Maxisets for μ-thresholding rules

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1 Test 008 7: DOI 0.007/s ORIGINAL PAPER Maxisets for μ-thresholding rules Florent Autin Received: 3 January 005 / Acceted: 8 June 006 / Published online: March 007 Sociedad de Estadística e Investigación Oerativa 007 Abstract In this aer, we study the erformances of a large class of rocedures, called μ-thresholding rules. At first, we exhibit the maximal saces or maxisets where these rules attain given rates of convergence when considering the Besov-risk. Then, we oint out a way to construct μ-thresholding rules for which the maxiset contains the hard thresholding rule s one. In articular, we rove that rocedures which consist in thresholding coefficients by grous, as block thresholding rules or thresholding rules with tree structure, outerform in the maxiset sense rocedures which consist in thresholding coefficients individually. Keywords Adative rocedures Besov saces and weak Besov saces Maximal sace Minimax risk Thresholding rules Mathematics Subject Classification 000 Primary 6G05 Secondary 6G0 65T60 Introduction Wavelet methods in nonarametric estimation of functions are renown for their adativity across a wide range of function classes. Frequently, the erformance of a wavelet estimator is measured thanks to the minimax aroach. The final outcome of this aroach is to exhibit estimates which attain the minimax rate of convergence of a functional sace V containing f. However, this oint of view seems to be essimistic, since it requires the knowledge of V. Recently, Cohen et al. 00 have suggested an alternative aroach to measure the erformance of an estimation rocedure which consists in exhibiting the functional sace maxiset over which an estimator attains a given rate of convergence. F. Autin Centre de Mathématiques et Informatique, Université d Aix-Marseille, 39 rue F. Joliot Curie, 3453 Marseille Cedex 3, France autin@cmi.univ-mrs.fr

2 Maxisets for μ-thresholding rules 333 This aroach is more otimistic than the minimax one, since it draws the strong connection between a given rocedure and a functional set for which the rocedure is well adated. In this way, Cohen et al. 00 and Kerkyacharian and Picard 000 have shown that Besov saces and weak Besov saces are directly connected with the hard thresholding rules out. More than this, they have roved that thresholding rules outerform linear rocedures, dealing with the density estimation model. In this aer, we investigate the erformances of a large class of rocedures: the μ-thresholding rules which can be viewed as a generalization of usual hard, global, and block thresholding rules. Each μ-thresholding rule is associated with a sequence of ositive functions μ jk j,k and consists in only keeing the emirical wavelet coefficients y jk for which μ jk λ, y λ where y λ reresents a articular set of emirical coefficients deending on λ are strictly larger than a threshold λ. Under the maxiset aroach and choosing the Besov risk, we exhibit the maximal saces where these rocedures attain given rates of convergence Theorem 3.. Then, we give a way to construct μ-thresholding rules at least as good as the hard thresholding rule in the maxiset sense Proosition 4. and give many examles of such rules. Particularly, we rove that block thresholding rules outerform in the maxiset sense the hard thresholding rule on condition that the length of their blocks are small enough Proosition 4.3. This result is imortant, since it allows us to give a theoretical exlanation about the good erformances of these estimators often observed in the ractical setting see Hall et al. 997, 999 and Cai 998, 999, 00. In the same way, we show that the thresholding rule using tree structure roosed by Autin 004 is another examle of a μ-thresholding rule which outerforms hard thresholding rule. In other words, thanks to the maxiset aroach, we rove in this aer that rules constructed with thresholding methods alied to grous of emirical coefficients are referable to rules based on thresholding methods alied to individual emirical coefficients. The aer is organized as follows. Section is devoted to the model and to the definition of μ-thresholding rules illustrated by some examles. In Sect. 3, we exhibit the maximal saces associated with such rocedures and discuss around. Section 4 aims at comaring the erformances of some articular μ-thresholding rules. Model and classes of estimators. Model We will consider a white-noise setting: X. is a random measure satisfying the following equation on [0, ]: where 0 <</ is the noise level f is a function defined on [0, ] W is a Brownian motion X dt = ftdt+ Wdt,

