Last exit times for a class of asymptotically linear estimators

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1 Las exi ies for a class of asypoically linear esiaors M. Alagh, M. Broniaowski 2, and G. Celan 3 Dépareen de Mahéaiques, Universié Louis Paseur, 4 Rue René Descares, Srasbourg, France 2 LSTA, Universié Paris 6, 4 Place Jussieu, Paris, France 3 Diparieno di Scienze Saisiche, Via C.Baisi 24, Padova, Ialy Absrac. We sudy he las exi ie for he Glivenko-Canelli saisics indexed by soe class of funcions. Also we provide upper bounds for is ail disribuion. Our firs exaple is he Glivenko-Canelli saisics indexed by a subclass of a Sobolev space; we nex consider las exi ies for adapive seiparaeric esiaes in he spiri of Klaassen, for which we provide he disribuion and ail bounds uniforly upon he nuisance paraeer. AMS 2000 Classificaion:Priary 62F35 Keywords: Las exi ie, Funcional Glivenko-Canelli saisics, Adapive esiaion, Seiparaeric esiaion. Inroducion Raes of convergence for saisical esiaors usually focus of asypoic equivalens for he disance beween he esiae and he paraeer i inends o approxiae. When he esiae is srongly consisen, which is o say ha i converges alos surely, hen he ie which is necessary in order ha i says in soe neighborhood of he rue value of he paraeer fro hen on is a well defined rando variable, which bears a very inuiive sense and which, soeies, can be evaluaed, a leas for sall neighborhoods of he paraeer. In his conex, he siuaion is quie siilar o he case when we consider a deerinisic sequence x n converging o x in a eric space: given soe sall ε, which is he order of agniude of he ineger Nε such ha, for all n larger han Nε, he disance beween x n and x reains forever saller han ε? This class of probles is usually referred o as las exi ies probles, considering ha he ers of he sequence of esiaes ay say ouside he ε-neighborhood of is i only when when n is saller han Nε. This noion has been presened for sequences of M- esiaes by [Sue, 983], and has been exended o he las exi ie for he Glivenko-Canelli saisics by [Hjor and Fensad, 992]; exensions for he case when he saple is drawn fro a Markov chain or fro a srongly ixing Work parly pored by Aeneo di Padova, n. CPDAO2345, Ialy

2 864 Alagh e al. sequence have been sudied by [Barbe e al., 999], and soe exensions for U-saisics have recenly been proposed by [Bose and Chaerjee, 200]. The presen paper follows his chain of works and provides soe insigh in he range of adapive sei paraeric esiaes; i also provides soe inforaion on he ail of he disribuion of las exi ies for hose ypes of esiaes. The ain ool o be ipored for he obenion of he law of Nε for such esiaes is uniforiy wih respec o he nuisance paraeer. This is achieved hrough Gaussian approxiaions for he so-called sequenial epirical process, a device which has been proposed in he for which is o be used here by [Sheehy and Wellner, 992]. The srucure of he paper is as follows: The firs secion is devoed o he obenion of a general resul on las exi ies for he Glivenko-Canelli saisics indexed by a class of funcions, following he line defined by [Sue, 983]. The second secion specializes his resul for various siuaions, wih an ephasis owards sei paraeric adapive esiaors in he spiri of [Klaassen, 987]. 2 Las exi ie for he funcional Glivenko-Canelli Saisics A saple X,..., X n is given, wih i.i.d. coponens following a coon disribuion P on soe space X. For f a real valued easurable funcion on X we denoe Pf he expecaion of f wih respec o P, i.e. Pf := fdp. Denoe P n he epirical easure peraining o he saple, P n := n n i= δ X i where δ x is he Dirac ass a poin x. In he sequel F denoes a subclass of funcions in L 2 P for which we assue ha for all x in X, he condiion fx Pf is finie f F holds. Define N ε := { n : P n Pf ε f F }, which denoes he las exi ie for he Glivenko Canelli saisics indexed by F. Since F is a Donsker class, i saisfies he Glivenko Canelli propery, naely P n Pf = 0 a.s., n f F which iplies ha N ε is a.s. finie. We consider he iing disribuion of ε 2 N ε when ε ends o 0. Following [Sue, 983], wih N ε f := {n : P n f Pf > ε}, we have, for fixed f in F, Proposiion Le f belongs o L 2 P. Then

