Adaptive global thresholding on the sphere. Claudio Durastanti a,1

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1 Adative global thresholding on the shere Claudio Durastanti a, a Fakultät für Matematik, Ruhr Universität, Bochum arxiv: v [math.st] 6 Jul 06 Abstract This work is concerned with the study of the adativity roerties of nonarametric regression estimators over the d-dimensional shere within the global thresholding framework. The estimators are constructed by means of a form of sherical wavelets, the so-called needlets, which enjoy strong concentration roerties in both harmonic and real domains. The author establishes the convergence rates of the L -risks of these estimators, focussing on their minimax roerties and roving their otimality over a scale of nonarametric regularity function saces, namely, the Besov saces. Keywords: Global thresholding, needlets, sherical data, nonarametric regression, U-statistics, Besov saces, adativity. 00 MSC: 6G08, 6G0, 65T60. Introduction The urose of this aer is to establish adativity for the L -risk of regression function estimators in the nonarametric setting over the d-dimensional shere S d. The otimality of the L risk is established by means of global thresholding techniques and sherical wavelets known as needlets. Let X, Y,..., X n, Y n be indeendent airs of random variables such that, for each i {,..., n, X i S d and Y i R. The random variables X,..., X n are assumed to be mutually indeendent and uniformly distributed locations on the shere. It is further assumed that, for each i {,..., n, Y i = f X i + ε i, where f : S d R is an unknown bounded function, i.e., there exists M > 0 such that su f x M <. x S d Moreover, the random variables ɛ,..., ɛ n in Eq. are assumed to be mutually indeendent and identically distributed with zero mean. Roughly seaking, address: claudio.durastanti@gmail.com Claudio Durastanti The author is suorted by Deutsche Forschungsgemeinschaft DFG - GRK 3, Highdimensional Phenomena in Probability Fluctuations and Discontinuity. Prerint submitted to Journal of Multivariate Analysis October 6, 08

2 they can be viewed as the observational errors and in what follows, they will be assumed to be sub-gaussian. In this aer, we study the roerties of nonlinear global hard thresholding estimators, in order to establish the otimal rates of convergence of L -risks for functions belonging to the so-called Besov saces... An overview of the literature In recent years, the issue of minimax estimation in nonarametric settings has received considerable attention in the statistical inference literature. The seminal contribution in this area is due to Donoho et al. [7]. In this aer, the authors rovide nonlinear wavelet estimators for density functions on R, lying over a wide nonarametric regularity function class, which attain otimal rates of convergence u to a logarithmic factor. Following this work, the interaction between wavelet systems and nonarametric function estimation has led to a considerable amount of develoments, mainly in the standard Euclidean framework; see, e.g., [3, 5, 4, 6, 7, 8, 30] and the textbooks [, 44] for further details and discussions. More recently, thresholding methods have been alied to broader settings. In articular, nonarametric estimation results have been achieved on S d by using a second generation wavelet system, namely, the sherical needlets. Needlets were introduced by Narcowich et al. [39, 40], while their stochastic roerties dealing with various alications to sherical random fields were examined in [, 6, 34, 35, 36]. Needlet-like constructions were also established over more general manifolds by Geller and Mayeli [8, 9, 0, ], Kerkyacharian et al. [5] and Pesenson [4] among others, and over sin fiber bundles by Geller and Marinucci [6, 7]. In the nonarametric setting, needlets have found various alications on directional statistics. Baldi et al. [] established minimax rates of convergence for the L -risk of nonlinear needlet density estimators within the hard local thresholding aradigm, while analogous results concerning regression function estimation were established by Monnier [38]. The block thresholding framework was investigated in Durastanti [9]. Furthermore, the adativity of nonarametric regression estimators of sin function was studied in Durastanti et al. [0]. In this case, the regression function takes as its values algebraical curves lying on the tangent lane for each oint on S and the wavelets used are the so-called sin ure and mixed needlets; see Geller and Marinucci [6, 7]. The asymtotic roerties of other estimators for sherical data, not concerning the needlet framework, were investigated by Kim and Koo [3, 3, 33], while needlet-like nearly-tight frames were used in Durastanti [8] to establish the asymtotic roerties of density function estimators on the circle. Finally, in Gautier and Le Pennec [5], the adative estimation by needlet thresholding was introduced in the nonarametric random coefficients binary choice model. Regarding the alications of these methods in ractical scenarios, see, e.g., [3, 4, 3], where they were fruitfully alied to some astrohysical roblems, concerning, for instance, high-energy cosmic rays and Gamma rays.

3 .. Main results Consider the regression model given in Eq. and let {ψ j,k : j 0, k =,..., K j be the set of d-dimensional sherical needlets. Roughly seaking, j and K j denote the resolution level j and the cardinality of needlets at the resolution level j, resectively. The regression function f can be rewritten in terms of its needlet exansion. Namely, for all x S d, one has f x = j 0 K j β j,k ψ j,k x, where {β j,k : j 0, k =,..., K j is the set of needlet coefficients. For each j 0 and k {,..., K j, a natural unbiased estimator for β j,k is given by the corresonding emirical needlet coefficient, viz. β j,k = n n Y i ψ j,k X i ; 3 i= see, e.g., Baldi et al. [] and Härdle et al. []. Therefore, the global thresholding needlet estimator of f is given, for each x S d, by ˆf n x = J n K Jn τ j β j,k ψ j,k x, 4 where τ j is a nonlinear threshold function comaring the given j-deendent statistic Θ j, built on a subsamle of < n observations, to a threshold based on the observational samle size. If Θ j is above the threshold, the whole j-level is ket; otherwise it is discarded. Loosely seaking, this rocedure allows one to delete the coefficients corresonding to a resolution level j whose contribution to the reconstruction of the regression function f is not clearly distinguishable from the noise. Following Kerkyacharian et al. [30], we consider the so-called hard thresholding framework, defined as τ j = τ j = { ˆΘ j B dj n /, where N is even. Further details regarding the statistic ˆΘ j will be discussed in Section 3.4, where the choice of the threshold B dj n / will also be motivated. For the rest of this section, we consider Θ j as an unbiased statistic of β j, + + β j,kj. The so-called truncation bandwidth J n, on the other hand, is the higher frequency on which the emirical coefficients ˆβ j,,..., ˆβ j,kj are comuted. The otimal choice of the truncation level is J n = ln B n /d ; for details, see Section 3. This allows the error due to the aroximation of f, which is an infinite sum with resect to j, to be controlled by a finite sum, such as the estimator ˆf n. 3