3 334 F. Autin Let φ 0k, ψ jk, j 0,k Z be a comactly suorted wavelet basis of L [0, ]. For the sake of simlicity, we suose that, for some a N, the suorts of φ and ψ are included in [0,a] and we denote ψ k to design φ 0k. Any f L [0, ] can be reresented as f = β jk ψ jk = f, ψ jk L ψ jk.. j a<k< j j a<k< j Let us suose that we disose of observations y jk = X ψ jk = β jk + ξ jk, where ξ jk are indeendent Gaussian variables N 0,. In the sequel, we set j λ λ to design the smallest integer j λ such that j λ λ and we denote t = log. In the following aragrah, we define the class of rocedures we shall study along the aer, the μ-thresholding rules.. Definition of μ-thresholding rules and examles For any λ>0, let us denote for any sequence y jk j,k and any sequence β jk j,k : y λ = y jk ; j, k I λ, β λ = β jk ; j, k I λ, where I λ = j, k; j<j λ, a <k< j and j λ λ. Remark. For any λ>0, the number #I λ of elements belonging to I λ is less than or equal to a j λ. This remark is imortant since it shall be often used in the roofs of results resented in the aer. Let us consider the following class of Kee-or-Kill estimators: F K = f ˆ = γ jk y jk ψ jk ; γ jk 0, measurable. j k Definition. We say that ˆ f μ F K is a μ-thresholding rule if j fˆ μ = j= μ jk λ,y λ >λ yjk ψ jk,. k where λ = mt, m>0, j λ, and μ jk λ, : R #I λ R + j,k is, for any λ>0, a sequence of ositive functions such that, for any t R and any y λ,β λ R #I λ R #I λ: μjk λ, y λ μ jk λ, β λ >t jo,k o I λ / y jo k o β jo k o >t..3

4 Maxisets for μ-thresholding rules 335 Let us notice that any μ-thresholding rule is a limited rocedure see Autin et al. 006 in the sense that the reconstruction of f by such a rocedure does not use the emirical coefficients y jk for which j j. The reconstruction of the signal f by a μ-thresholding rule consists in keeing the emirical coefficients y jk at level strictly less than j for which μ jk λ,y λ are strictly larger than the threshold λ, as we can see in the following scheme: There is no doubt that μ-thresholding estimates constitute a large sub-family of Kee-or-Kill estimates. Let us give some examles of such rocedures by choosing different choices of functions μ jk :. Hard thresholding rocedure belongs to the family of μ-thresholding rules. It corresonds to the choice μ jk λ,y λ = y jk. This rocedure has been roved to have good erformances in the minimax oint of view see Donoho et al. 995, 996, 997 and in the maxiset oint of view see Cohen et al. 00 and Kerkyacharian and Picard Block thresholding rocedures belong to the family of μ-thresholding rules. They corresond to the following choices: Mean-block thresholding μ jk λ,y λ = l j Maximum-block thresholding k P j k y jk, μ 3 jk λ,y λ = max k P j k y jk,

5 336 F. Autin Maximean-block thresholding where, for any j, k and any 0 <<, μ 4 jk λ,y λ = max y jk,μ jk λ,y λ, k P j k a,..., j #P j k = l j and k P j k P j k P j k = P j k Block thresholding estimators are known to have good erformances in the ractical setting. For examle, Hall et al. 997 considered mean-block thresholding. The goal was to increase estimation recision by utilizing information about neighboring wavelet coefficients. The method they roosed was to first obtain a near unbiased estimate of the sum of squares of the true coefficients within a block and then to kee or kill all the coefficients within the block based on the magnitude of the estimate. As well as the family blockwise James Stein estimators see Cai 998, 999, 00, on condition that the length of blocks is not exceeding C log C >0, this estimator was shown to have good erformances in the ractical setting see Hall et al. 997 and was roved to attain exactly the minimax rate of convergence for the L -risk without the logarithmic enalty over a range of erturbed Hölder classes Hall et al Maxisets associated with μ-thresholding rules Dealing with the Besov risk B, 0, we aim at exhibiting in this section the maximal saces where the μ-thresholding rules attain the rate of convergence uλ s/+s s > 0 and <, where u is an increasing transformation ma of R + into R + that is continuous and satisfies: 0 <</, uλ. 3. Remark 3. Even if the choice uλ = λ is often used, we choose here more general rates of convergence so as to integrate, for examle, logarithmic terms. 3. Functional saces To begin, we introduce the functional saces that will be useful throughout the aer when studying the maximal saces of μ-thresholding rules. Definition 3. Let s>0 and <. We say that a function f L [0, ] belongs to the Besov sace B, s u if and only if s su uλ j β jk <. λ>0 j j λ k Note that B, s Id R + is the classical Besov sace, which has been roved to contain the maximal sace of arbitrary limited rule for the rate λ s see Autin et al. 006.