3 Las exi ies for a class of asypoically linear esiaors 865 i ε 2 N ε f = d Wax 2 f := σ2 f W 2 s, ε 0 0 s where σ 2 f = Pf 2 PF 2, and Ws is he sandard Wiener process. P {ε 2 N ε f > λ} λ =, where ψ denoes he upper ail of he ψ σf sandard noral disribuion ψλ := P[N0, > λ]. ii λ ε 0 We obain an inforaion peraining o he oens of he r.v. ε 2 N ε f for sall ε. A sequence of r.v. s Y n is r-quick convergen o 0 whenever, for all ε > 0, EN r ε := E{n : X n ε} r is finie. This propery has been used by [Lai, 98] in order o assess opialiy properies of probabiliy raio ess in sequenial analysis. As a consequence of Proposiion we have Corollary Le f belong o L 2 P. The sequence P n Pf is r-quick convergen o 0 for all r > 0. Proof of Proposiion i for all f L 2 P, i holds N ε f = { n : n n i= fx i Pf ε } { } = n : νnf ε. n Le y be a posiive nuber. Then P { ε 2 N ε f y } = P { N ε f y ε 2 }. Define :=< y/ε 2 >, he salles ineger larger or equal y/ε 2. Then P {ε 2 N εf y} = P {N εf } = P n : nx fx i Pf n ε i= nx r = P fx i Pf n y0, n i= where y 0 := ε 2. We hus have 0 y 0 y ε 2, which enails ha y 0 ends o y as ε ends o 0. For all f in L 2 P, we herefore have P {ε 2 nx N εf y} P fx i Pf n n i= X = P fx i Pf i= = P σf σf X fx i Pf =: PE. i=

4 866 Alagh e al. For all r >, se and A r := B r := For all r >, and herefore σf r σf >r σf X fx i Pf i= σf X fx i Pf. i= PE = PA r B r, P ε 2 N ε f y = PE = PA r B r ε 0 r = PA r + PB r PA r B r. r If PB r = 0, 2 r hen PE = PA r. r The proof of 2 is easy. Following Donsker Invariance Principle, he processes σf fx i Pf; [, r] i= converge in disribuion in D[, r] o he Brownian process W. By coninuiy of he reu i herefore holds σf d fx i Pf = W. r i= r For all [, r], he process W = W is also a Brownian process. Therefore W r = d W s. r s Hence whenever 2 holds, we have! ε 0 Pε2 N εf y = P = P σf r Ws y r s ff j σf Ws y 0 s = P W 2 axf y, where W 2 axf = σ 2 f 0 s W 2 s.

5 Las exi ies for a class of asypoically linear esiaors 867 ii The axial variance of σfws equals σ 2 f and is obained when s =. Le ] I h := {s [0, ] : sσ 2 f σ 2 f h 2 } = [ h2 σ 2 f,, and noe ha Eσ 2 fw = σ 2 f. The unifor a.s. coninuiy of σfws on [0, ] enails ha { } h 0 h E σf Ws W = 0, s I h which proves ha he condiions in [Adler, 990], Theore 5.5 are fulfilled, proving he clai. Le us urn o he unifor case, ha is, consider he iing disribuion of ε 2 N ε, ε 0. We will assue H: F is a Donsker class H2: For all x X, f F fx Pf is finie H3: F is a perissible class of funcion, iplying ha f F P n f Pf is easurable. Define a Gaussian process Z P defined on [0, ] F, a version of which has uniforly bounded saple pahs which are uniforly coninuous on [0, ] F when equipped wih he ρ P pseudo-eric defined on [0, ] F defined hrough ρ P s, f,, g := s +Pf g 2. The exisence of such process is a consequence of hypohesis H above see [Sheehy and Wellner, 992]. The Kiefer-Müller process Z P is cenered for any s and f, and is covariance operaor is given by We hen have cov [Z P s, f, Z P, g] = s Pfg PfPg. Proposiion 2 When F saisfies H, H2 and H3, where F := [0, ] F. Proof ε 0 ε2 d N ε = Z P s, f 2, s,f F