4 Our objective is to estimate the global error measure for the regression estimator ˆf n. For this reason, we study the worst ossible erformance over a so-called nonarametric regularity class {F α : α A of function saces of the L -risk, i.e., R n ˆfn ; F α = su E ˆf n f. L f F S d α Recall that an estimator ˆf n is said to be adative for the L -risk and for the scale of classes {F α : α A if, for every α A, there exists a constant c α > 0 such that E ˆf n f c L S d α R n ˆfn ; F α ; see, e.g., [,, 30]. For r > 0 and for [, r], we will establish that the regression estimator ˆf n is adative for the class of Besov saces B s,q, where q and d/ s < r +. Finally, let R 0, be the radius of the Besov ball on which f is defined. The roer choice of r will be motivated in Section.. Our main result is described by the following theorem. Theorem.. Given r,, let [, r]. Also, let ˆf n be given by Eq. 4, with J n = ln B n /d. Then, for q, d/ s < r + and 0 < R <, there exists C > 0 such that su E f Br,q s R ˆf n f L S d Cn s s+d. The behavior of the L -risk function will be studied searately in Section 3 and the analogous result is described in Theorem 3.. Moreover, the details concerning the choice of r will be resented in Remark 3. and other roerties of L -risk functions, such as otimality, will be discussed in Remark Comarison with other results The bound given in Eq. is consistent with the results of Kerkyacharian et al. [30], where global thresholding techniques were introduced on R. As far as nonarametric inference over sherical datasets is concerned, our results can be viewed as an alternative roosal to the existing nonarametric regression methods see, e.g., [, 9, 0, 38], related to the local and block thresholding rocedures. Recall that in local thresholding aradigm, each emirical estimator β j,k is comared to a threshold τ j,k and it is, therefore, ket or discarded if its absolute value is above or below τ j,k resectively, i.e., the threshold function is given by { β j,k τ j,k. Tyically, the threshold is chosen such that τ j,k = κ ln n/n, where κ deends exlicitly on two arameters, namely, the radius R of the Besov ball on which the function f is defined and its suremum M; see, e.g., Baldi et al. []. An alternative and artially data-driven choice for κ is roosed by Monnier [38], i.e., here κ = κ 0 n n ψ j,k X i. i= 4

5 Even if this stochastic aroach is roved to outerform the deterministic one, the threshold still deends on both R and M, which control κ 0. Also according to the results established on R see Härdle et al. [], local techniques entail nearly otimality rates for the L -risks over a wide variety of regularity function saces. In this case, the regression function f belongs to B,q s R, where s d/r, {,, q {, and 0 < R < cf. [, 0, ]. However, these adative rates of convergence are achieved on the exense of having an extra logarithmic term and of requiring exlicit knowledge of the radius of the Besov balls on which f is defined, in order to establish an otimal threshold. As far as the block thresholding is concerned, for any fixed resolution level this rocedure collects the coefficients ˆβ j,,..., ˆβ j,kj into l = l n blocks denoted B j,,..., B j,l of dimension deending on the samle size. Each block is then comared to a threshold and then it is retained or discarded. This method has exact convergence rate i.e., without the logarithmic extra term, although it requires exlicit knowledge of the Besov radius R. Furthermore, the estimator is adative only over a narrower subset of the scale of Besov saces, the so-called regular zone; see Härdle et al. []. The construction of blocks on S d can also be a difficult rocedure, as it requires a recise knowledge of the ixelization of the shere, namely, the structure of the subregions on which the shere is artitioned, in order to build sherical wavelets. On the other hand, the global techniques resented in this aer do not require any knowledge regarding the radius of Besov ball and have exact otimal convergence rates even over the narrowest scale of regularity function saces..4. Plan of the aer This aer is organized as follows. Section resents some reliminary results, such as the construction of sherical needlet frames on the shere, Besov saces and their roerties. In Section 3, we describe the statistical methods we aly within the global thresholding aradigm. This section also includes an introduction to the roerties of the sub-gaussian random variables and of the U-statistic Θ j, which are key for establishing the thresholding rocedure. Section 4 rovides some numerical evidence. Finally, the roofs of all of our results are collected in Section 5.. Preliminaries This section resents details concerning the construction of needlet frames, the definition of sherical Besov saces and their roerties. In what is to follow the main bibliograhical references are [,, 7,,, 4, 37, 39, 40]... Harmonic analysis on S d and sherical needlets Consider the simlified notation L S d = L S d, dx, where dx is the uniform Lebesgue measure over S d. Also, let H l be the restriction to S d of 5

6 the harmonic homogeneous olynomials of degree l; see, e.g., Stein and Weiss [43]. Thus, the following decomosition holds L S d = H l. An orthonormal basis for H l is rovided by the set of sherical harmonics {Y l,m : m =,..., g l,d of dimension g l,d given by g l,d = l + η d l + ηd, η d = d η d l. For any function f L S d, we define the Fourier coefficients as a l,m := Y l,m x f x dx, S d such that the kernel oerator denoting the orthogonal rojection over H l is given, for all x S d, by P l,d f x = g l,d m= l=0 a l,m Y l,m x. Also, let the measure of the surface of S d be given by ω d = π d+// d + Γ. The kernel associated to the rojector P l,d links sherical harmonics to the Gegenbauer olynomial of arameter η d and order l, labelled by C ηq l. Indeed, the following summation formula holds P l,d x, x = g l,d m= Y l,m x Y l,m x = l + η d η d ω d C η d l x, x, where, is the standard scalar roduct on R d+ ; see, e.g., Marinucci and Peccati [37]. Following Narcowich et al. [40], K l = l i=0 H i is the linear sace of homogeneous olynomials on S d of degree smaller or equal to l; see also [, 37, 39]. Thus, there exist a set of ositive cubature oints Q l S d and a set of cubature weights {λ ξ, indexed by ξ Q l, such that, for any f K l, S d f x dx = ξ Q l λ ξ f ξ. In the following, the notation a b denotes that there exist c, c > 0 such that c b a c b. For a fixed resolution level j and a scale arameter B, let 6

