B 2 is the Borel σalgebra i R 2m, such that for all B, C B l m D {Zm B} {Zm k C} a.s. R k B C. If the limit measure is oradom Z is called a regular field i L. The followig propositio cf. Proof i [7] relates asymptotically measurable ad regular fields with asymptotically measurable subsets. Propositio 3. If Z = Z L is a asymptotically measurable field, B, B 2,...,B m are disjoit subsets of B ad A i = { L : Z B i } for i =,...,m, the, coditioally to Z the family {A,...,A m } is a asymptotically measurable family with Fk, A i, A j = R k B i, B j ad F0, A i, A j = R 0 B i δ ij, where δ i j is the Kroecker delta. Remarks: I the ext sectios we will assume further hypotheses o Z for the field described i 3 i order to prove the results, but the previous propositio is the tool for coditioig o level sets of the field Z ad applyig the results for statioary fields to the ostatioary case 3. 3 Results I the followig, we cosider a fixed x D ad we will study the asymptotic ormality of φ x give by 2. Proofs of the results are i Sectio 5. The estimator φ ca be also writte φ x = g x f x, where g x = f x = Y i K X i, ψh K X i 5 ψh
7 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data 7 with u x K u = K, u D. 6 ad ψh is give i assumptio A. We recall that we use the covetio 0/0 = 0. h I Subsectio 3., we give the assumptios uder which the asymptotic ormality of φ x is obtaied ad i Subsectio 3.2 we give our results. 3. Assumptios We make the followig assumptios o the distributio of X, Y. A There exist positive fuctios ψ, ψ,...,ψ m defied o R + D, fuctios c,...,c k defied o D ad a subset of {,...,m} such that for all h > 0 P [ ϕξ, z k x h] = c k xψ k h, x, ψ k h, x where lim h 0 ψh, x = if k, ad lim ψ k h, x h 0 ψh, x = 0 if k C. I the followig, to simplify the otatio, we replace ψh, x by ψh, but the quatity ψh still depeds of x. A 2 The fuctios u ψ k u, x are differetiable o R +, with differetial fuctio deoted ψ k u, x ad satisfy lim h 0 h ψ k h, x where the d k s are fuctios defied o D. 0 Kuψ k uh, xdu = d kx, A 3 The process Z is regular ad satisfies for k {,...,m} as teds to that ψh {Zi =z k } P PZ i = z k = 0, 7 where P = meas covergece i probability. For k {,...,m}, deote by p k the limit p k = lim PZ i = z k. Let the R 2m value radom process X = X,,..., X 2m, defied for i N as follows: for l {,...,m} X l, i = ψh K ϕξ i, z l φϕξ i, z l + ε i ψh E [K ϕξ i, z l φϕξ i, z l ],
8 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data 8 for l {m +,...,2m} X l, i = where K is defied by 6. ψh K ϕξ i, z l Moreover we assume the followig hypotheses. A 4 The R 2m value radom process X belogs to BN. A 5 The fuctio φ is cotiuous. ψh E [K ϕξ i, z l ], A 6 The estimator f x defied by 5 coverges i probability to fx > 0 where the fuctio f is defied by fu = k p k d k uc k u, u D. We suppose the followig hypothesis o the estimator φ. A 7 The fuctio K is a positive fuctio with support [0, ]. A 8 The badwidth h satisfies ad Remarks o the assumptios: lim h = 0 lim ψh =. The coditios A ad A 2 mea that, coditioally o Z, the distace betwee the observatio X i ad the estimatio poit x has a desity. The quatity ψ k h, x plays a role aalogous to h d for multidimesioal estimatio. The hypothesis A 3 meas that the variable Z does ot behave too irregularly. It is assumed that the process has some kid of statioarity i mea. However this is ot a strog assumptio, ad for example A 3 could be satisfied by periodic or semiperiodic radom variables. The relatio 7 is satisfied for example if the variables Zi =z k satisfy a cetral limit theorem for each k =,...,m. The hypothesis A 4 ca be replaced by various sets of hypothesis doig a tradeoff betwee coditios o the momets of Y ad coditios of mixig. Perera 997 [8] describes several sets of coditios that imply the previous hypotheses. I the hypothesis A 6, fx > 0 meas roughly that there is a positive probability of fidig observatios X i ear i the sese of the semiorm o D the poit x where we wat to compute the estimator. The fuctio f play the role of the desity of the whole process X = ϕξ, Z. Moreover the covergece i probability ca be obtaied uder the coditios A y A 2, coditios o the momets of Y i ad coditios of mixig cf. Proof i [6]. The hypothesis A 5, A 7 ad A 8 are classical for obtaiig asymptotic ormality.