6 Maxisets for μ-thresholding rules 337 Definition 3. Let 0 <r<<. We say that a function f belongs to the sace W μ,u r, if and only if r su uλ j λ>0 j<j λ k β jk μ jk λ, β λ λ <. The definitions of such saces in the case u = Id R + are close to the ones of weak Besov saces. Weak Besov saces have been roved to be directly connected with hard and soft thresholding rules see Cohen et al. 00 and Kerkyacharian and Picard 000. Definition 3.3 Let 0 <r<<. We say that a function f belongs to the sace Wμ,u r, if and only if: su λ uλ r log λ λ>0 j<j λ j k μ jk λ, β λ >λ <. In this aer, we shall see the strong relation between W μ,u r,, Wμ,u r, and μ-thresholding rules. The aim of the following aragrah is to exhibit the maxisets associated with the μ-thresholding rules. Undoubtedly, these maximal saces deend on the choice of the transformation ma u. 3. Main result The following theorem deals with the maximal saces associated to μ-thresholding rules. Theorem 3. Let < and m 4 +. Denote λ = mt and suose that ˆ f μ is a μ-thresholding rule such that μ jk jk are decreasing functions with resect to λ. If there exist K m > 0 and λ seuil > 0 such that 0 <λ<λ seuil, u4mλ K m uλ, 3. then we have equivalence between: i su uλ s/+s E fˆ μ f <, B 0<</, 0 ii f B, s/+s u W μ,u +s, Wμ,u +s,. With the usual maxiset notation, the revious equivalence can be formulated as follows: MS fˆ μ, uλ s/+s s/+s = B, u W μ,u + s, Wμ,u + s, Remark 3. When ut = t res. ut =, notice that 3. is satisfied by taking K m = 4m res. K m = 4 log log3m m and λ seuil = res. λseuil = 3m.

7 338 F. Autin Proof of Theorem 3. Here and later, we shall denote by C a constant which may be different from line to line. i ii It suffices to rove the result for 0 << seuil m 4, where seuil is such that t seuil = λ seuil. For any 0 << seuil m 4,wehave j j j k β jk E ˆ f μ f B 0, C uλ s/+s. So, using the continuity of λ in 0, we deduce that s/+s su uλ j λ>0 j j λ k β jk <. It comes that f B, s/+s u. Moreover, j β jk μ jk λ,β λ λ = A + A, where A = E j<j j k E j<j j k β jk μ jk λ,β λ λ μ jk λ,y λ λ β jk μ jk λ,y λ λ E ˆ f μ f B 0, C uλ s/+s, and, using.3, the concentration roerties of the Gaussian distribution and Remark., A = E j β jk μ jk λ,β λ λ μ jk λ,y λ >λ = j β jk P μ jk λ,y λ >λ μ jk λ,β λ λ j β jk P μ jkλ,y μ λ jkλ,β λ > λ = j β jk P j o,k o I λ y jo k o β jo k o > λ C j m 8 C uλ s/+s. The last inequality is due to the fact that m 8 +.