7 K j = card Q [B j+ ]. Therefore, {ξj,k : k =,..., K j is the set of cubature oints associated to the resolution level j, while {λ j,k : k =,..., K j contains the corresonding cubature weights. These are tyically chosen such that K j B dj and k {,...,Kj λ j,k B dj. Define the real-valued weight or window function b on 0, so that i b lies on a comact suort [ B, B ] ; ii the artitions of unity roerty holds, namely, j 0 b l/b j =, for l B; iii b C ρ 0, for some ρ. Remark.. Note that ρ can be either a ositive integer or equal to. In the first case, the function b can be built by means of a standard B-sline aroach, using linear combinations of the so-called Bernstein olynomials, while in the other case, it is constructed by means of integration of scaled exonential functions see also Section 4. Further details can be found in the textbook Marinucci and Peccati [37]. For any j 0 and k {,..., K j, sherical needlets are defined as ψ j,k x = l λ j,k b B j P l,d x, ξ j,k. l 0 Sherical needlets feature some imortant roerties descending on the structure of the window function b. Using the comactness of the frequency domain, it follows that ψ j,k is different from zero only on a finite set of frequencies l, so that we can rewrite the sherical needlets as ψ j,k x = λ j,k l Λ j b l B j P l,d x, ξ j,k, where Λ j = { u : u [ B j ], [ B j+] and [u], u R, denotes the integer art of u. From the artitions of unity roerty, the sherical needlets form a tight frame over S d with unitary tightness constant. For f L S d, f L S d = j 0 K j β j,k, where β j,k = f x ψ j,k x dx, 5 S d are the so-called needlet coefficients. Therefore, we can define the following reconstruction formula holding in the L -sense: for all x S d, f x = j 0 K j β j,k ψ j,k x. 7

8 From the differentiability of b, we obtain the following quasi-exonential localization roerty; for x S d and any η N such that η ρ, there exists c η > 0 such that c η B jd/ ψ j,k x { + B jd/ d x, ξ j,k, 6 η where d, denotes the geodesic distance over S d. Roughly seaking, ψ j,k x B jd/ if x belongs to the ixel of area B dj surrounding the cubature oint ξ j,k ; otherwise, it is almost negligible. The localization result yields a similar boundedness roerty for the L -norm, which is crucial for our uroses. In articular, for any [, there exist two constants c, C > 0 such that c B jd ψj,k L S d C B jd, 7 and there exist two constants c, C > 0 such that c B j d ψj,k L S d C B jd/. According to Lemma in Baldi et al. [], the following two inequalities hold. For every 0 <, K j β j,k ψ j,k cb jd βj,k l, 8 and for every, L S d β j,k l B jd c f L S d, where l denotes the sace of -summable sequences. The generalization for the case = is trivial. The following lemma resents a result based on the localization roerty. Lemma.. For x S d, let ψ j,k x be given by Eq... Then, for q, k i k i, for i i =,..., q, and for any η, there exists C η > 0 such that q B djq ψ j,ki x dx S d + B dj, ηq where i= = min dξ j,ki, ξ j,ki. i,i {,...,q,i i Remark.. As discussed in Geller and Pesenson [] and Kerkyacharian et al. [5], needlet-like wavelets can be built over more general saces, namely, over comact manifolds. In articular, let {M, g be a smooth comact homogeneous manifold of dimension d, with no boundaries. For the sake of simlicity, we assume that there exists a Lalace Beltrami oerator on M with resect to the 8

9 action g, labelled by M. The set {γ q : q 0 contains the eigenvalues of M associated to the eigenfunctions {u q : q 0, which are orthonormal with resect to the Lebesgue measure over M and they form an orthonormal basis in L M; see [0, ]. Every function f L M can be described in terms of its harmonic coefficients, given by a q = f, u q L M, so that, for all x M, f x = q a q u q x. Therefore, it is ossible to define a wavelet system over {M, g describing a tight frame over M along the same lines as in Narcowich et al. [40] for S d ; see also [, 5, 4] and the references therein, such as Geller and Mayeli [9, 0]. Here we just rovide the definition of the needlet scaling function on M, given by ψ j,k x = λ j,k j+ B b q=b j γq B j u q x ū ξ j,k, where in this case the set {ξ j,k, λ j,k characterizes a suitable artition of M, given by a ε-lattice on M, with ε = λ j,k. Further details and technicalities concerning ε-lattices can be found in Pesenson [4]. Analogously to the sherical case, for f L M and arbitrary j 0 and k {,..., K j, the needlet coefficient corresonding to ψ j,k is given by β j,k = f, ψ j,k L S d = λ j,k j+ B b q=b j γq B j a q u q ξ j,k. These wavelets reserve all the roerties featured by needlets on the shere: because, as shown in the following sections, the main results resented here do not deend strictly on the underlying manifold namely, the shere but rather they can be easily extended to more general frameworks such as comact manifolds, where the concentration roerties of the wavelets and the smooth aroximation roerties of Besov saces still hold... Besov sace on the shere Here we will recall the definition of sherical Besov saces and their main aroximation roerties for wavelet coefficients. We refer to [, 0,, 39] for more details and further technicalities. Suose that one has a scale of functional classes G t, deending on the q- dimensional set of arameters t T R q. The aroximation error G t f; concerning the relacement of f by an element g G t is given by G t f; = inf g G t f g L S d. Therefore, the Besov sace B,q s is the sace of functions such that f L S d and t {ts G t f; q <, t 0 9

