Asymptotic normality of a recursive estimator of a conditional hazard function with functional stationary ergodic data

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1 ,, ProbStat Forum, Volume, Aprl 08, Pages 36 5 ISSN ProbStat Forum s a e-joural. For detals please vst Asymptotc ormalty of a recursve estmator of a codtoal hazard fucto wth fuctoal statoary ergodc data Goutal A, Mechab B, Issa M A, Soudd Y Laboratory of Statstc ad Processus Stochastc, Uversty of Djllal Labes, L.P 89, Sd Bel Abbes 000, Algera. Abstract. I ths paper, we vestgate a recursve kerel estmator of the codtoal hazard fucto wheever fuctoal statoary ergodc data are cosdered. Uder the assumpto of ergodcty, the ovelty of our approach s that we do ot requre depedece of the observatos. It s show that, uder some wld codtos, the recursve kerel estmate of the three parameters codtoal desty, codtoal dstrbuto ad codtoal hazard are asymptotcally ormally dstrbuted.. Itroducto Recetly there has bee a creasg terest the study of fuctoal data. For a overvew of the preset state o oparametrc fuctoal data FDA, we refer to the works of 4 ad 3, ad the refereces there. Codtoal hazard estmato wth a fuctoal explaatory varable ad a scalar respose acqured cosderable terest the statstcal lterature. The frst work was proposed by Ferraty et al. 5, where they troduce a kerel estmator ad prove some asymptotc propertes wth rates varous stuatos cludg cesored ad/ or depedet varables. Qutela-del-Río exteded the results of Ferraty et al. 5 by calculatg the bas ad varace of these estmates, ad establshg ther asymptotc ormalty. I the case of completely observed data, aother estmators have bee proposed for the codtoal hazard fucto by dfferet approaches. I 04, Attouch ad Belabed have studed the oparametrc estmator of the codtoal hazard fucto usg the k Nearest Neghbors k-nn estmato method ad they have show ts asymptotc propertes the case of depedet data. Aother approach has bee proposed by Massm ad Mechab based o the local lear method, they have establshed the almost complete covergece of the proposed estmator. I the case of the fuctoal spatal data, the works o the codtoal hazard fucto s lmted ad we ca refer to 9 where they studed the almost complete covergece of the kerel type estmate. These authors studed 0 the mea squared covergece rate ad proved the asymptotc ormalty of the 00 Mathematcs Subject Classfcato. 6G05, 60G4, 6F. Keywords. Asymptotc ormalty, Codtoal hazard fucto, Fuctoal data, Recursve estmate, Statoary ergodc processes. Receved: 8 September 07 ; Revsed: 6 November 07, 04 March 08; Accepted: 6 March 08 mal addresses: amasara90@hotmal.com Goutal A, mechaboub@yahoo.fr Mechab B, bemht3@yahoo.com Issa M A, souddyoucef@yahoo.fr Soudd Y

2 proposed estmator. Goutal, Mechab, Issa ad Soudd / ProbStat Forum, Volume, Aprl 08, Pages I recet years, the statstcal modelg for fuctoal ergodc data has bee a creasg terest ad a great mportace varous felds. The geeral framework of ergodc fuctoal data has bee tated by Laïb ad Loua 7,8 who stated cossteces wth rates together wth the asymptotc ormalty of the regresso fucto estmate. I ths topc of ergodc data, several works have bee publshed, for example, Gherballah et al. 6 showed the almost complete covergece wth rate of a famly of robust oparametrc estmators for regresso fucto. More recetly, Ardjou et al. treated the almost complete covergece ad the asymptotc ormalty of the estmator of codtoal mode, Bezad et al. 3 studed the almost complete rate covergece of fuctoal recursve kerel of the codtoal quatles. Ths paper s orgazed as follows: Secto troduces the estmator of the codtoal hazard fucto. I secto 3 we wll defe some otatos ad hypothess. The asymptotc ormalty of the proposed estmator of the codtoal hazard fucto s gve Secto 4. Fally, the proofs of our results are gve the Appedx.. The recursve estmato of the codtoal hazard fucto Let X, Y,..., be a sequece of strctly statoary ergodc processes. Where X are values semmetrc space F, d ad Y are real-valued radom varables. N x wll deote a fxed eghborhood of x. We assume that the regular verso of the codtoal probablty of Y gve X exsts. Moreover, we suppose that, for all x N x the codtoal dstrbuto fucto of Y gve X x, the recursve kerel estmator of the codtoal hazard fucto hy x such that s hy x fy x, for y R ad F y x < F y x ĥy x fy x F, y R. y x We defe the recursve kerel estmator of the codtoal dstrbuto fucto by F y x Ka dx, X Hb y Y Ka dx, X where K s the kerel, H s a strctly creasg dstrbuto fucto ad a, b are a sequeces of postve real umbers such that lm a lm b The recursve kerel estmator f. x of f. x s gve by fy x b Ka dx, X H b y Y Ka dx, X. 3. Notatos ad hypothess The assumptos that we wll eed our study are the followg: The fuctoal ergodc data s carred out by the followg cosderato: for,...,, we put F k s the σ-