9 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data Asymptotic ormality of φ x The asymptotic ormality of φ x is obtaied i three stages: asymptotic ormality of the field X i Propositio 4, of g x Eg x, f x Ef x i Propositio 5 ad the of φ x i Theorem. Theorem 2 gives the asymptotic ormality of φ x φx which allows to costruct asymptotic cofidece iterval for φx. Propositio 4. For l {,...,m}, deote A l = {k N, Z k = z l } ad A l+m = A l. If Z is asymptotically measurable, the uder Assumptio A 4, coditioally o Z, we have, as teds to S A, X,,..., S A 2m, X 2m, w = N 2m 0, Σ, where for i, j {,...,2m} Σi, j = γi, j,0fi, j,0 + k N {γi, j, kfi, j, k + γj, i, kfj, i, k}, with γi, j, k, k N, defied by 4 for the field X ad Fi, j, k = lim card{a i A j k}. Propositio 5. Uder Assumptios A 3 ad A 4, we have, as teds to, that w ψh g x Eg x, f x Ef x = N 2 0, A, where with a x = A = Σi, j, a 2 x = i,j= a x a 2 x a 2 x a 3 x 2 i= j=m+ Σi, j, a 3 x = 2 i,j=m+ Σi, j. Theorem. Uder Assumptios A A 8, we have as teds to ψh φ x Eg x w = N 0, σ 2 x, Ef x where Theorem 2. We suppose that B the fuctio φ satisfies with β > 0 ad L a positive costat, σ 2 x = a x 2a 2 xφx + a 3 xφ 2 x f 2. 8 x φu φv L u v β,
10 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data 0 B 2 there exist M > 0 ad a fuctio α such that for each k {,...,m} ad h > 0, we have P [ ϕξ, z k x h] c k x ψ k h, x Mαh α > 0. B 3 the badwidth h satisfies ad lim αh ψh = 0 lim hβ ψh = 0 B 4 for each k {,...,m}, we have lim ψh PZ i = z k p k = 0. The uder Assumptios A A 8, B B 4, we have as teds to, where σ 2 x is defied by 8. ψh φ x φx w = N 0, σ 2 x, 4 Applicatio to edtoed quality of service estimatio 4. Motivatio ad model Nowadays ew services are offered over the Iteret like voice ad video. For these ew applicatios the eed to measure the etwork performace has icreased. Multimedia applicatios have several requiremets i terms of delay, losses ad other quality of service parameters. These costraits are stroger tha the oes for usual data trasfer applicatios mail, etc.. Measuremets of the Iteret performace are ecessary for differet reasos, for example to advace i uderstadig the behavior of the Iteret or to verify the quality of service assured to the ew services. There are several measurig techiques, most of them with software implemetatios, that ca be classified i active sedig cotrolled traffic called probe packets ad passive measures geerally measures at the routers. The aim of these techiques is to detect differet characteristics of the etwork that ca be the topology or some performace parameters delay, losses, available badwidth, etc.. There are also several problems to measure the performace parameters, for example the route of the packets ca chage, the traffic bit rate is ot costat ad ormally is i bursts, the probe packets ca be filtered or altered by oe ISP Iteret Service Provider i the path, there is ot clock sychroizatio betwee routers ad ed equipmets, etc.. Normally the iteral routers i the path betwee two poits of iterest are ot uder the cotrol of oly oe user or oe ISP. Therefore, it is ot very useful to have measurig procedures that deped o the iformatio of the iteral routers. For this reaso edtoed measures is oe of the most developed methodologies durig the last years cf. [9], [20],[2]. We cosider a sigle lik ad a voice or video traffic that we wat to moitor, that is we wat to kow the quality of service parameters whe this traffic goes betwee two poits over