10 which is equivalent to B j {G B j f; q <. j 0 The function f belongs to the Besov sace B s,q if and only if K j { β j,k ψ j,k L S d / = B js w j, 9 where w j l q, the standard sace of q-ower summable infinite sequences. Loosely seaking, the arameters s 0, and q of the Besov sace B,q s can be viewed as follows: given B >, the arameter denotes the -norm of the wavelet coefficients taken at a fixed resolution j, the arameter q describes the weighted q-norm taken across the scale j, and the arameter r controls the smoothness of the rate of decay across the scale j. In view of Eq. 7, the Besov norm is defined as f B s = f,q L S d + j 0 B jq{s+d/ / q/ K j β j,k = f L S d + B j{s+d/ / β j,k l lq, for q. The extension to the case q = is trivial. We conclude this section by introducing the Besov embedding, discussed in [, 9, 30] among others. For < r, one has or, equivalently, B s r,q B s,q and B s,q B s d/ /r r,q, K j K j β j,k β j,k r K /r j ; 0 K j β j,k r β j,k. K j Proofs and further details can be found, for instance, in [, 0]. /q 3. Global thresholding with sherical needlets This section rovides a detailed descrition of the global thresholding technique alied to the nonarametric regression roblem on the d-dimensional shere. We refer to [,, 30] for an extensive descrition of global thresholding methods and to [, 0] for further details on nonarametric estimation in the sherical framework. 0

11 3.. The regression model Recall the regression model given by Eq., i.e., for all i {,..., n, Y i = f X i + ε i. While {X,..., X n denotes the set of uniformly samled random directions over S d, {Y,..., Y n is the set of the indeendent observations which are related to {X,..., X n through the regression function f and affected by {ε,..., ε n, which is the set of the observational errors. The indeendent and identically distributed random variables ε,..., ε n are such that, for all i {,..., n, E ε i = 0, E ε i = σ ε <, and they are assumed to be sub-gaussian. Further details are given in Section 3.. Assume that f B,q, s d/ s < r +, r and q, where r is fixed, and that there exists R > 0 such that f B s,q R. As mentioned in Section. and Section, the regression function can be exanded in terms of needlet coefficients as f x = j 0 where β j,k are given in Eq. 5. K j β j,k ψ j,k x, Remark 3.. As discussed in Section.3, we do not require exlicit knowledge of the Besov radius R. Although it can be difficult to determine r exlicitly, we suggest the following criterion. Consider Remark.; if ρ <, we choose r = ρ see again [30]. If ρ =, we choose r = B djn+ emirically, using the so-called vanishing moment condition on S d, roerly adated for the needlet framework; see, e.g., Schröder and Sweldens [4]. 3.. The observational noise Following Durastanti et al. [0], we assume that ε,..., ε n follow a sub- Gaussian distribution; see also Buldygin and Kozachenko [4]. A random variable ε is said to be sub-gaussian of arameter a if, for all λ R, there exists a 0 such that Ee λε e a λ /. Sub-Gaussian random variables are characterized by the sub-gaussian standard, given by { ζ ε := inf a 0 : Ee λε e a λ /, λ R, which is finite. As roved in [4], ζ ε = su λ 0 { / ln E e λε { λ λ ; Ee λε ζ ε ex.

12 Following Lemma.4 in [4], for > 0 E ε = 0; E ε / ζ ε ; E ε ζ ε. ex Therefore, sub-gaussian random variables are characterized by the same moment inequalities and concentration roerties featured by null-mean Gaussian or bounded random variables. Remark 3.. In order to establish the robabilistic bounds described in Sections 3.3 and 3.4, it would be sufficient for ε,..., ε n to be null-mean indeendent random variables with finite absolute th moment. However, we include the notion of sub-gaussianity in order to be consistent with the existing literature; see [0]. Furthermore, sub-gaussianity involves a wide class of random variables, including Gaussian and bounded random variables and, in general, all the random variables such that their moment generating function has a an uer bound in terms of the moment generating function of a centered Gaussian random variable of variance a. Hence the term sub-gaussian The estimation rocedure We note again that the method established here can be viewed as an extension of global thresholding techniques to the needlet regression function estimation. In this sense, our results are strongly related to those resented in [, 0, 30], as discussed in Section.3. For any j 0 and k {,..., K j, the emirical needlet estimator is given by β j,k = n Y i ψ j,k X i, n and it is unbiased, i.e., E β j,k = n i= n [E{f X i ψ j,k X i + Eε i E{ψ j,k X i ] = β j,k. i= The emirical needlet coefficients are moreover characterized by the following stochastic roerty. Proosition 3.. Let β j,k and β j,k be as in Eq. 5 and Eq. 3, resectively. Thus, for, there exists c such that E ˆβ j,k β j,k c n /. Therefore, we define the global thresholding needlet regression estimator at every x S d by J n ˆf n x = τ j β j,k ψ j,k x ; see Eq. 4. Recall now the main results, stated in Section. k

13 Theorem.. Given r,, let [, r]. Also, let ˆf n be given by Eq. 4, with J n = ln B n /d. Then, for q, d/ s < r + and 0 < R <, there exists C > 0 such that su E f Br,q s R ˆf n f L S d Cn s s+d. In the nonarametric thresholding settings, the L -risk is generally bounded as follows E ˆf n f L S d E J n β j,k β j,k ψ j,k L S d + β j,k ψ j,k j J n = S + B, L S d where S is the stochastic error, due to the randomness of the observations and B is the deterministic bias error. The so-called truncation level J n is chosen so that B djn = n. In this case the bias term term does not affect the rate of convergence for s d/, r +. As far as S is concerned, its asymtotic behavior is established by means of the so-called otimal bandwidth selection, i.e., a frequency J s such that B djs = n s+d ; see [, 30]. Note that trivially, Js < J n. The meaning and the relevance of the otimal bandwidth selection is given in Section 5, in the roof of Theorem.. However, in the next section it will also be crucial for the construction of the threshold function. Consider now the case =. First, we have to modify the threshold function given in Eq. 3 slightly, in view of the exlicit deendence on. Hence, in the selection rocedure we use the statistic Θ j = Θ j = ˆβ j, + + ˆβ j,kj, which will be comared to B dj n /. Further details on the threshold will also be given in the next section. Under this assumtion, we obtain the following result. Theorem 3.. Let ˆf n by given by Eq. 4. Given r,, for any d < s < r +, there exists C > 0 such that su f Br,q s E ˆf n f L S d Cn s d s+d. Remark 3.3. As far as otimality is concerned, Eq. given in Theorem., achieves the same otimal minimax rates rovided on R by Kerkyacharian et al. [30], where the established otimal rate of convergence was given by n s/+. Moreover, this rate is consistent with the results rovided over the so-called regular zone by [, 9, 0, 38] for local and block thresholding estimates by needlets 3

