Adaptive estimation on anisotropic Hölder spaces Part II. Partially adaptive case

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1 Adaptive estimation on anisotropic Hölder spaces Part II. Partially adaptive case Nicolas Klutchnikoff To cite this version: Nicolas Klutchnikoff. Adaptive estimation on anisotropic Hölder spaces Part II. Partially adaptive case <hal > HAL Id: hal Submitted on 8 Apr 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Adaptive estimation on anisotropic Hölder spaces Part II. Partially adaptive case Nicolas Klutchnikoff March 8, 2006 Abstract In this paper, we consider a particular case of adaptation. Let us recall that, in the first paper Fully case, a large collecton of anisotropic Hölder spaces is fixed and the goal is to construct an adaptive estimator with respect to the absolutely unknown smoothness parameter. Here the problem is quite different: an additionnal information is known, the effective smoothness of the signal. We prove a minimax result which demonstrates that a knowledge of is type is useful because the rate of convergence is better than that obtained without knowledge of the effective smothness. Moreover we linked this problem with the maxiset theory. Introduction. Statistical model This paper is the second part of our paper Fully adaptive case. Further, we will refer to this paper as Part I). We consider the same model. Our observations X ε) = X ε u)) u [0,] d satisfies the same SDE: X ε du) = fu)du + εwdu), u [0, ] d, where f : R d R is an unknown signal to be estimated, W is a standartd Gaussian white noise from R d to R and ε is the noise level. Our main goal is to estimate f at a fixed point t 0, ) d. Université de Provence, LATP, UMR CNRS Mail: klutchni@cmi.univ-mrs.fr

3 .2 Our goal In this second part of our article, we study the Partially adaptive case. Let us recall that we are interrested in pointwise estimation among the class of anisotropic Hölder spaces. Let us recall some notations: l < l and b = b,...,b d ) are given. Moreover, we consider only Hölder spaces Hβ, L) defined in Part I) such that β B = d 0; b i ] and L I = [l ; l ]. i= Remark. Let us just recall that β = β,...,β d ) can be viewed as the smoothness parameter. Each β i represents the smoothness of a function in direction i. Moreover, L is a Lipschitz constant. We denote Σ = B I Hβ, L). Our goal is to answer this questions: Is it possible to guarantee a quality of estimation? On which space included in Σ)? With which procedure of estimation? For example, if we consider η ε γ) = ε 2γ/2γ+), it is well known that we can guarantee this quality on each space Hβ, L) such that β = γ because it is the minimax rate of convergence on this space) using the minimax on this space estimator. But one of our results implies that we cannot guarantee this quality simultaneousely on each such space. Now, we fix 0 < γ < b, and we consider η ε γ) = l ) 2γ+ K ε ln ln ε ) 2γ 2γ+. Our result is that there exists an estimator, namely fε γ ), such that η εγ) is the minimax rate of convergence of this estimator on Σγ) defined by Σγ) = Hβ, L) = Hβ, l ) where β,l) Bγ) I β Bγ) Bγ) = { β B : β = γ }. Thus, using f γ ε as procedure of estimation, we can guarantee that the quality is η ε γ), at least on Σγ). 2

4 .3 Result Theorem. Our result consists in two inequalities: lim sup lim inf sup f Σγ) inf f E f [ ηε γ) f γ ε t) ft) ) q] < +. sup E f [η ε γ) ft) ft) f Σγ) where the infimum is taken over all possible estimators. In words, fε γ is a minimax on Σγ) estimator. U.B.) ) q ] > 0, L.B.) This paper consists in the proof of this assertion. First, we construct the estimator fε γ. Next, we prove the corresponding lower bound. Remark 2. Let us remark that this result can be viewed as an adaptive result. Indeed, let us consider Σγ) as a family instead of an union of Hölder spaces Hβ, L) such that β = γ. It is well known that on each Hβ, L) there exists a minimax on this space estimator which depends explicitely on β, L) at least trough its bandwidth. Thus, question of adaptation arrizes naturally. Our lower bound proves that an optimal adaptive estimator f ) such that lim sup sup β,l) Bγ) I [ ) sup E f ε 2γ q ] 2γ+ f t) ft) < + f Hβ,L) does not exist. Our upper bound proves that fε γ ) is an adaptive estimator. Moreover the price to pay is only ln ln /ε which is to be compared with the classical loss ln /ε in other adaptive problems. Moreover we prove that our estimator is optimal in a minimax sense. 2 Procedure 2. Collection of kernel estimators Let us recall that kernels were defined in the first part of this paper: fully adaptive case. Here we have just to chose a good collection of kernel estimators. Let us define 4 b γ) n ε γ) = ln 2 2 b + )2γ + ) ln l K ε 2γ + ln ln ln ). ε 3