6 868 Alagh e al. Le y be soe posiive nuber, and :=< y/ε 2 >. I holds, seing y 0 = ε 2, Pε 2 N ε y = PN ε = P! nx fx i Pf n f F n i=! X = P fx i Pf f F i= = P f F Z, f «y 0, where Z is he sequenial epirical process, an eleen of l [0, F defined hrough Z, f := i= fx i Pf. Noe ha for all fixed, by, Z,. belongs o l F, he se of all bounded sequences defined fro F ono R. Following [Sheehy and Wellner, 992], Theore, for any r > 0, Z converges in disribuion in l [0, r] F o Z P. For all r, i hus holds Z, f = d Z P, f. f F r f F r Define Z ZP,f, f :=, for [, r] and f F. The cenered Gaussian process Z is a Kiefer-Müller process indexed by [, r] L 2 X; is covariance operaor is defined, for f, g in L 2 X and s, in [, r], by s E Z, f Z, g = s Pfg PfPg, whence P Z P, f = d Z r, f d = Z s, f. r s r I follows by coninuiy ha for any r, Z, f y 0 = P f F r r s Z s, f y 0 which, since y ends o 0 as equals P r s Z s, f 2 y. In order o prove Proposiion 2, i reains o prove ha P fx i Pf r f F r y 0 = 0. This follows, as 2, selecing soe f in F and noing ha f F r i= fx i Pf [r] [r] i= fx i Pf. i=,

7 Las exi ies for a class of asypoically linear esiaors Las exi ies for adapive esiaes Le X n := X,..., X n be an i.i.d. saple wih X disribued by P θ,g on R k. The paraeer of ineres is θ, wih θ Θ, an open se, an g G, he se of nuisance paraeers. A locally asypoically linear esiaor T n of θ saisfies n T n θ n n JX i, θ n, g = o θn,g. 3 n i= In 3 θ n is any sequence such ha nθn θ = O 4 and J saisfies Jx, θ, gdp θ,g x = 0 5 ogeher wih Jx, θ, g 2 dp θ,g x <, 6 for all θ Θ and g G. All he o and O noaion are ean in probabiliy where rando variables are involved. The funcion J in 3 is he influence funcion for T n. An esiae S n of θ is adapive and efficien whenever here exiss a funcion Jx, θ, g such ha, for all sequence θ n saisfying 4, i holds n nsn θ n d = N 0, Σ θ,g where Σ θ,g is he covariance arix of JX, θ, g for fixed regular θ and fx,θ,g g. When he J funcion coincides wih he usual score funcion fx,θ,g wih f he densiy of P θ,g, 7 is he classical noral i behavior for ML esiaes under coniguiy of he sequence of easures P θn,g o P θ,g for all θ n saisfying 7 and g G, which we will assue fro now on. Regulariy of θ and g is defined in [Bickel, 982]. Adapive esiaes are efficien for all g G, even hough he knowledge of g ay no be used in he consrucion of he esiaes. Under he above seing, consrucions of efficien adapive esiaes have been proposed in [Beran, 978], [Schick, 986] and [Klaassen, 987] aong ohers. We will follow Klaassen s approach based on influence funcions and provide soe insigh on las exi ies for his esiaes, srenghening his assupions when needed. We jus need soe kind of uniforiy wih respec o g as saed now. Assue ha here exiss a sequence of funcions J n x, θ n, X n defined on R k, Θ, R kn and a funcion Jx, θ, g defined on R k, Θ, G, such ha J n x, θ n, X n dp θ n,gx = o θn 8 g G 7