14 on the shere, where the rates are nearly otimal due to the resence of a logarithmic term. Regarding the L -risk function, according to Theorem 3., the rate established is not otimal; see, e.g., Baldi et al. []. In the global thresholding aradigm, a straightforward generalization of the thresholding function Θ j given by Eq. 3 is not available see Remark 3.4. Therefore, the uer bound for the case = is established in a different framework, which can be reasonably assumed to cause the lack of otimality The threshold The construction of the threshold function τ j is strictly related to Efromovich [] and Kerkyacharian et al. [30], where analogous results were established in the real case. Let K j Θ j = β j,k. Using Eq. 9, it follows immediately that, if f B s,q, Θ j CB j{s+d. Consider now the otimal bandwidth selection J s. If j J s, B j{s+d Bdj n /. Thus, even if j J s doesn t imly Θ j > B dj /n /, according to [, 30], one has that Θ j B dj /n / imlies j J s. Clearly, the case { Θ j B dj /n /, j J s rovides no guarantee of a better erformance if comared to the linear estimate, whose error is of order B dj /n / ; see Härdle et al. []. Thus, the natural choice is to construct a threshold function that kees the level j if and only if Θ j Bdj n /. As ointed out in [, 30], the natural estimator ˆβ j, + + ˆβ j,kj for Θ j yields an extremely large bias term and, therefore, undersmoothing effects. Hence, following the rocedure as suggested in [, 30] see also [] but roerly adated to the needlet framework see Lemma., we roose an alternative method as described below. Let N be even, Σ denoting the set of -dimensional vectors chosen in {,..., n and also let υ = i,..., i be the generic element belonging to Σ. Define the U-statistic Θ j by Θ j = n K j υ Σ Ψ j,k X υ, ε υ, 3 4

15 where Ψ j,k X υ, ε υ = {f X ih + ε ih ψ j,k X ih. h= Given that the sets of variables {X,..., X n and {ε,..., ε n are indeendent, it can be easily seen that K j E{ ˆΘ j = β j,k = Θ j. 4 Remark 3.4. As mentioned in the revious section, if we consider the case =, we lack a straightforward extension of Θ j. Hence, we choose K j Θ j = ˆβ j,k. so that the threshold function is given by τ j = {Θ j Bdj n / Our urose is to establish two robabilistic bounds related to the mth moment and to the mth centered moment Θ j, resectively. We have that m E[{ ˆΘ n { m j m ] = Ψ j,k l X υl, ε υl. υ,...,υ m Σ E. k,...,k m l= For any fixed configuration υ,..., υ m, let the sequence c,..., c m denote the cardinality of the E [{f X + ɛψ j,k X] l of size l aearing in E{ Θ j. Observe that m lc l = m. l= Following Kerkyacharian et al. [30], the next results hold. Proosition 3.3. Let Θ j be given by Eq. 3. Also, let be an even integer. Then, for m N, there exists C such that E[{ ˆΘ j m ] C c,...,c m Γ m, B jd where Γ m, = {c,..., c m : m l= lc l = m. n m m l= c l B j{c s d d γ n m/, Proosition 3.4. Let Θ j and Θ j be given by 3 and 4 resectively. Also, let, m be even integers. Then, there exists C such that E{ ˆΘ j Θ j m C B jd mh {Θ j h/m. h= n / 5

16 Remark 3.5. According to [30], this rocedure can be easily extended to the case of is not being an even natural number, by means of an interolation method. Indeed, by fixing, N, both even, we can rewrite = δ + δ, as Θ j = { ˆΘ j δ { ˆΘ j δ. The following lemma is crucial for the alication of our interolation method. As in [30], we consider for the sake of simlicity just the case =, so that 0 0, 0 = 0 +. Lemma 3.5. For any even m N, { E n ˆΘ m j 0 Θ j 0 C 0 h= {Θ j 0 h m n mh/. We conclude this section with a result regarding the behavior of Θ j. Proosition 3.6. Let Θ j and Θ j be given by 3 and 4 for =, resectively. Then, there exists C such that E Θ j Θ j C B dj /n. 4. Simulations In this section, we resent the results of some numerical exeriments erformed over the unit circle S. In articular, we are mainly concerned with the emirical evaluation of L -risks obtained by global thresholding techniques, which are then comared to the L -loss functions for linear wavelet estimators. As in any ractical situation, the simulations are comuted over finite samles and therefore, they can be considered as a reasonable hint. Furthermore, they can be viewed as a reliminary study to ractical alications on real data concerning estimation of the ower sectrum of the Cosmic Microwave Background radiation; see, e.g., Faÿ et al. [4]. The needlets over the circle used here are based on a weight function b which, analogously to Baldi et al. [], is a roerly rescaled rimitive of the function x e / x ; see also Marinucci and Peccati [37]. Following Theorem., we fix B = and n = 6, 7, 8 and J n = 6, 7, 8, resectively. The U-statistic Θ j, corresonding to the L risk considered here, results in considerable comutational effort, because it is built over,06, 8,8 or 3,640 ossible combinations of needlet emirical coefficients for J n = 6, 7, 8, resectively. By choosing a test function F and fixing the set of locations X,..., X n, we obtain the corresonding Y,..., Y n by adding to F X i a Gaussian noise, with three different amlitudes, i.e., the noise standard deviation σ ε is chosen to be equal to 0.5M, 0.5M or 0.75M, where M is the L -norm of the test function. Therefore, the following three numerical exeriments are erformed. 6