5 Let us denote Z ε γ = Zn εγ)). Let us recall the definition of this set: { } d Zn) = k = Z d : k i + ) = n and i, k i Cb)n +, where i= Cb) = 2 b + 2 b ln ln2. ln2 Finally, we consider the following collection { ˆf k )} k Z ε γ. 2.2 Notations Let us recall the following notation: for all k Zγ ε, we have σ ε k) = d ε K ) /2, i= hk) i where h k) = h k),..., h k) ) is defined by: d h k) i = K ε) 2 b 2 b+ b i 2 k i+). It is clear that for all k and l in Zγ ε, σ εk) = σ ε l) σ ε γ) and moreover that: σ ε γ) lnln ε η εγ). Following the same strategy as in the first part of our paper, let us define the set A as follows: an index k Zγ ε belongs to A if it satisfies: ˆf k l t) ˆf l t) Cσ ε γ) ln ln ε, l k, l Zε γ, where k l denote the index k i l i ) i=,...,d. 2.3 Definition of our procedure First of all, let us reformulate one of our result obtained in the first part of this paper: there exists an estimator, namely fε Φ ), sucht that ) lim sup sup E f ε ln 2 β 2q 2 β+ fε Φ t) ft) < +. ε sup β B f Hβ,l ) 4

6 Now, let us define our new estimator: If the random set A is non-empty, we chose arbitrary any index which belongs to this set. We denote ˆk a such index. Then we construct: ) = ˆfˆk ). On the other hand, if A is empty, we define f γ ε f γ ε ) = f Φ ε ). Remark 3. This procedure is closed to the adaptive one. The main difference consists in the following: when the set A is empty, we estimate using a best estimator than 0. In fact the probability P f [A = ] is too large to use a trivial estimator. 3 Proof of U.B) 3. Method First of all, let us recall that our minimax on Σ γ estimator is in fact an adaptive procedure of estimation because the real smoothness parameter is unknown. Thus, the mechanism of the proof is very closed to the previous one. We will compare the estimator chosen by our procedure with respect to the best estimator among our class but depending on the unknown parameter. First, we have to define correctly all indexes we need. Next, we will be able to prove the result. Moreover, as the class Σ γ depends only in L though l because Hβ, L) Hβ, l )), we will assume that l = l = to make the proof simpler. Consequently we will denote Hβ) instead of Hβ, ). 3.2 Indexes Let us suppose that our unknown signal in Σ γ belongs to Hβ) with β = γ. Clearly, if we consider the kernel estimator defined using bandwidth ) 2γ h i β, ε) = K ε ln ln 2γ+ β i, ε it achieves the expected rate η ε γ). We consider the bandwidth h 2 b i ε) = h ε) = K ε) 5 i=,...,d ) 2 b+ b i i=,...,d