11 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data the etwork. The lik is shared by this multimedia traffic ad may other traffics i the etwork that are ukow ad that is called cross traffic ad could be modelled as a stochastic process. The performace parameter for packets of a multimedia traffic could be delay, packets losses, delay variatio or all of them together ad we will cosider it as a radom variable Y. This performace parameter could be modeled as a fuctio of the cross traffic stochastic process, the video or voice stochastic process, the lik capacity ad the buffer size. We wat to estimate Y without sedig video or voice traffic durig log time periods. Active measuremets, that sed traffic, are useful if they do ot charge the etwork, so they cosist o small probe packets set durig short time periods. The lik capacity ad the buffer size are ot kow but it is assumed that they are costats durig the moitorig process. The multimedia traffic is also kow therefore we ca suppose that Y = φx where X is a characterizatio of the cross traffic stochastic process. We ca ot measure the cross traffic process but we ca obtai data closely related with this process. This data is described i the followig subsectio. The cross traffic process o the Iteret is a depedet ostatioary process c.f [4] or [5]. To take ito accout ostatioary behaviors we will suppose that X satisfies the relatio Measuremet procedure I what follows we describe the procedure to estimate the fuctio φ. We divide the experimet i two phases: first, we sed a burst of small probe packets pp of fixed size spaced by a fixed time. Immediately after the burst we sed durig a short time a video stream. We repeat the previous procedure durig some time lapse, sedig a ew burst ad a video probe after a time iterval measured from the previous ed of the video stream as it is show i the scheme i figure. pp video Figure : Measuremet procedure scheme With the probe packets burst we ifer the cross traffic of the lik. We measure at the output of the lik the iterarrival time betwee cosecutive probe packets. This time serie is strogly correlated with the cross traffic process that shares the lik with the probe traffic. We compute the empirical distributio fuctio of probe packets iterarrival times because this distributio is related with the queue behavior i the etwork. Usig this cross traffic estimatio ad measurig the delay Y over the video, we wat to estimate the fuctio φ. From each probe packet burst ad video sequece j we have a pair X j, Y j, where X j is the empirical distributio fuctio of iterarrival times ad Y j is the performace metric of iterest measured from the video stream j. Therefore, our estimatio problem has bee trasformed ito the problem of iferrig a fuctio φ : D R where D is the space of the probability distributio fuctios ad R is the real lie such that the X j, Y j satisfy.