3 Goutal, Mechab, Issa ad Soudd / ProbStat Forum, Volume, Aprl 08, Pages algebra geerated by X, Y,..., X k, Y k. We pose B k s the σ-algebra geerated by X, Y,..., X k, Y k, X k+. We suppose that the strctly statoary ergodc process X, Y N satsfes H The fucto φx, h : PX Bx, h > 0, h > 0, where Bx, h : {x F/dx, x < h}. For all,..., there exsts a determstc fucto φ x,. such that almost surely 0 < PX Bx, h F φ x, h, h > 0 ad φ x, h 0 as h 0. PX Bx, h F For all sequece h,..., > 0, φx, h H Let S be a compact set of R, the codtoal dstrbuto fucto F. x s such that, y S, β > 0, f y S F y x > β, y,y S S, x,x N x N x wth C > 0, β > 0, β > 0.. F y x F y x C dx, x β + y y β The desty f. x s such that, y S, α > 0 fy x < α, y,y S S, x,x N x N x fy x fy x C dx, x β + y y β wth C > 0, β > 0, β > 0. H3 y, y R H j y H j y C y y, for j 0, t β H tdt < ad H tdt <. H4 K s a fucto wth support 0, such that 0 < C < Kt < C <. H5 The badwdths a, b satsfed : t 0, ad lm + lm + φ x ϕ x where ϕ x φx, a. Commets o hypotheses: φx, ta φx, a β xt a β φx, a + b β φx, a The codto H volves the ergodc ature of the data ad the small ball techques used ths paper. The hypothess H s a drect cosequece of Beck s theorem. The assumpto H presets the Lpschtz s codto to the codtoal dstrbuto fucto ad codtoal desty fucto, t meas that the both fuctos are cotuous wth respect to each varable ad permts us to evaluate the bas term wthout usg the dfferetablty. Hypothess H3 mpose some regularty codtos upo the dstrbuto fucto H used our estmates. The codto H4 s very stadard oparametrc fucto estmato. The assumptos H5 s a techcal codto. 0

4 Goutal, Mechab, Issa ad Soudd / ProbStat Forum, Volume, Aprl 08, Pages Ma result: Asymptotc ormalty Theorem 4.. Uder hypotheses H-H5, we have for all x A / ϕ x D σh ĥy x hy x N 0, as 3 x, y where A {x, σh x, y 0} ad σ h x, y α hy x α F y x wth α K 0 K sβ x sds ad α K 0 K s β x sds. The proof of ths theorem s based o the followg decomposto ad lemmas below: ĥy x hy x F y x fy x fy x + hy x F y x F y x F. 4 y x Lemma 4.. Uder hypothess of the theorem 4., we have / ϕ x σf x, y F y x F y x D N 0, as 5 where σf x, y α F y x F y x. α Lemma 4.3. Uder hypothess of the theorem 4., we have / ϕ x x, y fy x fy x D N 0, as 6 σ f where σf x, y α fy x α. The proof of lemma 4. s based o the followg decomposto where we put, for ay x F, ad,..., ; K Ka dx, X ad H Hb Y y. We start by wrtg F y x F y x B, x, y + R, x, y F D x + Q, x, y F D x 7 where Q, x, y F N y x F N y x F y x F D x F D x, B, x, y F N y x F D x ad R, x, y B, x, y F D x F D x