8 Maxisets for μ-thresholding rules 339 Using the continuity of λ in 0, we deduce that s/+s su uλ j β jk μ jk λ, β λ λ <. λ>0 j<j λ k It comes that f W μ,u +s,. Let us now consider the following lemma: Lemma 3. For all 0 < m 4, λ logλ / m. Proof Let 0 < m 4. Since log,wehavem log m. Hence, log λ = log m log log m So, λ log m 4 log. log λ / λ log / m. Using Lemma 3.,wehave λ log λ / j μ jk λ,β λ >λ C j μ jk λ,β λ >λ = CE j<j j k y jk β jk μ jk λ,β λ >λ = CA 3 + A 4 with A 3 = E j<j j k y jk β jk μ jk λ,β λ >λ μjk λ,y λ >λ, A 4 = E j<j j k A 3 = E j<j j k E j<j j k y jk β jk μ jk λ,β λ >λ μjk λ,y λ λ, y jk β jk μ jk λ,β λ >λ μjk λ,y λ >λ y jk β jk μ jk λ,y λ >λ E ˆ f μ f B 0, C uλ s/+s.

9 340 F. Autin Using the Cauchy Schwartz inequality,.3, and Remark., we get E yjk β jk μ jk λ,y λ λ μjk λ,β λ >λ E y jk β jk P μjk λ,y λ μ jk λ,β λ >λ E y jk β jk P j o,k o I λ y jo k o β jo k o >λ a j E y jk β jk P y jk β jk >λ, where E y jk β jk = C and P y jk β jk >λ m /. So, since m 4 +, from the concentration roerties of the Gaussian distribution and Remark. one gets A 4 = E j<j j k y jk β jk μ jk λ,β λ >λ μjk λ,y λ λ j<j j k E / y jk β jk P / μ jk λ,y λ λ μ jk λ,β λ >λ j<j j k E / y jk β jk P / j o,k o I λ y jo k o β jo k o >λ C j / m /4 C uλ s/+s. Using the continuity of λ in 0, we deduce that su λ uλ s/+s log λ / λ>0 j<j λ j k μ jk λ, β λ >λ <. It comes that f Wμ,u +s,. ii i For any 0 << seuil m 4,wehave E fˆ μ f = E j B, 0 yjk μ jk λ,y λ >λ βjk + j j j k β jk. Since f B s/+s, u, the second term can be bounded by Cuλ s/+s.

10 Maxisets for μ-thresholding rules 34 The first term E j<j j k y jkμ jk λ,y λ >λ β jk can be decomosed in B + B, where B = E j<j j k β jk μ jk λ,y λ λ, B = E j<j j k y jk β jk μ jk λ,y λ >λ. We slit B into B + B as follows: B = E j<j j k β jk μ jk λ,y λ λ μjk λ,β λ λ, B = E j<j j k β jk μ jk λ,y λ λ μjk λ,β λ >λ. Since f B s/+s, u W μ,u +s, and μ jk j,k are decreasing functions with resect to λ,using3., one gets: B = E j β jk μ jk λ,y λ λ μjk λ,β λ λ j β jk μ jk λ,β λ λ + j β jk j<j 4 k j j 4 k j β jk μ jk 4λ,β 4λ λ + j β jk j<j 4 k j j 4 k C u4λ s/+s C uλ s/+s. Using.3 and Remark., wehave B = E j<j j k β jk μ jk λ,y λ λ μjk λ,β λ >λ = j<j j k = j<j j k β jk P μ jk λ,y λ λ μjk λ,β λ >λ β jk P j o,k o I λ y jo k o β jo k o >λ C j j<j j k β jk P y jk β jk >λ C j m / C m / C uλ s/+s. We have used here the concentration roerty of the Gaussian distribution and the fact that m +.

11 34 F. Autin We slit B into B + B as follows: B = E j<j j k B = E j<j j k y jk β jk μ jk λ,y λ >λ μ jk λ,β λ λ y jk β jk μ jk λ,y λ >λ μ jk λ,β λ > λ. For B we use the Cauchy Schwartz inequality and Remark.: E y jk β jk μ jk λ,y λ >λ μ jk λ,β λ λ E y jk β jk P μ jkλ,y μ λ jkλ,β λ > λ E y jk β jk P j o,k o I λ y jo k o β jo k o > λ a j E y jk β jk P y jk β jk > λ, where E y jk β jk = C and P y jk β jk > λ m /8 using the concentration roerties of the Gaussian distribution. So, choosing m such that m 6 +, B = E j y jk β jk μ jk λ,y λ >λ μ jk λ,β λ λ C j/ j μ jk λ,β λ λ m /6 C j +/ m /6+ C uλ s/+s. Since f W μ,u B = E j<j j k +s,, we can bound B as follows: y jk β jk μ jk λ,y λ >λ μ jk λ,β λ > λ μ jk λ,β λ > λ C j λ 4 / C log j μ jk λ,β λ > λ 4 λ j<j +4 k s/+s λ C u C uλ s/+s. 4 The revious theorem oint out the maximal saces where μ-thresholding rules attain the rate of convergence uλ s/+s. We notice that the bigger are the,

