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


 Dwayne Lynch
 1 years ago
 Views:
Transcription
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.
Properties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationI. Chisquared Distributions
1 M 358K Supplemet to Chapter 23: CHISQUARED DISTRIBUTIONS, TDISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad tdistributios, we first eed to look at aother family of distributios, the chisquared distributios.
More informationOutput Analysis (2, Chapters 10 &11 Law)
B. Maddah ENMG 6 Simulatio 05/0/07 Output Aalysis (, Chapters 10 &11 Law) Comparig alterative system cofiguratio Sice the output of a simulatio is radom, the comparig differet systems via simulatio should
More information3. Covariance and Correlation
Virtual Laboratories > 3. Expected Value > 1 2 3 4 5 6 3. Covariace ad Correlatio Recall that by takig the expected value of various trasformatios of a radom variable, we ca measure may iterestig characteristics
More informationChapter 7 Methods of Finding Estimators
Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of
More information7. Sample Covariance and Correlation
1 of 8 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 7. Sample Covariace ad Correlatio The Bivariate Model Suppose agai that we have a basic radom experimet, ad that X ad Y
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationAn example of nonquenched convergence in the conditional central limit theorem for partial sums of a linear process
A example of oqueched covergece i the coditioal cetral limit theorem for partial sums of a liear process Dalibor Volý ad Michael Woodroofe Abstract A causal liear processes X,X 0,X is costructed for which
More informationCase Study. Normal and t Distributions. Density Plot. Normal Distributions
Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca
More informationChapter 7  Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:
Chapter 7  Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries
More informationNPTEL STRUCTURAL RELIABILITY
NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationCharacterizing EndtoEnd Packet Delay and Loss in the Internet
Characterizig EdtoEd Packet Delay ad Loss i the Iteret JeaChrysostome Bolot Xiyu Sog Preseted by Swaroop Sigh Layout Itroductio Data Collectio Data Aalysis Strategy Aalysis of packet delay Aalysis of
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationOur aim is to show that under reasonable assumptions a given 2πperiodic function f can be represented as convergent series
8 Fourier Series Our aim is to show that uder reasoable assumptios a give periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationHypothesis testing. Null and alternative hypotheses
Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More information5: Introduction to Estimation
5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample
More informationModified Line Search Method for Global Optimization
Modified Lie Search Method for Global Optimizatio Cria Grosa ad Ajith Abraham Ceter of Excellece for Quatifiable Quality of Service Norwegia Uiversity of Sciece ad Techology Trodheim, Norway {cria, ajith}@q2s.tu.o
More information0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5
Sectio 13 KolmogorovSmirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More informationNonlife insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring
Nolife isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy
More informationChapter 6: Variance, the law of large numbers and the MonteCarlo method
Chapter 6: Variace, the law of large umbers ad the MoteCarlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationFIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
More informationIrreducible polynomials with consecutive zero coefficients
Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem
More informationLECTURE 13: Crossvalidation
LECTURE 3: Crossvalidatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Threeway data partitioi Itroductio to Patter Aalysis Ricardo GutierrezOsua Texas A&M
More informationOverview of some probability distributions.
Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability
More informationClass Meeting # 16: The Fourier Transform on R n
MATH 18.152 COUSE NOTES  CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationPROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM
PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUSMALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics
More informationPSYCHOLOGICAL STATISTICS
UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics
More informationJoint Probability Distributions and Random Samples
STAT5 Sprig 204 Lecture Notes Chapter 5 February, 204 Joit Probability Distributios ad Radom Samples 5. Joitly Distributed Radom Variables Chapter Overview Joitly distributed rv Joit mass fuctio, margial
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the BruMikowski iequality for boxes. Today we ll go over the
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationPlugin martingales for testing exchangeability online
Plugi martigales for testig exchageability olie Valetia Fedorova, Alex Gammerma, Ilia Nouretdiov, ad Vladimir Vovk Computer Learig Research Cetre Royal Holloway, Uiversity of Lodo, UK {valetia,ilia,alex,vovk}@cs.rhul.ac.uk
More informationAnnuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.
Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio  Israel Istitute of Techology, 3000, Haifa, Israel I memory
More informationARTICLE IN PRESS. Statistics & Probability Letters ( ) A Kolmogorovtype test for monotonicity of regression. Cecile Durot
STAPRO 66 pp:  col.fig.: il ED: MG PROD. TYPE: COM PAGN: Usha.N  SCAN: il Statistics & Probability Letters 2 2 2 2 Abstract A Kolmogorovtype test for mootoicity of regressio Cecile Durot Laboratoire
More informationZTEST / ZSTATISTIC: used to test hypotheses about. µ when the population standard deviation is unknown
ZTEST / ZSTATISTIC: used to test hypotheses about µ whe the populatio stadard deviatio is kow ad populatio distributio is ormal or sample size is large TTEST / TSTATISTIC: used to test hypotheses about
More informationKey Ideas Section 81: Overview hypothesis testing Hypothesis Hypothesis Test Section 82: Basics of Hypothesis Testing Null Hypothesis
Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, Pvalue Type I Error, Type II Error, Sigificace Level, Power Sectio 81: Overview Cofidece Itervals (Chapter 7) are
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS200609 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationApproximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find
1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationNotes on Hypothesis Testing
Probability & Statistics Grishpa Notes o Hypothesis Testig A radom sample X = X 1,..., X is observed, with joit pmf/pdf f θ x 1,..., x. The values x = x 1,..., x of X lie i some sample space X. The parameter
More informationProbabilistic Engineering Mechanics. Do Rosenblatt and Nataf isoprobabilistic transformations really differ?
Probabilistic Egieerig Mechaics 4 (009) 577 584 Cotets lists available at ScieceDirect Probabilistic Egieerig Mechaics joural homepage: wwwelseviercom/locate/probegmech Do Roseblatt ad Nataf isoprobabilistic
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationChapter 14 Nonparametric Statistics
Chapter 14 Noparametric Statistics A.K.A. distributiofree statistics! Does ot deped o the populatio fittig ay particular type of distributio (e.g, ormal). Sice these methods make fewer assumptios, they
More informationLIMIT DISTRIBUTION FOR THE WEIGHTED RANK CORRELATION COEFFICIENT, r W
REVSTAT Statistical Joural Volume 4, Number 3, November 2006, 189 200 LIMIT DISTRIBUTION FOR THE WEIGHTED RANK CORRELATION COEFFICIENT, r W Authors: Joaquim F. Pito da Costa Dep. de Matemática Aplicada,
More informationTHE ABRACADABRA PROBLEM
THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected
More information1. C. The formula for the confidence interval for a population mean is: x t, which was
s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : pvalue
More informationUniversity of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chisquare (χ ) distributio.
More informationCenter, Spread, and Shape in Inference: Claims, Caveats, and Insights
Ceter, Spread, ad Shape i Iferece: Claims, Caveats, ad Isights Dr. Nacy Pfeig (Uiversity of Pittsburgh) AMATYC November 2008 Prelimiary Activities 1. I would like to produce a iterval estimate for the
More informationA Faster ClauseShortening Algorithm for SAT with No Restriction on Clause Length
Joural o Satisfiability, Boolea Modelig ad Computatio 1 2005) 4960 A Faster ClauseShorteig Algorithm for SAT with No Restrictio o Clause Legth Evgey Datsi Alexader Wolpert Departmet of Computer Sciece
More informationTHE HEIGHT OF qbinary SEARCH TREES
THE HEIGHT OF qbinary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average
More informationGregory Carey, 1998 Linear Transformations & Composites  1. Linear Transformations and Linear Composites
Gregory Carey, 1998 Liear Trasformatios & Composites  1 Liear Trasformatios ad Liear Composites I Liear Trasformatios of Variables Meas ad Stadard Deviatios of Liear Trasformatios A liear trasformatio
More informationInference on Proportion. Chapter 8 Tests of Statistical Hypotheses. Sampling Distribution of Sample Proportion. Confidence Interval