5 wth F N y x F N y x F D x F D x Goutal, Mechab, Issa ad Soudd / ProbStat Forum, Volume, Aprl 08, Pages ϕ ϕ ϕ ϕ Ka dx, X Hb Y y, Ka dx, X Hb Y y F, Ka dx, X, Ka dx, X F. The proof of the lemma 4. s a cosequece of the followg lemmas, whose proofs are gve the Appedx Lemma 4.4. Uder the hypothess of theorem 4. ϕ x Lemma 4.5. Uder the hypothess H ad H5, we have σ F / Q,x, y D N 0, as. F D y x o p. Lemma 4.6. Uder the hypothess H, H ad H3 we have / ϕ x B,x, y o p as. σ F Lemma 4.7. Uder the hypothess H, H ad H5 we have / ϕ x R,x, y o p as. σ F The proof s based o the followg decomposto ad lemmas below. fy x fy x B, x, y + R, x, y F D x + Q, x, y F D x 8 where wth Q, x, y f N y x f N y x fy x F D x F D x, B, x, y f Ny x F D x fy x ad R, x, y B, x, y fn y x f N y x f N y x f N y x ϕ ϕ b Ka dx, X H b Y y Ka dx, X, b Ka dx, X H b Y y F Ka dx, X.

6 Goutal, Mechab, Issa ad Soudd / ProbStat Forum, Volume, Aprl 08, Pages Lemma 4.8. Uder the hypothess of theorem 4. / ϕ x Q,x, σf y D N 0, as. Lemma 4.9. Uder the hypothess H, H ad H3 we have / ϕ x B,x, σf y o p as. Lemma 4.0. Uder the hypothess H, H ad H5 we have / ϕ x R,x, σf y o p as. 5. Cofdece bads A usual applcato of asymptotc ormalty s to establsh cofdece bads for the estmates. Our goal ths secto s the applcato of our asymptotc ormalty result Theorem 4. to buld the cofdece tervals for the true value of hy x for a gve curve X x. I oparametrc estmato, the asymptotc varace depeds o certa ukow fuctos. I our case, we have σ hx, y α α hy x F y x where hy x, F y x, α ad α are ukow a pror ad have to be estmated practce. The oe ca obta a cofdece bads eve f σh x, y, s fuctoally specfed. Now a plug- estmate for the asymptotc stadard devato σh x, y, ca be easly obtaed usg the estmators ĥy x, F y x, α ad α of hy x, F y x, α ad α respectvely, that s σ hx, y α ĥy x α F y x. We estmate emprcally the costats α ad α, as follows: α φx, a Ka dx, X, α φx, a K a dx, X. Remark that the specal case whe K I 0,, t becomes mmedately that α α. Now the asymptotc cofdece bad at asymptotc level ζ for hy x s gve by ĥy x u ζ σ / h x, y, ĥy x + u ϕ x ζ where u ζ deotes the ζ quatle of the stadard ormal dstrbuto. 6. Cocludg remarks σ / h x, y ϕ x Ths artcle provdes a theoretcal framework about recursve codtoal hazard fucto estmator wth fuctoal statoary ergodc data. The resultg recursve codtoal hazard fucto estmator has

7 Goutal, Mechab, Issa ad Soudd / ProbStat Forum, Volume, Aprl 08, Pages bee show to be cosstet ad asymptotcally ormally dstrbuted uder approprate codtos. To prove the results, the methodology s based upo the martgale approxmato used 7. The ergodc hypothess used the paper avods complcated probablstc calculatos of the mxg codto. The hypothess H plays a mportat role, whch volve the recursve estmate, the ergodc ature of the data ad the small ball techques. It s clear that ths hypothess s qute mlder tha Lab ad Loua. Ideed, ulke Lab ad Loua 7, t s ot ecessary to wrte approxmately the cocetrato fucto PX Bx, h ad the codtoal cocetrato fucto PX Bx, h F a product of two depedet oegatve fuctos of the ceter ad of the radus. The recursve estmate s very fast practce because the smoothg parameter s lked to the observato X, Y, whch permts to update our estmator for each addtoal observato. It should be oted that our work s a geeralzato of the results obtaed by other authors two dfferet drectos the estmato method ad the data correlato. Ideed, the classcal kerel method used by Ferraty et al. 5 ad Qutela-del-Río ca be cosdered as a specal case of ths study by cosdered that a h K ad b h H. O the other had, the depedet case, codto H s automatcally verfed ad for all,..., take φ x,. φx,. ad we obta the same result gve by Qutela-del-Río whe X, Y are depedet. 7. Appedx Proof of Lemma 4.4. We use the same deas Lab ad Loua 0. For all,...,, we defe ad, we defe ξ η η F. It s easly see that ϕ x η H y F y x K x ϕ x σ F ϕ x σ F Q, x, y ξ as ξ s a tragular array of martgale dffereces accordg the σ-algebra F, we are posto to apply the cetral lmt theorem based o ucodtoal ldeberg codto to establsh the asymptotc ormalty of Q, x, y. Ths ca be doe f we establsh the followg statemets: a b ξ F probablty, ξi ξ >ɛ 0 holds for ay ɛ > 0 Ldeberg codto. Proof of part a ξ F η η F η F η F. The statemet a follows the f we show that:

