1 Introduction. Bertrand Maillot. Abstract

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1 Conditional density estimation for continuous time processes with values in functional spaces, with applications to the conditional mode and the regression function estimators Bertrand Maillot Abstract In this chapter two continuous time processes are considered : a vectorial valued one explained by a process valued in a semi-metric vectorial space. We rst obtain the uniform convergence of the conditional density estimate from which we deduce the almost sure convergence of the conditional mode estimator. As corollaries, we have the convergence of the conditional distribution function and the regression function estimators. 1 Introduction Since the pioneer work of Parzen 1962, many articles have been dealing with the density estimation from which they have deduced the mode estimation for real or vectorial variables see e.g. Cherno 1964, Van Ryzin 1969 and Samanta Later, in view of ecient prediction, there has been an important interest for the conditional mode estimation with vectorial explanatory variables see Collomb et al. 1987, Berlinet et al and Louani et Ould-Saïd 1999 and, more recently, convergence of the conditional mode estimator with a functional covariable has been obtained Ferraty et al. 25, Ferraty et al. 26, Dabo-Niang et Laksaci 27 and Ezzahrioui 28. But, to our knowledge, the conditional mode has never been estimated for continuous time processes and all the results with functional explicative variables pose that the variable of interest is real valued. However, it is obvious that, if a result for a real valued response directly implies the same result for a vectorial response when we study the regression, it is not the case with the conditional bertrand.maillot@upmc.fr 1

2 mode estimation. In this chapter, we consider a vectorial process of interest and a covariable taking values in a metric vectorial space, both observed at continuous time. It's easily seen that many physical phenomenons can be seen as functional variables that are observable at continuous time, as seismic waves or Earth's magnetic eld. We rst establish the uniform almost sure convergence of the conditional density estimator from which we deduce the convergence of the conditional mode estimator and then obtain as corollaries the almost sure convergence of the conditional repartition function estimator and that of the regression function estimator for real bounded variables. Let {X t, Y t } t R + be a mixing-process see Bradley 25 for a presentation of mixing coecients dened on a probability space Ω, F, P and observed for t [, T ], where Y t is vectorial valued and X t takes values in a semi-metric vectorial space H equipped with the semi-metric d.,.. The law of X t, Y t is posed not to depend on t. Let C and S be respectively compact sets of H and R d, and assume that for any x C and any y S the conditional density of Y knowing X = x applied in y, denoted by fy x, and the marginal density fy of Y exist. In the following, we use two positive kernels H. and K. with port [, 1] and two bandwidths h H,T and h K,T. The conditional density function estimator is given by f T y x = T t= H H,T Kh 1 K,T dx, X tdt T t= Kh 1 K,T dx, X tdt y Y t R dh 1 h d H,T when the denominator is not null, otherwise, we take T f H y Y t= t R dh 1 H,T dt T y x = T h d H,T where. R d stands for the Euclidean norm on R d. 2 Hypotheses and results We need at rst to introduce some notations. For x H and h >, we denote by Bx, h the ball of center x and radius h. In the following, for a real valued variable 1 Z, we set Z p := E Z p p and Z := x R P Z > x >. We rst present the hypotheses about the distribution of the processes. D.1 a There exist c 1 >, c 2 >, C <, β >, β > such that for any x C, y S, x 1, x 2 Bx, c 1 2 and y 1, y 2 By, c 2 2