17 Examle 4. Global Linear J n \σ ε 0.5M 0.50M 0.75M 0.5M 0.50M 0.75M Table : Examle 4. - Values for L risk Examle 4.. According to Baldi et al. [], we use the uniform test function defined, for all x S, by F x = 4π. In this case, for every j, k, we get β j,k = 0. The erformance of our model can be roughly evaluated by simly controlling how many resolution levels ass the selection rocedure. For all the choices of n and σ, we get τ j = 0 for all j {,..., J n. On the other hand, we consider a finite number of resolution levels and therefore of frequencies. Thus, it is ossible that higher resolution levels, involving higher frequencies, could be selected by the thresholding rocedure. Examle 4.. In this examle, we choose the function defined, for all x S, by F x = cos 4x. In this case, the test function corresonds to the real art of one of the Fourier modes over the circle with eigenvalue 4. This choice allows us to establish whether the thresholding rocedure is able to select only the suitable corresonding resolution levels, as the amlitude of the noise increases. As exected, for every n and for every σ ε, we get τ j = for j = containing the frequency k = 4 and 0 otherwise. Table resents the value for L -risks for different values of J n and σ ε, while Figure illustrates the grahical results for the case J n = 8 and σ ε = 0.5M. Examle 4.3. A more general function is chosen here, which is defined, for all x S, by { F 3 x = e x 3π/ + e x sin x, deending on a larger set of Fourier modes. In this case, Table gives the values of L -risks corresonding to different J n and σ ε, while Figure resents the grahical results for the case J n = 8 and σ ε = 0.5M. Table 3 contains, for every air J n, σ ε, the resolution levels selected by the rocedure. 7

18 Test function Test function + noise Linear needlet estimator Global thresholding needlet estimator Figure : Examle 4. J n = 8, σ ε = 0.5M Examle 4.3 Global Linear J n \σ ε 0.5M 0.50M 0.75M 0.5M 0.50M 0.75M NA NA Table : Examle 4.3 Values for L risk Examle 4.3 Selected j J n \σ ε 0.5M 0.50M 0.75M 6,,,3 7,4, 8,,3,5 Table 3: Examle Values of the function τ j 8

19 Test function Test function + noise Linear needlet estimator Global thresholding needlet estimator Figure : Examle J n = 8, σ ε = 0.5M 5. Proofs In this section, we rovide roofs for the main and auxiliary results. 5.. Proof of the main results The roofs of Theorem. and Theorem 3. follow along the same lines as the roof of Theorem 8 in Baldi et al. []. Proof of Theorem.. Following, for instance, [, 7, 9, 0,, 30] and as mentioned in Section 3.3, the L -risk E ˆf n f can be decomosed as the L S d sum of a stochastic and a bias term. More secifically, E ˆf J n f L S d E n K j τ j βj,k β j,k ψ j,k L S d K j + β j,k ψ j,k. j>j n L S d Using the definition of Besov saces, we obtain for the bias term the following inequality: K j β j,k ψ j,k K j β j,k ψ j,k CB sjn Cn s s+d. j>j n L S d j>j n L S d 9

20 Following Baldi et al. [], the stochastic term can be slit into four terms, i.e., J n K j E τ j βj,k β j,k ψ j,k 4 Aa + Au + Ua + Ua, where Aa = E Au = E Ua = E Uu = E J n K j L S d τ j βj,k β j,k ψ j,k { Θ j Bdj J n K j n / τ j βj,k β j,k ψ j,k { Θ j Bdj J n K j n / τ j βj,k β j,k ψ j,k { Θ j Bdj J n K j n / τ j βj,k β j,k ψ j,k { Θ j Bdj n / {Θ j Bdj n / {Θ j Bdj n / {Θ j Bdj n / {Θ j Bdj n / L S d L S d L S d L S d Following Durastanti et al. [0], the labels A and U denote the regions where Θ j is larger and smaller than the threshold B dj n / resectively, whereas a and u refer to the regions where the deterministic Θ j are above and under a new threshold, given by B dj n / for a and B dj n / for u. The decay of Aa and Uu deends on the roerties of Besov saces, while the bounds on Au and Ua deend on the robabilistic inequalities concerning β j,k and Θ j, given in Proositions 3., 3.3 and 3.4. Let N be even. Then, using the definition of the otimal bandwidth selection, we have J n K j { Aa C J n + ψ j,k L S d E ˆβ j,k β j,k Θ j Bdj C C J s J n K j B jd/ n / + n / B dj/ Θ j B dj n / j=j s J n K j BJsd/ n / + β j,k ψ j,k L S d j=j s C B Jsd/ n / + B Jss = C n s s+d,,,, n /. 0

21 given that where d B Jsd/ n / = n s+d / = n s+d. 5 Similarly, for the region Au we obtain K j Au C Au + Au, J s Au = ψ j,k L S d E ˆβ j,k β j,k, Au = J n K j j=j s { ψ j,k L S d [ E ˆβ j,k β j,k Θ j Using Eq.s 7 and 5, it is easy to see that Au C n s s+d. s ] Bd j. n / Regarding Au, using Hölder inequality with /α + /α =, the generalized Markov inequality with even m N and Proosition 3.3, we obtain Au C C J n K j j=j s J n K j j=j s ] ψ j,k L S d [ E ˆβ j,k β j,k { Θ j Bdj n / B jd/ { E ˆβ j,k β j,k α [ ]] /α /α Pr [{ Θ j Bdj n / ] J n C B jd/ n / E [{ Θj m j=j B dj /n / m s J n d C B jd n Bj n / j=j s Au, + Au,, c,...,c m Γ m, /α m m l= c l B j{cs d B dj{ m γ where Au, and Au, are defined by slitting Γ m, into two subsets, Γ + m, and Γ m,. These subsets are defined as { Γ + m, := Γ m, := c,..., c m : m m c m 0 ; i=i { c,..., c m : m m c m 0. i=i /α