12 Maxisets for μ-thresholding rules 343 functions μ jk, the larger are the saces W μ,u +s, and so the thinner are the saces Wμ,u +s,. In the next section, we give assumtions on the choices of u and μ jk j,k to be sure that we have the following embedding: B, s/+s u W μ,u + s, 3.3 Conditions for embedding inside maximal saces B, s/+s u Wμ,u + s,. Theorem 3. Let 0 <r<<, and let μ jk jk be a sequence of decreasing functions with resect to λ. Assume that there exist C seuil > 0 and λ seuil > 0 such that, for any 0 <λ<λ seuil, the following conditions are satisfied: j j<j λ k j<j λ j k μ jk λ, β λ >λ C seuil log λ β jk > n λ μ jk λ, β λ +n λ. 3.3 n N, C n > 0not deending on λ, u +n λ C n uλ, and C r n n <. 3.4 Then B r/, u W μ,u r, B, r/ u Wμ,u r,. Remark 3.3 It is easy to see that condition 3.4 imlies condition 3.. Once again, condition 3.4 is clearly satisfied when ut = t or ut =. Proof of Theorem 3. Let u satisfy condition 3.4, and let μ jk j,k satisfy condition 3.3. Fix 0 <λ<λ seuil and set, for any n N, j λ,n +n λ. Let f in. be such that f B, r/ u W μ,u r,. Using3.3, we have μ jk λ, β λ >λ j<j λ j k j μ jk λ, β λ >λ j<j λ k C log λ β jk > n λ μ jk λ, β λ +n λ C log λ C + C, j<j λ j k n λ j<j λ j k β jk μ jk λ, β λ +n λ

13 344 F. Autin where C = C log λ n λ j<j λ,n j k β jk μ jk λ, β λ +n λ and C = C log λ Since f W μ,u r,, C = C log λ C log λ k n λ n λ j<j λ,n j n λ j<j λ,n j j j λ,n j k β jk μ jk +n λ,β +n λ +n λ C log λ n λ u +n λ r C log λ λ u λ r C r C log λ λ uλ r. n n k β jk. β jk μ jk λ, β λ +n λ The two last inequalities use condition 3.4. Now, since f B, r/ u, C = C log λ n λ C log λ j j λ,n j k n λ u +n λ r C log λ λ uλ r C r n n β jk C log λ λ uλ r. Last inequalities use condition 3.4. By adding u C and C,wehave j j<j λ k μ jk λ, β λ >λ C log λ λ uλ r, which roves that f Wμ,u r, and ends the roof.

14 Maxisets for μ-thresholding rules 345 Corollary 3. Let s>0, <, and m 4 +. Let MS fˆ μ,uλ s/+s be the maximal set of any μ-thresholding rule fˆ μ for the rate of convergence uλ s/+s. Under the conditions of Theorem 3., we have MS fˆ μ, uλ s/+s s/+s = B, u W μ,u + s,. To rove this, it suffices to aly Theorems 3. and 3. with r = +s. It is clear that μ jk satisfies condition 3.3 of Theorem 3.. Consequently, if u satisfies 3.4, the maximal sace where the rocedure fˆ μ attains the rate of convergence uλ s/+s is B, s/+s u W μ,u + s,. Notice that, for u = Id R +, we identify B, s/+s u with the usual Besov sace B, s/+s = f ; su Js+ β jk <. J j J k For the same choice of u, W μ,u +s, reresents the weak Besov sace W + s, = f ; su λ r j β jk β jk λ <. λ>0 j<j λ k Let us recall that the maxiset of the hard thresholding rule λ s/+s ˆ f μ for the rate has already been studied by Cohen et al. 00 and Kerkyacharian and Picard Alications of results for articular μ-thresholding rules The aim of this section is twofold. First of all, we give a way to construct μ- thresholding rules with better erformances in the maxiset sense than the hard thresholding one fˆ μ. Then, we give some examles of such rules. Let us state the following roosition: Proosition 4. Let <. Under conditions of Theorem 3., the maximal sace for the rate uλ s/+s of any μ-thresholding rule satisfying, for any λ>0 and any β λ R #I λ, μ jk λ, β λ λ β jk λ, 4. is larger than the hard thresholding one. Moreover, if C n O n for all n N, then the maximal sace contains the Besov sace B s, u.