3 fy 1 x 1 fy 2 x 2 C dx 1, x 2 β + dy 1, y 2 β. b There exist two positive constants f and δ such that for any s, t R 2 +with s t > δ, the conditional density f t,s.,..,. of the couple Y t, Y s knowing X t, X s exists and maxfy 1 x 1, f t,s y 1, y 2 x 1, x 2 f. c There exist a function φ and two constants β 1, β 2 R 2 + such that, for any x C and any h ], c 1 ], < β 1 φh P X Bx, h β 2 φh. d β 3 >, s, t R 2 + with s t > δ, x C P X s Bx, h, X t Bx, h β 3 φh 2 h ], c 1 ]. e The process X t, Y t t is α-mixing, and αu cu a where c >, a > 3. Remark : The bound imposed to the conditional density of Y knowing X in condition D.1b was already provided by the Hölderian hypothesis D.1a. We write it again in hypothesis D.1b in order to unify the notation for the bound. Hypotheses Dc d are very classic in non-parametric statistics dealing with functional data since the pioneer work of Ferraty et Vieu 2. Finally, we introduce in hypothesis D.1e an α-mixing condition, which is less restrictive than β-mixing or φ-mixing conditions see Doukhan 1994 and Bradley 25 for references on mixing coecients Then, as usual, in order to obtain uniform results, we need an hypothesis on the compact C. C There exist two positive constants C 1 and d 1 such that for any < ν < 1, C can be covered by L ν C 1 ν d 1 balls of radius ν. We introduce now the conditions on the kernels and the windows. H a The kernels H and K have port D [, 1] with D, are non-increasing, have bounded derivatives and z R d H z R ddz = 1. b For any n N, the bandwidths h H,T and h K,T are continue

4 functions of T [n, n + 1[, dierentiable on ]n, n + 1[ with h H,T h H,T + h K,T 1 h K,T = O, T and h d H,T φh K,T T is increasing and veries the relation h d H,T φh K,T T lnt n ξ 2 n d d n a+1 φh K,T h 2 K,T hd H,T φh K,T h d+2 2 H,T where a and d 1 are respectively dened in D.1e and C, and ξ > 1. We are now in position to state our main result which is captured in Theorem 2.1. Theorem 2.1 Under D.1, H and C, there exists L > such that lim T f T y x fy x h β H,T + hβ K,T + lnt T φh K,T h d H,T L a.s. 1 We can now focus on the conditional mode θ x := argmax fy x that is posed to be unique. We estimate it by We then have to introduce another condition. θ x := argmax fy x. 2 D.2 There exist c 3 >, β 4 >, L 5 > such that for any x C and any y S [θ x c 3, θ x + c 3 ] fθ x x fy x > L 5 y θ x β 4 R d. Corollary 2.1 Under the conditions of Theorem 2.1, if D.2 holds, then there exists a constant L 2 such that, almost surely, lim T θ x θ x h β H,T + hβ K,T + lnt T φh K,T h d H,T 1 L 2. 3 β 4 For the following results, we consider the case when Y is real valued d=1 and we pose that S = [a 1, b 1 ], with P Y S X = x = 1 for any x C. We turn our

5 attention on the conditional distribution function F y x = P Y y X = x that we estimate by T t= F T y x = 1I ],y] Yt Kh 1 K,T dx, X tdt T t= Kh 1 K,T dx, X tdt when the denominator is not null. Here, for any set A, 1I A is the indicator function of A. If T t= Kh 1 K,T dx, X tdt =, we take F T y x = T t= 1I ],y] Yt dt T Corollary 2.2 Under the conditions of Theorem 2.1, we have, for some L 3 lim F T y x F y x T h β K,T + lnt T φh K,T L 3 a.s. 4 Using the last corollary, we immediately obtain the convergence of the estimator r T x of the regression function rx = EY X = x, dened by r T x = t [,T Y ] tkh 1 K,T dx, X tdt T t= Kh 1 K,T dx, X tdt if T t= Kh 1 K,T dx, X tdt otherwise, r T x = t [,T ] Ytdt T. 5 Corollary 2.3 Under the conditions of Corollary 2.2, we have, for some constant L 4 lim T r T x rx h β K,T + lnt T φh K,T L 4 a.s. 6 Note that a direct integration of the conditional density estimator gives us results for two other estimators of the conditional density and the regression function. 3 Proofs Set T,t x = Kh 1 K,T d 1x, X t and Υ T,t y = 1 Hh 1 h d H,T y Y t R d H,T