Chapter 8 Tests of Statistical Hypotheses 8. Tests about Proportios HT  Iferece o Proportio Parameter: Populatio Proportio p (or π) (Percetage of people has o health isurace) x Statistic: Sample Proportio
More information1 Introduction to reducing variance in Monte Carlo simulations
Copyright c 007 by Karl Sigma 1 Itroductio to reducig variace i Mote Carlo simulatios 11 Review of cofidece itervals for estimatig a mea I statistics, we estimate a uow mea µ = E(X) of a distributio by
More informationEstimating the Mean and Variance of a Normal Distribution
Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers
More informationVladimir N. Burkov, Dmitri A. Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT
Keywords: project maagemet, resource allocatio, etwork plaig Vladimir N Burkov, Dmitri A Novikov MODELS AND METHODS OF MULTIPROJECTS MANAGEMENT The paper deals with the problems of resource allocatio betwee
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationLecture 10: Hypothesis testing and confidence intervals
Eco 514: Probability ad Statistics Lecture 10: Hypothesis testig ad cofidece itervals Types of reasoig Deductive reasoig: Start with statemets that are assumed to be true ad use rules of logic to esure
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationUniversal coding for classes of sources
Coexios module: m46228 Uiversal codig for classes of sources Dever Greee This work is produced by The Coexios Project ad licesed uder the Creative Commos Attributio Licese We have discussed several parametric
More informationThe Euler Totient, the Möbius and the Divisor Functions
The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship
More informationTHE TWOVARIABLE LINEAR REGRESSION MODEL
THE TWOVARIABLE LINEAR REGRESSION MODEL Herma J. Bieres Pesylvaia State Uiversity April 30, 202. Itroductio Suppose you are a ecoomics or busiess maor i a college close to the beach i the souther part
More informationarxiv:1506.03481v1 [stat.me] 10 Jun 2015
BEHAVIOUR OF ABC FOR BIG DATA By Wetao Li ad Paul Fearhead Lacaster Uiversity arxiv:1506.03481v1 [stat.me] 10 Ju 2015 May statistical applicatios ivolve models that it is difficult to evaluate the likelihood,
More informationResearch Article Sign Data Derivative Recovery
Iteratioal Scholarly Research Network ISRN Applied Mathematics Volume 0, Article ID 63070, 7 pages doi:0.540/0/63070 Research Article Sig Data Derivative Recovery L. M. Housto, G. A. Glass, ad A. D. Dymikov
More informationDistributions of Order Statistics
Chapter 2 Distributios of Order Statistics We give some importat formulae for distributios of order statistics. For example, where F k: (x)=p{x k, x} = I F(x) (k, k + 1), I x (a,b)= 1 x t a 1 (1 t) b 1
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationApproximating the Sum of a Convergent Series
Approximatig the Sum of a Coverget Series Larry Riddle Ages Scott College Decatur, GA 30030 lriddle@agesscott.edu The BC Calculus Course Descriptio metios how techology ca be used to explore covergece
More informationDiscrete Random Variables and Probability Distributions. Random Variables. Chapter 3 3.1
UCLA STAT A Applied Probability & Statistics for Egieers Istructor: Ivo Diov, Asst. Prof. I Statistics ad Neurology Teachig Assistat: Neda Farziia, UCLA Statistics Uiversity of Califoria, Los Ageles, Sprig
More informationA CUSUM TEST OF COMMON TRENDS IN LARGE HETEROGENEOUS PANELS
A CUSUM TEST OF COMMON TRENDS IN LARGE HETEROGENEOUS PANELS JAVIER HIDALGO AND JUNGYOON LEE A. This paper examies a oparametric CUSUMtype test for commo treds i large pael data sets with idividual fixed
More informationAMS 2000 subject classification. Primary 62G08, 62G20; secondary 62G99
VARIABLE SELECTION IN NONPARAMETRIC ADDITIVE MODELS Jia Huag 1, Joel L. Horowitz 2 ad Fegrog Wei 3 1 Uiversity of Iowa, 2 Northwester Uiversity ad 3 Uiversity of West Georgia Abstract We cosider a oparametric
More informationSequences II. Chapter 3. 3.1 Convergent Sequences
Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,
More informationTHIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E. MCCARTHY, SANDRA POTT, AND BRETT D. WICK
THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E MCCARTHY, SANDRA POTT, AND BRETT D WICK Abstract We provide a ew proof of Volberg s Theorem characterizig thi iterpolatig sequeces as those for
More informationBASIC STATISTICS. f(x 1,x 2,..., x n )=f(x 1 )f(x 2 ) f(x n )= f(x i ) (1)
BASIC STATISTICS. SAMPLES, RANDOM SAMPLING AND SAMPLE STATISTICS.. Radom Sample. The radom variables X,X 2,..., X are called a radom sample of size from the populatio f(x if X,X 2,..., X are mutually idepedet
More informationx(x 1)(x 2)... (x k + 1) = [x] k n+m 1
1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,
More informationTaking DCOP to the Real World: Efficient Complete Solutions for Distributed MultiEvent Scheduling
Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed MultiEvet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) types of data scatter plots measure of directio measure of stregth Computatio covariatio of X ad Y uique variatio i X ad Y measurig
More informationON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE
Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE by Maria E Gageoea ad Silvia Moldoveau
More informationLecture 4: Cauchy sequences, BolzanoWeierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, BolzaoWeierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationSystems Design Project: Indoor Location of Wireless Devices
Systems Desig Project: Idoor Locatio of Wireless Devices Prepared By: Bria Murphy Seior Systems Sciece ad Egieerig Washigto Uiversity i St. Louis Phoe: (805) 6985295 Email: bcm1@cec.wustl.edu Supervised
More informationAQA STATISTICS 1 REVISION NOTES
AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if
More informationConfidence Interval Estimation of the Shape. Parameter of Pareto Distribution. Using Extreme Order Statistics
Applied Mathematical Scieces, Vol 6, 0, o 93, 4674640 Cofidece Iterval Estimatio of the Shape Parameter of Pareto Distributio Usig Extreme Order Statistics Aissa Omar, Kamarulzama Ibrahim ad Ahmad Mahir
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationConfidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.
Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).
More informationOn the Capacity of Hybrid Wireless Networks
O the Capacity of Hybrid ireless Networks Beyua Liu,ZheLiu +,DoTowsley Departmet of Computer Sciece Uiversity of Massachusetts Amherst, MA 0002 + IBM T.J. atso Research Ceter P.O. Box 704 Yorktow Heights,
More informationChapter 9: Correlation and Regression: Solutions
Chapter 9: Correlatio ad Regressio: Solutios 9.1 Correlatio I this sectio, we aim to aswer the questio: Is there a relatioship betwee A ad B? Is there a relatioship betwee the umber of emploee traiig hours
More informationInstitute of Actuaries of India Subject CT1 Financial Mathematics
Istitute of Actuaries of Idia Subject CT1 Fiacial Mathematics For 2014 Examiatios Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig i
More informationA note on weak convergence of the sequential multivariate empirical process under strong mixing SFB 823. Discussion Paper.
SFB 823 A ote o weak covergece of the sequetial multivariate empirical process uder strog mixig Discussio Paper Axel Bücher Nr. 17/2013 A ote o weak covergece of the sequetial multivariate empirical process
More informationCentral Limit Theorem and Its Applications to Baseball
Cetral Limit Theorem ad Its Applicatios to Baseball by Nicole Aderso A project submitted to the Departmet of Mathematical Scieces i coformity with the requiremets for Math 4301 (Hoours Semiar) Lakehead
More informationStudy on the application of the software phaselocked loop in tracking and filtering of pulse signal
Advaced Sciece ad Techology Letters, pp.3135 http://dx.doi.org/10.14257/astl.2014.78.06 Study o the applicatio of the software phaselocked loop i trackig ad filterig of pulse sigal Sog Wei Xia 1 (College
More information