7 and define the following indexes: for all i [; d ] and β B such that b = γ, we construct k i β, ε) = ln 2 ln h iε). h i β, ε) If h k) = h k) i ) i denote the bandwidth defined by h k) i = h i ε)2 k i+), we obtain clearly that the kernel estimator defined using bandwidth h kβ,ε)) is asymptotically as good as that one defined using hβ, ε). Now, let us define: { ki β, ε) if i =,...,d k i β, ε) = n ε γ) d i= k i β, ε) + ) otherwise It is easy to prove that k d β, ε) k d β, ε) d. Thus, asymptotically, estimator defined using h kβ,ε)) is as good as that one defined using h kβ,ε)) and thus as good as that one defined by hβ, ε). Moreover it is simple, by producing similar arguments than in the first part of this paper, to obtain that kβ, ε) belongs to Zn ε γ)). 3.3 Proof We want to prove that, for all ε < : sup sup β Bγ) f Hβ) E f [ η ε γ) f γ ε t) ft) ) q] < Mq γ) where M q γ) is an explicit constant given in the proof. Set ε < and β B such that β = γ. Let us suppose that f Hβ) Σ γ. Set q a fixed parameter. First, let us suppose that A is non empty. A) A is non empty Let us denote κ = kβ, ε). Our goal is to majorate the following quatiy: [ ) = E f ˆfˆkt) ] ft) q. Let us consider I = ˆfˆkt) ˆfˆk κ I 2 = ˆfˆk κ t) ˆf κ t) = ˆf κ t) ft) I 3 6

8 Let us remark that, if ˆk = κ, then I = I 2 = 0. Thus we can suppose that ˆk κ. a) Let us control of E f [I3]. q Using lemma??, we have: [ E f [I3] q = E f ˆf ] κ t) ft) q E f [ b κ t) ft) + σ ε γ) ξκ) ) q ] E f [ B β κ) + σ ε γ) ξκ) ) q] E f [C S ε κ) + σ ε γ) ξκ) ) q ] ) σ ε γ) ln ln q q E f C + ξκ) ε ln ln ε b) Let us control E f [I q 2 ]. Our procedure control itself this expectation. We have: E f [I q 2] C q σ ε γ) ln ln ε) q. Let us remark that we use the fact that κ belongs to Zγ ε. c) Finally, let us control E f [I q ]. Using lemma??, we ontain: [ E f [I] q = E f ˆfˆkt) ] ˆfˆk κ q ) q ] E f [2C S ε κ) + σ ε ˆk) ξˆk) + σ ε ˆk κ) ξˆk κ) q S ε κ) q E f 2C + ξˆk) + ξˆk κ) ln ln ε q E f 2C + ξˆk) + ξˆk κ) σ ε γ) ln ln ln ln ε ε Finally, we obtain the following inequelity: ) 3 q ) E f [I q ] + E f[i q 2 ] + E f[i q 3 ]) 3 q ) {C q + 2 q + )C ) q + o/ε)} σ ε γ) ) q ln ln ε) q, where o/ε) tends to 0 where ε tends to 0. It is clear by applying Lebesgue s theorem. B) A is empty 7

9 As A is empty, in particular κ does not belong to this set. Thus, we obtain: E f [ fε γ t) [ ] ft) q ] E f f Φ ε t) ft) q {κ/ A} E f [ fε Φ t) ft) 2q ]P f [κ / A] Using the upper bound of the first part of this paper we obtain: ) 2γ E f [ fε Φt) ft) 2q ] Cte ε ln 2γ+ q. ε Thus, we have to control P f [κ / A]. If κ / A, there exists l Zε γ, l κ, such that: ˆf κ l t) ˆf l t) > Cσ ε γ) ln ln ε. And, consequently, we obtain that: ] P f [κ / A] l κ P f [ ˆf κ l t) ˆf l t) > Cσ ε γ) lnln ε 2 κ l ln ε )C 2C ) 2 8 ) C 2C ) 2 2#Zγ) ε 8 ln. ε Moreover, it is easy to prove that there exists a constant C b depending only on b such that: #Z ε γ C b ln ε) d. On the other hand our choice of C implies that C 2C ) 2 8 = d + 2γ 2γ + 2q). Thus, we obtain: E f [ f γ ε t) ft) q ] Cte ε 2γ 2γ+ q 8