22 Note that γ [0, ]. Hence we choose m, α so that m > + α/. It can be easily verified that Au, C J n On the other hand, Au, C and therefore, B j d j=j s C d Js B C n s s+d c,...,c m Γ m, C B dj s n C B dj s n C n s s+d, n / [ n / B Js c s d/ α / c,...,c m Γ m, / c,...,c m Γ m, Consider now Ua. We have that J s Ua C 3 K j β j,k ψ j,k = Ua + Ua. It is easy to see that + c,...,c m Γ + m, c,...,c m Γ + m, B dj{ m γ] /α B djs m J n B B jd/ n / jd n j=j s B Js c α s d B J sd n /α m m m m l= c l/αb m γ α l= c l/α B m γ α n c d s αs+d {n s+dα m m l= c d m γ l n αs+d Au C n L S d J n j=j s s s+d. [ ] E { Θ j Bdj {Θ n / j Bdj n / K j β j,k ψ j,k L S d Ua C 3 B Jss = C 3 n s s+d. On the other hand, using the generalized Markov inequality, Proosition 3.4 dj s dj

23 with m = and Eq. 5, we have that Ua = and therefore, J s [ ] B dj/ E Θ j { Θ j Bdj {Θ n / j Bdj n / J s { C B dj/ Θ j Pr ˆΘ j Θ j Θ j {Θ j Bdj n / J s C B dj/ Θ m j E ˆΘ j Θ j m {Θ j Bdj n / J s C B dj/ Θ j l= J s dj/ Bdj C B n / Cn s s+d, Θ j B dj n / Ua C 3 n s s+d. ml Finally, in view of Eq. 7 and Eq. 5, we have that J s Uu C 4 B jd/ Θ j {Θ j Bdj + n / J s J n C 4 B jd/ n / + C 4 n s s+d. j=j s B js J n j=j s K j β j,k ψ j,k L S d We now need to extend these results to any, r using the interolation method described in Remark 3.5. The two terms that have to be studied searately are Au and Ua, in articular Au and Ua, since they involve the robabilistic inequalities described in Proositions 3.3 and 3.4, holding only for even N. According to [30], the generalization in the case of Au is obtained by bounding [{ m ] [{ m ] δ [{ m ] δ E ˆΘj CE ˆΘj E ˆΘj 3

24 Indeed, Au C J n j=j s B j d n c,...,c m Γ m, B jd ] ] /α δ [ B jc s d B dj{ m γ B j d n B jd c,...,c m Γ m, n n m m l= c l B jc s d The result above follows from Eq.s 0 and, so that B s,q B s,q; B s,q B s d,q. m m l= c l B dj{ m γ /α δ Straightforward calculations lead to the claimed result. On the other hand, in order to study Ua, we aly Lemma 3.5 to obtain J s [ ] Ua C B dj/ E Θ j { Θ j Bdj {Θ n / j Bdj n / J s C [ { B dj/ E Θ j ˆΘ j 0 Bdj n 0 { ˆΘ j 0 Bdj {Θ ] j Bdj n 0 n / {Θ j Bdj n / J s { C B [Pr dj/ ˆΘj 0 Θ j 0 Θ j 0 { + Pr n ˆΘ j 0 Θ j 0 Θ j ] 0, { { because Θ j Bdj Θ n / j 0. Bdj Finally, by alying Markov 0 n. 4

25 inequality with m >, m N even, we have that J s Ua C 0 B dj/ h= {Θ j 0 h m n mh { {Θ j 0 h m Θ j 0 Bdj J s C 0 B dj/ h= + mh 0 0 B dj J s C Cn 0 B dj/ s s+d n 0 mh {Θ j 0 0 n mh { Θ j 0 Bdj h= B dj n 0 mh 0 n 0 n mh B dj + mh 0 0 Proof of Theorem 3.. Similarly to the revious roof, note that E ˆf J n f L S d C E n K j τ j βj,k β j,k ψ j,k K j + β j,k ψ j,k. j>j n L S d L S d If f B,, s β j,k MB js+ d for any k =,..., Kj, then by Eq.s 7 and 8 with =, we get K j β j,k ψ j,k K j β j,k ψ j,k j>j n L S d j>j n L S d C su β j,k ψ j,k L S d j>j,...,k j n C B js = O n s d = O n s s+d. j>j n As far as the other term is concerned, following the same rocedure as described in the roof of Theorem., see also [, 0], we obtain J n K j E τ j βj,k β j,k ψ j,k C Aa + Au + Ua + Uu, L S d 5

26 where Aa= Au= Ua= Uu= J n j= J n j= J n j= J n j= K j { E τ j βj,k β j,k ψ j,k Θ j Bdj n K j { E τ j βj,k β j,k ψ j,k Θ j Bdj n K j { E β j,k ψ j,k Θ j Bdj {Θ j Bdj n n K j { E β j,k ψ j,k Θ j < Bdj {Θ j < Bdj n n {Θ j Bdj n {Θ j < Bdj n L S d L S d L S d L S d Now Θ j B dj /n imlies j J s see Section 3.4 and in view of Eq.s 7 and 8 with =, we get J s Aa C B d j E su ˆβ j,k β j,k,...,k j j= CB d Js J s + n =O n s s+d Consider now Au. Using J s, we slit this term into Au = Au + Au, as in the roof of Theorem.. Trivially, we get Au = O n s+d. On the other hand, using Eq. 8 and Proosition 3.6, we get Au =O J n J n B d j E su ˆβ j,k β j,k,...,k j B d j j + n B d j J n n = o n s s+d s Pr Θ j Θ j Bdj n As far as Ua is concerned, again Θ j B dj /n imlies j J s see Section 3.4, 6

27 so that J s Ua j= B j d K j β j,k ψ j,k J s B j d M Pr Θ j j= J s B j d B dj M j= J s n, n L S d Θ j Pr Θ j Θ j Bdj n E Θ j Θ j Bdj n where we used Eq. 8 and Proosition 3.6. Finally, we have that J n K j { Uu E β j,k ψ j,k Θ j < Bdj {Θ j < Bdj n n j= Uu + Uu, L S d where and Note that J s K j Uu β j,k ψ j,k j= L S d J s B 3d j n = O B 3d Js n j= Uu = j>j s B d B 3d Js n {Θ j < Bdj n j su β j,k.,...,k j = n d s s+d, as claimed. 5.. Proofs of the auxiliary results The roof of Lemma. can be viewed as a generalization of the roof of Lemma 5. in Durastanti et al. []. Proof of Lemma.. Using the needlets localization roerty given in Eq. 6, we have that q q ψ j,ki x dx C η B jdq { + B jq d x, ξ j,ki η. S d i= 7 S d i=