8 lm Goutal, Mechab, Issa ad Soudd / ProbStat Forum, Volume, Aprl 08, Pages η F 0 probablty, lm η F probablty. To prove, we put A η F η F ϕx σ F, we have A ϕ x H y F y x K x F A H y B F y x K x F ϕ x A ϕ x uder H ad H4, we have Cφ x, a K x F C φ x, a H y X F y xk x F Next, a tegrato by parts ad a chage of varable allow to get H y X H tf y b t X dt. Thus, we have H y X F y x C a β combg ths results, we have R η F AC φ x, a ϕ x, a aβ η F a β σ F a β φ x, a ϕ x, a Now, we have to prove, to do ths, we observe that η F A ϕ x, a A ϕ x, a 0 Uder H3. H y F y x K x F K xh y F F y x K xh y F + F y x K x F.

9 Goutal, Mechab, Issa ad Soudd / ProbStat Forum, Volume, Aprl 08, Pages We put I I I 3 We wrte K xh y F, K xh y F, K x F. I F y x F y x F y x K x F + K xh y F F y x K x F K x F + H y X K x F K x F H y X K x F F y x K x F. Usg the same argumet as those used proof of the part, we obta For I 3, we get so uder H we have ϕ x, a ϕ x, a I o. K x F K φ x, a K x F By combg ths results, we deduce that ϕ x, a K ϕ x, a K u o. lm Proof of part b The Ldeberg codto whch mples that K u φ x, a du φ x, a K φ x, ua du K u φ x, a du ϕ x, a η F, whch complete the proof of part a. ξ I ξ>ɛ 4 η I η>ɛ/. Let a > ad b > such that a + b makg use the Holder s ad Markov s equaltes oe ca wrte

10 for all ɛ > 0 ηi η>ɛ/ Goutal, Mechab, Issa ad Soudd / ProbStat Forum, Volume, Aprl 08, Pages η a ɛ/ a/b. Takg C 0 R + ad a + δ, from some δ > 0, such that H y +δ < ad H y F y x +δ X u W +δ u s a cotuous fucto, we obta 4 ηi η>ɛ +δ ϕ x, a C 0 σf ϕ x, a +δ +δ ϕ x, a C 0 σf C 0 ϕ x, a σ F H y F y x +δ K +δ H y F y x +δ K +δ x X ϕ x, a +δ +δ K +δ xw +δ x ϕ x, a +δ x +δ ϕ x, a K +δ x W +δ x W +δ x + W +δ x K +δ x C 0 σf ϕ x +δ +δ C 0 σf K +δ x W +δ x + o. Cosequetly ξi ξ>ɛ 0 as whch completes the proof of lemma. Proof of Lemma 4.5 Observe that F D x ϕ x, a K x K x F + K x F K x K x F + K x F. ϕ x, a ϕ x, a }{{}}{{} T T The proof of the lemma follows the f show that T o as ad T 0 probablty as. For T, Uder H3 ad H we prove that ϕ x, a K x F o as. So, t s easly see that T 0 probablty as. For T, observe that T x L x, where {L x} s a tragular array of martgale dffereces wth respect to the σ-algebra F. Combg the Burkholder equalty see P.H. All ad C.Heyde p3, 980 ad Jese s equalty see

11 Goutal, Mechab, Issa ad Soudd / ProbStat Forum, Volume, Aprl 08, Pages Lab ad Laou p365,0, we obta for ay ɛ > 0, there exsts a costat C 0 > 0 such that P T x > ɛ K C x 0 ɛ ϕ x o ɛ ϕ x, a + o sce ϕ x as, we coclude the that T o probablty as. Whch completes the proof of lemma 4.5. Proof of Lemma 4.6 We have B, x, y F N y x F D x. We wrte B, x, y K x F K x H y B F F y x K x F K x F K x H y X F F y x K x F K x F K x H y X F y x F. Next, a tegrato by parts ad chage of varable allow to get H y X H tf y b t X dt thus, we have H y X F y x R uder H, we obta that I Bx,aX H y X F y x C ad uder H4 we prove that We acheve the proof of lemma 4.6. Proof of Lemma 4.7 We wrte R H t F y b t X F y x dt 9 R H ta β K x F o. R, x, y B, x, y F y x FN x, y F N x, y F N x, y F y xf D x, y FN x, y F N x, y. F D x, y + t β b β dt 0