15 346 F. Autin Remark 4. For ut = t res. ut =, notice that, using Remark 3., thelast condition on C n is satisfied by taking C n = +n res. C n = 5 +n. Proof If fˆ μ is a μ-thresholding rule satisfying 4., then we have for any 0 <r< : W μ,u r, W μ,ur,. So, using Corollary 3. to characterize the maxisets for the rate uλ s/+s associated with fˆ μ and fˆ μ, one gets that the maximal sace for the rate uλ s/+s of fˆ μ is larger than the hard thresholding one. To rove now that if, for any n N, C n O n, the Besov sace B, s u is contained in the maxiset of fˆ μ, it suffices to rove that B s, u W μ,u + s,. Fix 0 <λ<λ seuil and set j λ,u λ u := uλ 4s/+s λ res. j λ λ.for any f B, s u, wehave j j<j λ k β jk β jk λ C jλ,u/ λ + j β jk j j λ,u k C uλ s/+s + uλu s = C uλ s/+s + D. Now, there exists n λ N such that +n λ uλ s/+s +n λ. Therefore, since C n O n for all n N, one gets: uλ u = u uλ s/+s λ u +n λ λ C n λ uλ Cuλ /+s. So D = uλ u s C uλ s/+s. Finally, we deduce that f W μ,u r,. 4. On block thresholding rules In the following roosition, we comare the maximal saces associated with the four examles of μ-thresholding rules defined in Sect... We rove that hard thresholding rules are outerformed by block thresholding rules when the length of the blocks are correctly chosen. Indeed: Proosition 4. For any < and any m 4 +, let MS ˆ f μ i, uλ s/+s, i 4,

16 Maxisets for μ-thresholding rules 347 be resectively the maximal sets of rocedures fˆ μ i for the rate of convergence uλ s/+s. Under the conditions of Theorem 3., we have the following inclusions of saces: MS fˆ μ, uλ s/+s MS f ˆ μ i, uλ s/+s, i 3, 4, 4. MS fˆ μ, uλ s/+s MS f ˆ μ 4, uλ s/+s MS ˆ f μ 3, uλ s/+s. 4.3 Proof Using Corollary 3.,wehave,forany i 4, MS fˆ μ i, uλ s/+s s/+s = B, u W μ i,u + s,. Now, for any f as in., we have max k P j k β jk λ β jk λ, max β jk, β jk λ β jk λ. l j k P jk So, using Proosition 4., the inclusion of saces 4. holds. In the same way, the inclusion of saces 4.3 also holds, since one has the both roerties: max β jk, β jk λ β jk λ, l j l k j P jk k P jk max β k jk λ max β jk, β jk λ. P j k l j k P jk The revious roosition is imortant. Indeed, we see that block thresholding rules can outerform hard thresholding ones when conditions 3.3 and 3.4 are satisfied. In articular, condition 3.3 is satisfied if the lengths of the blocks are chosen small enough, as we can see in the following roosition: Proosition 4.3 Under the maxiset aroach associated with the rate uλ s/+s, we have the following result: [Block thresholding rules] For any <, maximean- and maximumblock thresholding rules such that the lengths l j of the blocks P jk do not exceed Clog / for some C>0 outerform hard thresholding rules in the maxiset sense.