10 4 Proof of L.B.) 4. Method The method is classical. Our goal is to minorate the minimax risk by a bayesian risk taken on a large number ln /ε)of functions. In our mind, these functions are chosen because they represent the most difficult functions to be estimated in the considered class. This assertion is explained by lemma Notations Let us intoduce some basic notations. Let us fix 0 < γ < b. We say that a function g : R d R belongs to Gγ) if it satisfies: Here and later, we fix g Gγ). Let us denote d i=2 δ = b i d i=2 j i b = j We consider a = g0) > 0. g < + g β Bγ) Hβ) supp g [ a; a] d. /β /β d. γ ) < b, δ and we denote n ε = ln /ε. Now, let us consider a family of vectors {β k) } k indexed by k = 0,...,n ε and defined as follows: β k) = a + k b a β k) i = b i δ ) n ε ) i = 2,..., d. 2) γ β k) Lemma. For all k = 0,...,n ε the vector β k) belongs to Bγ). This lemma will be proved later. 9

11 Finally, let us introduce some functions. First of all, let us consider: i =,...,d, k = 0,..., n ε, h k) i = κε lnln ε ) 2γ 2γ+ β k) i where κ < / 2 g ). Then, we can define: f 0 0 f k x) = κ 2γ 2γ+ ηε γ)g x t h k),..., x d t d h k) d ), k. 4.3 Proof Now, let us prove our result. We will denote P k instead of P fk consider the likelyhood ratio: Z ε = n ε dp k X ε) ). n ε dp 0 k= and we This ratio satisfies the following lemma which will be proved further: Lemma 2. For all 0 < α <, we have: lim supp 0 [ Z ε > α] = 0. Let us consider for any arbitrary estimator f, the following quantiy: ) R ε f) = sup E f κε ln ln 2γ q 2γ+ f Σ γ ε ft) ft). It is a well known result that, using bayesian method, for all 0 < α < we obtain: ) q R ε f) g0) α) P 0 [ Z ε > α]). 2 Thus, we have: lim inf ) q R ε f) g0) α). 2 This inequality is equivalent to the following: lim inf [ sup E f ηε γ) ft) ) q ] ft) α) f Σ γ 0 κ 2γ 2γ+ g0) 2 ) q.

12 Now, if κ tends to 2 g ) and α tends to we obtain the lower bound: ) [ lim inf sup E f ηε γ) ft) ) q ] q ft) 2 +γ/2γ+)) g0) sup. f Σ γ g Gγ) g A Proof of lemma First of all, let us prove that a < b. In fact: But it is clear that Result follows. Let us fix β {β k) } k. Step. Let us calculate: d i= a < b b < γ δ. b + δ = b < γ. β i = β + = γ. d i=2 δ b i γ β Step 2. Let us prove that, for all i, β i > 0. First, we have β > a > 0. Next, for i 2, β i > 0 if /γ > β. But clearly we have β > a > γ. Result follows. Step 3. Let us prove that, for all i, β i b i. This inequality is equivalent to: δ γ ), β i.e. β a. Finally, β Bγ). ) B Proof of lemma 2 First, let us remark that: P 0 [ Z ε > α] α 2 E 0 [ Zε ) 2]

13 and, if, denotes the scalar product in L 2, E 0 [ Zε ) 2] = n 2 ε n ε k,l= It is enough to prove the following assertions: and n 2 ε lim sup n ε k= n 2 ε ) fk exp n ε k l First, let us prove Equation 3). Let us calculate f k 2 for all k. We have: Thus, we obtain: ε 2 ) fk, f l exp. ε 2 0, 3) ) fk, f l exp 4) f k 2 = g 2 κ 2 ε 2 lnln ε = 2 g 2 κ 2 ε 2 ln n ε. n 2 ε n ε k= ) fk exp ε 2 ε 2 = n 2 g 2 κ 2 ε. Thus, the choice of κ implies the result because 2 g 2 κ 2 < 0. Now, let us prove Equation 4). Let us fix k < l n ε. By an easy computation we obtain: f k, f l κ 4γ 2γ+ η 2 ε γ) g 2 VolC k C l ), where Vol is the standard volume in R d and C k denotes the support of f k : C k = d i= [ ah k) i ; ah k) i ]. Clearly, h k) < h l) and, for any i 2, we have h k) i conclude that: VolC k C l ) = 2a) dhk) h l) d i= h l) i ) > h l) i. Thus,w e can d ) 2a) d hk) h l) h k+) i. i= 2

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