12 Goutal, Mechab, Issa ad Soudd / ProbStat Forum, Volume, Aprl 08, Pages Clearly, t s suffces to show that a F N x,y F y xf D x,y o, F D x,y b FN x, y F N x, y o. The proof of the frst had uses argumets smlar to those used the proof of lemma 4.6 of the secod part, wll be establshed FN x, y F N x, y 0, V ar FN x, y F N x, y 0 as. For all,...,, we put δ x, y ϕ x, a K xh y K xh y F where δ x, y s a tragular array of martgale dffereces accordg to the σ algebra F ext by H ad H4 we obta F N x, y F N x, y δ x, y δ x, y 0 by defto of δ x, y, we wrte δ x, y δ x, y. Furthermore, by Jese s equalty we have δ x, y ϕ x, a K xh y F ϕ x, a K x F ϕ x, a P X x, a F ϕ x, a φ x, a. So, we obta δ x, y φ x, a ϕ x, a. We deduce uder H that var FN x, y F N x, y 0 as. Proof of lemma 4.8 We use the deas Lab ad Loua For all,...,, we defe η ad, we defe ξ η η F. It s easly see that ϕ x σf ϕ x b σf H y fy x K x ϕ x Q, x, y ξ.

13 Goutal, Mechab, Issa ad Soudd / ProbStat Forum, Volume, Aprl 08, Pages We follow the same dea the proof of lemma 4. to prove ths, we establsh the followg statemets: ξ F probablty, a b ξ I ξ >ɛ 0 holds for ay ɛ > 0 Ldeberg codto. Proof of part a ξ F η η F η F η F. The statemet a follows the f we show that lm η F 0 probablty, lm To prove, we put A η F probablty. ϕx σ f, we have η F η F A ϕ x b H y f y x K x F A b H ϕ x y B fy x K x F A ϕ x b H y X fy xk x F uder H ad H4, we have Cφ x, a K x F C φ x, a. Next, a tegrato by parts ad a chage of varable allow to get H y X b H tfy b t X dt. Thus, we have H y X b fy x C a B combg ths results, we have R η F A ϕ x, a C φ x, a a β / η F ϕ x φ x, a σ f ϕ x, a a β η F aβ a β φ x, a 0 Uder H3. ϕ x, a σ f

14 Goutal, Mechab, Issa ad Soudd / ProbStat Forum, Volume, Aprl 08, Pages Now, we have to prove, to do ths, we observe that We put I I I 3 We wrte η F A ϕ x, a A ϕ x, a b K xh y F, b K xh y F, K x F. I fy x fy x fy x fy x b H y fy x K x F b K xh y F fy x b K xh y F + fy x K x F. K x F + b K xh y F K x F K x F + b K x F b H y X K x F fy x H y X K x F K x F usg the same argumet as those used proof of the part, we have ϕ x, a I o. For I 3, t s the same I 3 proved proof of lemma 4. ad by the combg ths results, we deduce that lm η F whch complete the proof of part a. Proof of part b The Ldeberg codto whch mples that ξ I ξ >ɛ 4 η I η >ɛ/. Let a > ad b > such that a + b makg use the Holder s ad Markov s equaltes oe ca wrte

15 for all ɛ > 0 η I η >ɛ/ Goutal, Mechab, Issa ad Soudd / ProbStat Forum, Volume, Aprl 08, Pages η a ɛ/ a/b. Takg C 0 R + ad a +δ, from some δ > 0, such that W +δ u s a cotuous fucto, we obta 4 η I η >ɛ C 0 C 0 C 0 +δ ϕ x, a σf +δ ϕ x, a σf ϕ x, a σ f Y y +δ < ad b +δ b ϕ x, a +δ H y fy x +δ K +δ b H y fy x +δ K +δ x X +δ K +δ xw +δ x ϕ x, a +δ ϕ x, a +δ H y F y x +δ X u x +δ ϕ x, a K +δ x W +δ x W +δ x + W +δ x K +δ x C 0 σf ϕ x +δ +δ C 0 K +δ x W +δ x + o. Cosequetly σ f Proof of Lemma 4.9 We have We wrte B, x, y ξ I ξ >ɛ 0 as whch completes the proof of lemma. K x F B, x, y f N y x F D x. fy x K x F K x F fy x K x F K x F K x b H x B F K x b H x X F K x b H x X b fy x F.