12 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data 2 We did the estimatios both with simulated ad real traffic. Real data are obtaied with a software that do the active measures described before betwee a server ad its users. For simulated data, the variable Y is the oeway delay which is the delay betwee two poits. For real data, we measure roud trip time RTT delay that is the delay i the whole trip betwee two poits i a path ad i the reverse path. For simulated traffic, we have 360 observatios X j, Y j. For each j {,...,359}, we calculate the estimator φ j defied by 2 usig the observatios X, Y,...,X j, Y j ad we represet o figures 2 ad 3 the values of Y j+ ad of its predictio φ j X j+. Moreover for each j {,...,359}, we calculate a cofidece iterval aroud φ j X j+ usig a blockbootstrap. The estimator φ j is calculated usig the kerel Kx = x 2 2 [,] x, the badwidth that miimizes the sum of the relative error i the estimatio before the time j ad the L orm. The figure 4 represets the relative error of the estimatio. The results are relatively accurate for sample of size larger tha 00. For real data, we have 50 observatios X j, Y j ad we realize the same type of estimatio. I figure 5, we represets for each j {,...,49}, the values of Y j+ ad of its predictio φ j X j+ ad a cofidece a cofidece bad aroud φ j X j+ usig a blockbootstrap. The figure 6 represets the relative error of the estimatio. The results are less accurate tha i the case of simulated data but the estimatios are calculated with samples of small size Delayms Sample umber Figure 2: Estimated delays for all simulated data
13 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data Predictio delay Measured delay Upper boud cofidece Lower boud cofidece Delayms Sample umber Figure 3: Estimated delays for simulated data observatios betwee 230 ad Relative error Sample umber Figure 4: Relative error for simulated data
14 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data Predictio delay Measured delay Upper boud cofidece Lower boud cofidece 200 Delayms Sample umber Figure 5: Estimated RTT for real data Relative error Sample umber Figure 6: Relative error for real data
15 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data 5 5 Proof of the propositios ad theorems 5. Proof of Propositios ad 2 This result is obtaied by meas of Bershtei method cf. [8], [22] ad it is a geeralizatio of the theorem 4.5 o triagular arrays for Rvalued radom fields obtaied by Tablar 2006 [23], p 86 to R m valued radom fields. To prove that the vectorial field S A, X,,..., S A m, X m, w = N m 0, Σ, it is sufficiet to prove that for all λ R m w λ, S = N0, λ t Σλ where S = S A, X,,..., S A m, X m, ad λ t = λ,...,λ m. We have that S, λ = = λ i S A i, X i, = i= l /2 k D i= i= l /2λ i λ i X i, k {k A i } Let us cosider a real valued field X λ where X i, k λ = λ ix i, k if k A i. The S, λ = S D, X λ. As the field X λ verifies the hypotheses i the work of [23] we ca apply the theorem for a real valued radom field i order to obtai the result of propositios ad Proof of Propositio 4 Sice Z is asymptotically measurable, coditioally to Z, the subset family A,...,A 2m is a asymptotically measurable family i N. The applyig Propositio to X which belogs to BN, we deduce the result. 5.3 Proof of Propositio 5 Before provig Propositio 5, here we prove the followig lema. Lemma. Let k {,...,m}. Uder the Assumptios A, A 2, A 3, A 5, A 7 ad A 8, we have the followig limit lim k A i X i, k ψh E [K ϕξ, z k ] = d k xc k x {k }, 2 for β > 0, lim ψh E [ ϕξ, z k x h β K ϕξ, z k ] d k xc k x, 3 that the estimator f x satisfies lim Ef x = fx,
16 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data 6 4 ad the estimator g x satisfies lim Eg x = φxfx. Proof. The results ad 2 of this lemma come from the fact that uder Assumptio A, the desity of ϕξ, z k x is the fuctio u c k xψ k u, x. The uder Assumptio A 7 ad [ ϕξ, z k x E h E [K ϕξ, z k ] = h c k x β K ϕξ, z k ] 0 = h c k x Kuψ k uh, xdu ad it is sufficiet to use Assumptios A 2 ad A 8 to coclude. Here we prove 3 ad 4. We have for i {,...,}, EK X i = E{EK X i Z i } = 0 Kuu β ψ k uh, xdu E{K ϕξ i, z k }PZ i = z k. k= As ϕξ i, z k i=,..., is idetically distributed, we obtai that, Ef x = ψh E K ϕξ, z k PZ i = z k. 9 k= Now applyig A 3 ad the assertio of this lemma i 9, we obtai 3. Similar calculus give that Eg x = ψh E φϕξ, z k K ϕξ, z k PZ i = z k. The we have where R = k= Eg x = φxef x + R, ψh E {φϕξ, z k φx} K ϕξ, z k PZ i = z k k= sup u: x u h φu φx Ef x ad we obtai 4 usig the cotiuity of fuctio φ Assumptio A 5 ad usig that lim Ef x = fx. Coditioally to Z, we have ψh g x Eg x Z, f x Ef x Z = S A j, X,j, j= 2 j=m+ S A j, X,j.