6 and introduce the decomposition f T y x = f 2,T f 1,T where and f 1,T x = f 2,T y x = 1 T T,t xdt T E T, t= 1 T Υ Tt y T,t xdt. T E T, t= So that, when f 1,T x >, f T y x = f 2,T y x f 1,T x. Proof of Theorem 2.1 Since the technics are the same to treat the convergence of f 1,T and that of f 2,T, we will only treat the estimator f 2,T. We rst need the two following lemmas. Lemma 3.1 Under the conditions D.1a and D.1c, we have, for T large enough, E f 2,T y x fy x C h β K,T + hβ H,T. 7 Lemma 3.2 Under the conditions of Theorem 2.1, for some L 6 >, we have that lim T f 2,T y x E f 2,T y x lnt T φh K,T h d H,T < L 6 a.s. 8 Proof of lemma 3.1 We x x B et y S, and when no confusion is possible, we will write Υ T,t and T,t instead of Υ T,t y and T,t x. The bias can be written as follows E f 2,T y x fy x = E E Υ T, y fy x X T, x. E T, x

7 For any u Bx, h K,T, making use of the hypotheses D.1a, a substitution leads us to the following relations E Υ T, y fy x X = u H z R d fy h H,T z u fy x dz z R d H z R d fy h H,T z u fy u dz z R d + fy u fy x C h β H,T + hβ K,T 9 which nishes the proof of Lemma 3.1. To prove Lemma 3.2, we need the following result. Lemma 3.3 Under the conditions of Theorem 2.1 we have CovΥ T,t T,t, Υ T,s T,s dtds = O [,T ] 2 T φ hk,t h d H,T 1 Proof of Lemma 3.3 We x x B et y S. Set Γ := [, T ] 2, v T = φh K,T h d 1 H,T and s T := CovΥ [,T ] 2 T,t T,t, Υ T,s T,s. Making use of Fubbini, we get following decomposition s T = CovΥ T,t T,t, Υ T,s T,s dtds Γ { t s <δ} + CovΥ T,t T,t, Υ T,s T,s dtds Γ {δ t s v T } + CovΥ T,t T,t, Υ T,s T,s dtds Γ {v T < t s } =: V 1 + V 2 + V 3. For the rst term V 1, we can write V 1 Γ { t s <δ} EΥ T,t T,t 2 dtds.

8 But under conditions D.1a c, and Ha, we have, for a large enough T EΥ T,t y T,t 2 = E EΥ 2 T, X 2 T,i H 2 f E 2 T,i h d H,T 11 β 2H 2 K 2 φh K,T f. h d H,T Thus, we obtain the bound For the second term V 2, we can write V 1 2T δβ 2H 2 K 2 φh K,T f. h d H,T V 2 EΥ T,s Υ T,t T,s T,t + EΥ T, T, The conditions D.1b c and Ha imply EΥ T, T, = E EΥ T, X T, f β 2 Kφh H,T. 13 Moreover, from D.1b, D.1d and Ha, we have for t s δ EΥ T,s Υ T,t T,s T,t = E EΥ T,s Υ T,t X s, X t T,s T,t β 3 f K 2 φh K,T 2 From which we obtain, for some C 1 V 2 C 1T φh K,T. 14 h d H,T Finally, the assumptions D.1e and Hb allow us to use the Billigsley's inequality

9 see 1.11 in Bosq 1998 as follows V 3 x 4 Υ T, T, 2 4c Υ T, 2 K 2 Γ {v T < t s } Γ {v T < t s } αv T dtds s t a dtds 4c Υ T, 2 K 2 T v a+1 T a 1 4cT φh K,T 2 K 2 H a 1 So, from 12, 14 and 15, there exists a constant C 1 which does not depend on x and y, such that, T φh K,T s T x C 1 h d H,T 16 which concludes the proof of Lemma 3.3. Proof of Lemma 3.2 T φh Set V T := K,T lnt. From Hb, V h d T h d H,T H,T large enough n N and any constant A, is increasing, so we have, for any