16 Goutal, Mechab, Issa ad Soudd / ProbStat Forum, Volume, Aprl 08, Pages Next, a tegrato by parts ad chage of varable allow to get H y X b H tfy b t X dt thus, we have b H y X fy x Uder H, we obta that I Bx,aX H y X b fy x Cb ad uder H4 we prove that R R H t F y b t X fy x dt. R H ta β K x F o. We acheve the proof of lemma. Proof of Lemma 4.0 We wrte R, x, y B, x, y fn x, y f N x, y f N x, y fy xf D x, y fn x, y f N x, y. F D x, y Clearly, t s suffces to show that: + t β b β dt a f N x,y fy xf D x,y F D x,y o, b fn x, y f N x, y o. The proof of the frst had uses argumets smlar to those used the proof of lemma 4.6. Of the secod part, wll be establshed fn x, y f N x, y 0, V ar fn x, y f N x, y 0 as For all,...,, we put δ x, y ϕ x, a b K xh y b K xh y F where δ x, y s a tragular array of martgale dffereces accordg to the σ algebra F ext by H ad H4 we obta f N x, y f N x, y δ x, y δ x, y 0 by defto of δ x, y, we wrte δ x, y δ x, y.

17 Goutal, Mechab, Issa ad Soudd / ProbStat Forum, Volume, Aprl 08, Pages Furthermore, by Jese s equalty we have δ x, y ϕ x, a b K xh y F ϕ x, a b K x F ϕ x, a P X x, a Cb φ x, a ϕ x, a H y B F so, we obta δ x, y C b φ x, a ϕ. x, a We deduce that var fn x, y f N x, y 0 as sce H. Ackowledgmet. The authors would lke to thak the referee for hs/her costrctve commets ad suggestos, whch mproved the qualty ad presetato of the paper. Refereces Ardjou, F.Z., At Hea, L. ad Laksac, A. 06. A recursve kerel estmate of the fuctoal modal regresso uder ergodc depedece codto, Joural of Statstcal Theory ad Practce, 0, Attouch, M.k., Belabed, F.Z. 04. The k Nearest Neghbors estmato of the codtoal hazard fucto for fuctoal data, RVSTATStatstcal Joural,, Bezad, F., Laksac, A. ad Tebboue, F. 06. Note o codtoal quatles for fuctoal ergodc data, C. R. Acad.Sc.Pars,Ser.I, 354, Ferraty, F., ad Veu, P Noparametrc fuctoal data aalyss, Sprger Seres Statstcs, Sprger New York. 5 Ferraty, F., Rabh, A., ad Veu, P stmato o-paramétrque de la focto de hasard avec varable explcatve foctoelle,rev. Roumae Math. Pures Appl, 53, 8. 6 Gherballah, A., Laksac, A., ad Sekkal, S. 03. Noparametrc M-regresso for fuctoal ergodc data, Statstcs ad Probablty Letters 83, Laïb, N., ad Loua, D. 00. Noparametrc Kerel regresso estmato for fuctoal statoary ergodc data: Asymptotc proprtes, J. Mult. Aal., 0, Laïb, N., ad Loua, D. 0. Rates of strog cossteces of the regresso fucto estmator for fuctoal statoary ergodc data, J. of Stat. Pla. ad If., 4, Laksac, A., ad Mechab, B. 00. stmato o-paramtrque de la focto de hasard avec varable explcatve foctoelle : cas des doées spatales, Rev. Roumae Math. Pures Appl, 55, Laksac, A., ad Mechab, B. 04. Codtoal hazard estmate for fuctoal radom felds, Joural of Statstcal Theory ad Practce, 8, Massm, I., ad Mechab, B. 06. Local lear estmato of the codtoal hazard fucto, Iteratoal Joural of Statstcs & coomcs 7, -. Qutela-del-Río, A Hazard fucto gve a fuctoal varable: No-parametrc estmato uder strog mxg codtos, J. Noparametr. Stat., 0, Ramsay, J.O., ad Slverma, B.W. 00. Appled fuctoal data aalyss; Methods ad case studes, Sprger-Verlag, New York.