17 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data 7 Usig Propositio 4, we obtai that, coditioally to Z, S A j, X,j, 2 S A j, X,j = w N 2 0, A. j= j=m+ Sice Z is regular, the matrix A is ot radom ad the we have w ψh g x Eg x Z, f x Ef x Z = N 2 0, A. Now g x Eg x, f x Ef x = g x Eg x Z, f x Ef x Z We have Ef x Z Ef x = 2 j= + Eg x Z Eg x, Ef x Z Ef x. EK ξ, z j ψh carda j P[Z i = j] Uder Assumptio A 3 ad usig assertio of Lemma, we have that ψh Ef x Z Ef x. coverges i probability to 0 as teds to. Usig Assumptio A 3, ψh Eg x Z Eg x coverges i probability to 0 as teds to. Usig the lemma of Slutsky, we obtai the result of the propositio. 5.4 Proof of Theorem We have where ad φ x Eg x Ef x = Q x B xf x E f x, f x B x = Eg x Ef x φx Q x = g x Eg x φxf x Ef x. Usig 3 y 4 of Lemma, we deduce that lim B x = 0. The usig A 6, we have that f x coverges i probability to fx > 0 ad the it implies that B xf x E f x 0 f x coverges i probability to 0. Fially, usig A 6, 0 ad Propositio 5, applyig Slutsky Lemma, we deduce the result of the propositio.
18 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data Proof of Theorem 2 Theorem 2 is obtaied usig Theorem ad the followig lemma. Lemma 2. Uder Assumptios B B 4 ad Assumptios A, A 2, A 6, A 7 ad A 8, we have that Eg x lim ψh Ef x φx = 0. Proof. We have the followig decompositio Eg x Ef x φx = Ef x [Eg x fxφx] + φx Ef x [fx Ef x]. Here we study first the quatity Ef x fx. As i Lemma, we have Ef x = ψh E K ϕξ, z k PZ i = z k ad the Ef x fx = + k= k= ψh E K ϕξ, z k d k xc k x {k } PZ i = z k d k xc k x {k } PZ i = z k p k. k= Uder the Assumptios B 2, B 3 ad B 4 imply that ψh Ef x fx = 0. 2 lim Here we study the term Eg x fxφx. It satisfies Eg x fxφx φx Ef x fx + Eg x Ef xφx. We have Eg x Ef xφx = E [K ϕξ, z k φϕξ, z k φx] PZ i = z k ψh k= [ Lh β ψh E K ϕξ, z k ϕξ, z k x ] β h PZ i = z k k= L h β + o, 3 where the secod lie comes from Assumptio B ad the third is a cosequece of assertio 2 of Lemma. From 2, 3 ad Assumptio B 3, we deduce that ψh Eg x fxφx = 0. 4 lim Assertio of Lemma, 2 ad 4 applyig i imply the propositio.