8 870 Alagh e al. and n [J n x, θ n, X n Jx, θ n, g] 2 df θngx = o θn. 9 g G Assue furher ha we can consruc a sequence of preinary esiaes S n of θ such ha nsn θ = O θ. 0 Then we can consruc a uniforly locally asypoically linear adapive esiae T n of θ, saisfying herefore n T n θ n n JX i, θ n, g = o θn g G n i= for all sequence θ n saisfying 4. Display proves ha he esiae J n of he Influence funcion J, ogeher wih an iniial esiae of θ, provides explici esiaes of θ enjoying asypoic noraliy and second order efficiency in he sense of Rao. We now sae addiional condiions which enail soe knowledge n he las exi ie for he above adapive esiae T n. We assue ha Jx, θ, g is regular on a bounded open subse S of he iage of X, say I X, uniforly upon θ and g. Also J is assued o be consan ouside S. Such condiions enail robusness for T n. Precisely, assue H There exiss q > k/2 such ha θ,g J, θ, g W q 2 <, where W q 2 is he L 2 Sobolev nor of order q on S. H2 There exiss K > 0 such ha for all a in I X \ S, for all θ, g, Ja, θ, g = K. H3 I X is convex or is a counable union of convex ses wih non inersecing closures. Under H, H2 and H3 he class J of funcions J, θ, g is Donsker. When I X = [0, ] k, Theore 7.7. in [Dudley, 982], enails ha for soe K > 0, for any ε > 0, he enropy nuber of J saisfies log Nε, W q 2, J K e k/q. Denoe N ε := {n : T n θ > ε} where is he Euclidian nor. Applying Proposiion 3 yields Corollary Under all he above assupions, plus H, H2 and H3, i ε 0 ε 2 N ε d = θ,g 0 s Zs, J, θ, g 2 where Zs, f is he funcional Kiefer-Müller Process as defined in Secion. ii When I X = [0, ] k, hen here exiss soe posiive consan K such ha for all λ > 0, ε 0 Pε2 N ε > λ 2K λ +k/q ψ λ/σ y where σ 2 y := θ,g J 2 x, θ, gdp θ,g x.

9 References Las exi ies for a class of asypoically linear esiaors 87 [Adler, 990]R.J. Adler. An Inroducion o Coninuiy, Exrea and Relaed Topics for General Gaussian Processes. Lecure Noes Monograph Series IMS, 990. [Barbe e al., 999]P. Barbe, M. Doisy, and B. Garel. Las passage ie for he epirical ean of soe ixing processes. Sa. Prob. Leers, 403: , 999. [Beran, 978]R. Beran. An efficien and robus adapive esiaor of locaion. Annals of Saisics, 62:293 33, 978. [Bickel, 982]P.J. Bickel. On adapive esiaion. Annals of Saisics, 03:647 67, 982. [Bose and Chaerjee, 200]A. Bose and S. Chaerjee. Las passage ies of iniu conras esiaors. Jour. Ausral. Mah. Soc., 7: 0, 200. [Dudley, 982]R.M. Dudley. A Course on Epirical Processes. Lecure Noes in Maheaics, Springer, 982. [Hjor and Fensad, 992]N.L. Hjor and G. Fensad. On he las ie and he nuber of ies an esiaor is ore han ɛ fro is large value. Annals of Saisics, 820: , 992. [Klaassen, 987]A.J. Klaassen. Consisen esiaion of he influence funcion of locally asyoically linear esiaors. Annals of Saisics, 54: , 987. [Lai, 98]T.L. Lai. Asypoic opialiy of invarian sequenial probabiliy raio ess. Annals of Saisics, 9:38 333, 98. [Schick, 986]A. Schick. On asypoically efficien esiaion in seiparaeric odels. Annals of Saisics, 43:39 5, 986. [Sheehy and Wellner, 992]A. Sheehy and J.A. Wellner. Unifor Donsker classes of funcions. Annals of Saisics, 204: , 992. [Sue, 983]W. Sue. Las passage ies of M-esiaors. Scand. Jour. Saisics, 0:30 305, 983.