10 P P +1I ] A +P +P +P 6, [ +1I ] A 6, [ T [n,n+1[ 1 T V T 1 T [n,n+1[ V n T [n,n+1[ n T [n,n+1[ V n Υ T,ty T,tx E Υ T,ty T,tx dt > A Υ n,t y T,tx n,t x dt > A 6 n E Υ n,t y n,t x E Υ n,t y T V,tx n 1 n Υ n,t y n,t x E Υ n,t y n,t x dt > A V n 6 1 Υ n,t y T T [n,n+1[ V,tx E Υ n,t y T,tx > A n 6 1 T h d H,T Υ T [n,n+1[ V n h d T,ty Υ n,t y T,tx > A H,n 6 1 T h d H,T E Υ T,ty Υ n,t y T,tx dt h d H,n = A 1 + A 2 + A 3 + A 4 + A 5 + A 6 17 To treat the term A 1, we rst use the condition Hb to obtain T,t T = h K,T dx, X h 2 t K h 1 K,T dx, X t K,T h K,T K = O 1 h K,T T 18 which implies that there exists C 3 such that and T,tx n,t x C 3 n 19 C3 A 1 P nv n n Υ n,t y dt > A. 2 6 So, since the kernel H is bounded, the condition Hb, implies that, for n large enough,

11 A 1 =. 21 Then, from 19 we easily have that, for a large enough A and a large enough n, From the condition C, we can cover C with L n = center x k, k [1, L n ], and radius ν n A 3 P +1I ] A +P 18, [ k [1,L n] k [1,L n] x Bx k,ν n =: B 1 + B 2 + B 3. k [1,L n] x Bx k,ν n A 2 =. 22 C 1 L n 1/d1 such that d1 /2 n φh K,n h balls of 2 K,n hd H,n 1 n Υ n,t y n,t x k n,t xdt > A V n 18 1 n E Υ n,t y n,t x k n,t x dt V n 1 n Υ n,t y n,t x k E Υ n,t y n,t x k dt > A V n 18 For some m >, K is a m-lipschitz function, so for any x Bx k, ν n, n,t x k n,t x mνn h K,n, and for n large enough Thus, for a large enough n, we have We can cover S with L n = length 2l C 6 L 1/d n B 3 P +P +P n φh K,n h d+2 H,n B 1 =. 23 B 2 =. 24 d 2 intervals of center y k, k [1, L n] and where C 6 is independent of n, and decompose the term B 3 as follows 1 n n,t x k Υ n,t y Υ n,t y k dt > A k [1,L n] k [1,L n] y [y k ±l] V n 54 1 n E n,t x k Υ n,t y Υ n,t y k dt > A k [1,L n] y [y k ±l] V n 54 1 n Υ n,t y k n,t x k E Υ n,t y k n,t x k dt > A V n 54 k [1,L n] k [1,L n ] k [1,L n] =: B 1 + B 2 + B 3.

12 For the rst term, we have L n B 1 k =1 n K P V n y [y k ±l] Υ n,t y Υ n,t y k dt > A 54 Since H is a Lipschitz function, for n large enough, we have. Thus, for a large enough n, B 1 =. 25 B 2 =. 26 Making use of the Fuk-Nagev inequality like in chapter 4 statement??, we have B 3 C 8 n µ. 27 Thus, from 23, 24, 25, 25, 27, for a large enough A, there exists µ 1 > 1 and n N such that, for all n > n, Then, it is easily seen that, for a large enough n, A 3 n µ A 4 =. 29 Finally, we treat the terms A 5 and A 6 like A 1 and A 2. So from 21, 22, 28 and 29, using the Borel-Cantelli Lemma there exists a constant L' such that lim T T φh K,T h d H,T lnt f 2,T y x E f 2,T y x L a.s. It is obvious that E f 1,T x = 1, so we immediately have for some L lim T T φh K,T h d H,T lnt f 1,T x 1 L a.s. 3 This implies that there almost surely exists T R + such that, T > T, f T y x = f 2,T y x f 1,T x.