19 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data 9 6 Ackowledgemet This work has bee supported by the ARTES group. Karie Berti has bee supported by Project DIPUV 33/05, Project FONDECYT 0684 ad the Laboratory ANESTOC ACT 3. Laura Aspirot has bee partially supported by PDT Programa de Desarrollo Tecológico S/C/OP/46/. The authors would like to thak Yoaa Dooso for her cotributios to the simulatios programs. Refereces [] Nadaraya, E. A., 989, Noparametric estimatio of probability desities ad regressio curves, Mathematics ad its Applicatios Soviet Series, 20. Dordrecht: Kluwer Academic Publishers Group. [2] Roseblatt, M., 969, Coditioal probability desity ad regressio estimators. Multivariate Aalysis, II Proc. Secod Iterat. Sympos., Dayto, Ohio, 968, New York: Academic Press. [3] Robiso, P. M., 983, Noparametric estimators for time series. J. Time Ser. Aal., 4, [4] Ferraty, F., Goia, A. ad Vieu, P., 2002, Fuctioal oparametric model for time series: a fractal approach for dimesio reductio. Test,, [5] Ferraty, F. ad Vieu, P., 2004, Noparametric models for fuctioal data, with applicatio i regressio, timeseries predictio ad curve discrimiatio. J. Noparametr. Stat., 6, 25. [6] Aspirot, L., Belzarea, P., Bazzao, B. ad Perera, G., 2005, EdToEd Quality of Service Predictio Based O Fuctioal Regressio. Proc. Third Iteratioal Workig Coferece o Performace Modellig ad Evaluatio of Heterogeeous Networks HETNETs 2005, Ilkley, UK, July [7] Masry, E., 2005, Noparametric regressio estimatio for depedet fuctioal data: asymptotic ormality. Stochastic Process. Appl., 5, [8] Ferraty, F., Mas, A. ad Vieu, P., 2007, Noparametric regressio o fuctioal data: iferece ad pratical aspects. Australia ad New Zealad Joural of Statistics,. [9] Gasser, T., Hall, P. ad Presell, B., 998, Noparametric estimatio of the mode of a distributio of radom curves. J. R. Stat. Soc. Ser. B Stat. Methodol., 60, [0] DaboNiag, S., Ferraty, F., ad Vieu, P., 2004, Estimatio du mode das u espace vectoriel semiormé. C. R. Math. Acad. Sci. Paris, 339, [] Oliveira, P., 2005, Noparametric desity ad regressio estimatio fuctioal data. Pré Publicaçoes do Departameto de Matemática, Uiversidade de Coimbra, [2] Ramsay, J. O. ad Silverma, B. W., 2005, Fuctioal data aalysis 2d ed, Spriger Series i Statistics. New York: Spriger.
20 Asymptotic ormality of the NadarayaWatso estimator for ostatioary data 20 [3] Ferraty, F. ad Vieu, P., 2006, Noparametric Fuctioal data aalysis: Theory ad Practice, Spriger Series i Statistics. New York: Spriger. [4] Karagiais, T., Molle, M., Faloutsos, M. ad Broido, A., 2004, A ostatioary Poisso view of Iteret traffic. Proc. of INFOCOM Twetythird Aual Joit Coferece of the IEEE Computer ad Commuicatios Societies, 3, [5] Zhag, Y., Paxso, V. ad Sheker, S., 2000, The Statioarity of Iteret Path Properties: Routig, Loss, ad Throughput. ACIRI Techical Report. [6] Zhag, Y. ad Duffield, N., 200, O the costacy of iteret path properties. IMW 0: Proceedigs of the st ACM SIGCOMM Workshop o Iteret Measuremet, [7] Perera, G., 2002, Irregular sets ad cetral limit theorems. Beroulli, 8, [8] Perera, G., 997, Geometry of Z d ad the cetral limit theorem for weakly depedet radom fields. J. Theoret. Probab., 0, [9] Bolot, J., 993, Edtoed packet delay ad loss behavior i the iteret. SIGCOMM 93: Coferece proceedigs o Commuicatios architectures, protocols ad applicatios, [20] Jai, M. ad Dovrolis, C., 2002, Pathload: A measuremet tool for edtoed available badwidth. Proc. of Passive ad Active Measuremets PAM Workshop, Mar [2] Corral, J., Texier, G. ad Toutai, L., 2003, Edtoed Active Measuremet Architecture i IP Networks SATURNE. PAM 2003, A workshop o Passive ad Active Measuremets. [22] Perera, G., 997, Applicatios of cetral limit theorems over asymptotically measurable sets: regressio models. C. R. Acad. Sci. Paris Sér. I Math., 324, [23] Tablar, A., Estadística de procesos estocásticos y Aplicacioes a Modelos Epidémicos, PhD tesis, submitted to Departameto de Graduados de la Uiversidad Nacioal del Sur for evaluatio, december 2006.
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