17 348 F. Autin Proof Note that it is just a consequence of the revious roosition, since the condition l j Clog / ensures that condition 3.3 of Theorem 3. is satisfied when dealing with block thresholding rules. This result is very interesting, since it allows one to give a theoretical exlication about the better erformances of block thresholding rules comaratively to hard thresholding rules which have been observed in the ractical setting see Hall et al. 997 and Cai 998, 999, 00 and which were not exlained with the minimax aroach. 4. On μ-thresholding rules based on hereditary constraints In this last aragrah, we aim at underlining that, aart from the examles of block thresholding rules mentioned above, there exist other μ-thresholding rules able to outerform the hard thresholding one. Indeed, Autin 004 has studied the maxiset erformance of a new rocedure called hard tree rocedure, combining thresholding rules and hereditary constraints. By making use of the dyadic structure of the wavelet bases, the author has ointed out the relationshi between this rocedure and Leski 99 s rocedure and has roved that the maxisets of this rocedure for usual rates are larger than the hard thresholding rocedure ones. There is no difficulty to rove that the hard-tree rocedure belongs to the class of μ-thresholding rules. To be more exlicit, let us use the same notation as in Autin 004 and denote, for any j,k: I jk the dyadic interval corresonding to the suort of ψ jk l ψ the maximal size of the suorts of the scaling function and the wavelet used for the reconstruction Then the hard-tree rule corresonds to the μ-thresholding rule associated with the sequence of ositive functions μ 5 jk j,k defined as follows: μ 5 jk λ,y λ = max y j k,i j k T jkλ, where, for any >0, T jk λ is the binary tree containing the set of dyadic intervals such that the following roerties are satisfied: I jk T jk λ I T jk λ I I jk and I >l ψ λ Two distinct dyadic intervals of T jk λ with same length have their interiors disjointed The numbers of dyadic intervals of T jk λ of length l ψ j j j <j is equal to j j Any set of all dyadic intervals of T jk λ with same length is forming a artition of I jk According to its definition, the hard-tree rule satisfies condition 3.3 of Theorem 3. and condition 4. of Proosition 4. see Autin 004. Hence, dealing with the maxiset aroach associated with any rate satisfying condition 3.4 of Theorem 3., the hard-tree rocedure outerforms the hard thresholding one in the maxiset sense.

18 Maxisets for μ-thresholding rules 349 As a conclusion, thanks to the maxiset aroach, we rove in this aer that rules constructed with thresholding methods alied to grous of emirical coefficients are referable to rules based on thresholding methods alied to individual emirical coefficients. References Autin F 004 Maxiset oint of view in non arametric estimation. PhD Thesis, Université Paris 7 Denis Diderot, France Autin F, Picard D, Rivoirard V 006 Large variance Gaussian riors in Bayesian nonarametric estimation: a maxiset oint of view. Math Methods Stat 54 Cai T 998 Numerical comarisons of BlockJS estimator with conventional wavelet methods. Unublished manuscrit Cai T 999 Adative wavelet estimation: a block thresholding and oracle inequality aroach. Ann Stat 73: Cai T 00 On block thresholding in wavelet regression: adativity, block size and threshold level. Stat Sin 4:4 73 Cohen A, De Vore R, Kerkyacharian G, Picard D 00 Maximal saces with given rate of convergence for thresholding algorithms. Al Comut Harmon Anal :67 9 Donoho D, Johnstone IM, Kerkyacharian G, Picard D 995 Wavelet shrinkage: asymtotia? J Roy Stat Soc Ser B 57: with discussion and rely by the authors Donoho D, Johnstone IM, Kerkyacharian G, Picard D 996 Density estimation by wavelet thresholding. Ann Stat 4: Donoho D, Johnstone IM, Kerkyacharian G, Picard D 997 Universal near minimaxity of wavelet shrinkage. In: Festschrift for Lucien Le Cam. Sringer, New York, Hall P, Kerkyacharian G, Picard D 999 On the minimax otimality of block thresholded wavelet estimators. Stat Sin 9:33 49 Hall P, Penev S, Kerkyacharian G, Picard D 997 Numerical erformance of block thresholded estimators. Stat Comut 7:5 4 Kerkyacharian G, Picard D 000 Thresholding algorithms, maxisets and well concentrated bases. Test 9: with comments, and a rejoinder by the authors Leski O 99 Asymtotically minimax adative estimation : uerbounds. Otimally adative estimates. Theory Probab Al 36:68 697