13 Then, the equality f 2,T y x f 1,T x E f 2,T y x = E f 2,T y x 1 f f 1,T x 1,T x + 1 f f 2,T y x E f 2,T y x 1,T x 31 concludes the proof of Lemma 3.2. Proof of Corollary 2.1 From Theorem 2.1, we know there exists a Ω 1 Ω with P Ω 1 = 1 such that, for any ω Ω 1, there exists T ω such that, for any T > T ω and any y S, with W T := L h β H,T + hβ K,T + lnt T φh K,T h d H,T f T y xω fy x < W T 32 1 We x ω Ω 1, and set W T := 2W β T 4 L 5. For a large enough T,. and This implies that f T θ x xω fθ x x < W T 33 f T θ x ω xω f θ x ω x < W T. 34 inf fθ x x f θ x ω x < 2W T. 35 The conditional mode is posed to be unique on S for any x C, so, since C and S are compact sets, we have fθ x x Thus, for T large enough, fy x >. 36 \[θ x c 3 ;θ x+c 3 ]

14 2W T < inf fθ x x fy x. 37 \[θ x c 3 ;θ x+c 3 ] So, for T large enough, θ x ω [θ x c 3 ; θ x + c 3 ] Then, in view of 35, condition D.2 implies the following relation which concludes the proof Proof of Corollary 2.2 θ x ω θ x < W T. 38 We rst remark that, if we do not want to have a uniform result on y, Lemma 3.2 remains valid if we choose H. = 1I [, 1 ]. and h H,T = 1. With these choices, there 2 exists n N such that, for any y S and x C F T y x = n i= f T y 1 i, x Then, in view of the proof of Lemma 3.1, we immediately have that which concludes the proof of Corollary 2.2 Proof of Corollary 2.3 E F T y x F y x Ch β K,T 4 Corollary 2.3 arises from a direct application of Corollary 2.2. References Berlinet, A., Gannoun, A., et Matzner-Løber, E Normalité asymptotique d'estimateurs convergents du mode conditionnel. Canad. J. Statist., 262, Bradley, R. C. 25. Basic properties of strong mixing conditions. A survey and some open questions. Probab. Surv., 2, electronic. Update of, and a plement to, the 1986 original. Cherno, H Estimation of the mode. Ann. Inst. Statist. Math., 16, 3141.

15 Collomb, G., Härdle, W., et Hassani, S A note on prediction via estimation of the conditional mode function. J. Statist. Plann. Inference, 152, Dabo-Niang, S. et Laksaci, A. 27. Estimation non paramétrique de mode conditionnel pour variable explicative fonctionnelle. C. R. Math. Acad. Sci. Paris, 3441, Doukhan, P Mixing, volume 85 of Lecture Notes in Statistics. Springer-Verlag, New York. Properties and examples. Ezzahrioui, M.and Ould-Saïd, E. 28. Asymptotic normality of the kernel estimators of the conditional mode for functional data. J. Nonparametric Statist., 2, 318. Ferraty, F. et Vieu, P. 2. Dimension fractale et estimation de la régression dans des espaces vectoriels semi-normés. C. R. Acad. Sci. Paris Sér. I Math., 332, Ferraty, F., Laksaci, A., et Vieu, P. 25. Functional time series prediction via conditional mode estimation. C. R. Math. Acad. Sci. Paris, 345, Ferraty, F., Laksaci, A., et Vieu, P. 26. Estimating some characteristics of the conditional distribution in nonparametric functional models. Stat. Inference Stoch. Process., 91, Louani, D. et Ould-Saïd, E Asymptotic normality of kernel estimators of the conditional mode under strong mixing hypothesis. J. Nonparametr. Statist., 114, Parzen, E On estimation of a probability density function and mode. Ann. Math. Statist., 33, Samanta, M Nonparametric estimation of the mode of a multivariate density. South African Statist. J., 7, Van Ryzin, J On strong consistency of density estimates. Ann. Math. Statist., 4,