28 Let S = { x S d : d x, ξ j,k /, so that S d S S. Therefore, q S d i= { + B jq d x, ξ j,ki η S + q { + B jq d x, ξ j,ki η q { + B jd d x, ξ j,ki η. i= S i= From the definition of S and following Lemma 5. in Durastanti, Marinucci and Peccati [], it follows that q ηq { + B jd d x, ξ j,ki η + B jd ηd ηq + B jd ηd S i= On the other hand, S i= = πd ηq + B jd ηd πd ηq B dj + B jd ηd { + B jd d x, ξ j,k η dx S { + B jd d x, ξ j,k η dx S d π sin ϑ + B jd ϑ η dϑ 0 C η π d ηq B dj. + B jd ηd q η { + B jd d x, ξ j,ki η +B jd η Let S = {x S : d x, ξ j,k /. Then, S i= S S S and S d S S S. 0 y + y η dy q dx { + B jd d x, ξ j,ki+ η. As far as S is concerned, we aly the same chain of inequalities as those used for S. The integral over S can be bound by the factor η + B jd η multilied by the integral of the roduct of q localization bounds of the needlets. By re-iterating the rocedure, we obtain, a set of nested S g = { x Sg, d x, ξ j,kg /, g =,..., q so that S d S q q g= S g, which yields the claimed result. The roof of Proosition 3. is a simle modification of Proosition 6 in Durastanti et al. [0] concerning comlex random sin needlet coefficients. Many technical details are omitted for the sake of brevity. Proof of Proosition 3.. For we aly the classical convexity inequality such that for a set of indeendent centered random variables {Z i with finite 8

29 -th absolute moment, n E Z i E n Z i i= i= For >, we aly the Rosenthal inequality see for instance Härdle et al. [], that is, there exists a constant c > 0 such that n { n n E Z i c E Z i + E Zi / i= i= On the other hand, since B dj n, we have that E { f X + ε ψ j,k X β j,k [E { f X ψ j,k X β j,k i= / +E { εψ j,k X ] c {M + E ε ψ j,k L S d c B jd/ c n /. Hence, E ˆβ j,k β j,k n / c n + n / = c n / The roof of Proosition 3.3 can be considered as the counterart in the needlet framework of the roof of Lemma in Kerkyacharian et al. [30]. Remark 5.. Any element c l can be decomosed as the sum of integers c i,...,i l ;l, where the l-dimensional vector {i,..., i l {,..., m secifies the sherical needlets involved in each configuration of size l given by [ ] E {f X + ε l ψ j,ki X..., ψ j,kil X. The notation [i,..., i l ] denotes the set of all the ossible combinations of {i,..., i l such that c i,...,i l ;l = c l. [i,...,i l ] Proof of Proosition 3.3. Note that m ] E [{ Θj = m n υ,...,υ m Σ E m k,...,k m l= j,k l X υl, ε υl 6 Ψ 9

30 where E m k,...,k m l= j,k l X υl, ε υl Ψ = m k,...,k m l=[i,...,i l ] = k,...,k m h= m l=[i,...,i l ] E [ { m E f X E [ l {f X + ε l ψ j,kih X l ψ j,kh X h= h= ch; l {f X + ε l ψ j,kih X h= ] ci,...,i l ;l ] ci,...,i l ;l Using Eq. and the indeendence of the noise ε, for any l we have that [i,...,i l ] E [ l {f X + ε l ψ j,kih X h= ] ci,...,i l ;l { l C M,,l E ψ j,kh X where C M,,l = { M l + E ε l. In view of Lemma., we obtain m l= [i,...,i l ] { l E ψ j,kh X h= ci,...,i l,l C C m l= [i,...,i l ] m l= C B jd m l= =C B jd m l= h= ψ j,k lc i,...,ı l ;l L l S d {k =k i =...=k il. ci,...,i l,l ψ j,k l [i,...,i l ] c i,...,ı l ;l L l S d k, l lc l m l= c l lc l m l= c l k, m B jd c k, m, Note that [i,...,i l ] {k =k i =...=k il imlies that al least l k h indexes are equal. Thus, using Eq. 0, we obtain k,...,k m m h= E {f X ψ j,kh X ch, = k m h= β c h, j,k CB j m h= c h,{s+d B jd = CB jc{s+d B jd γ { min, m h= c h;, 30

31 where γ = min, c /. Hence, [i,...,i l ] E [ l {f X + ε l ψ j,kih X h= ] ci,...,i l ;l C m B jd m l= c l B jcs d B jd γ Finally, for any fixed configuration c,..., c m, the number of ossible combinations is bounded by m n n C cm n l=... c l, c m c m c where C denotes the ossible choices of {c i,...,i l ;l for any l and does not deend on n. In view of the Stirling aroximation, n n, the number of ossible combinations is bounded by Cn m l= c l. Using the aforementioned results, Eq. 6 is bounded by m ] E [{ Θj C as claimed. c,...,c m Γ m, B jd n m m l= c l B j{cs d d γ The roof of Proosition 3.4 can be viewed as the counterart of the roof of Lemma 3 in Kerkyacharian et al. [30] in the needlet framework. n m Proof of Proosition 3.4. Following Kerkyacharian et al. [30], note that x i β = i= h= β h t <...,<t h i= h x ti β., Let so that Ψ j,k X, ε := {f x + ε ψ j,k x β j,k. { E Ψj,k X, ε = 0. 7 We therefore obtain Θ j Θ j = K n j υ Σ h= β h j,k and, reversing the order of integration, we have that Θ j Θ j = K j h= n h h n β h j,k ι υ,ι Σ h Ψ h j,k X ι, ε ι, Ψ h j,k X ι, ε ι, ι Σ h 3

32 Hence, we can rewrite m E { Θj Θ j m h= ι,...,ι m Σ h E { n h h n { m m l= k,...,k m Ψ h j,k l X ιl, ε ιl m h β j,kl l= Similar to the roof of Proosition 3.3, we fix a configuration of indexes ι,..., ι m Σ h, corresonding to the set of coefficients {c, c..., c h. Because, in this case, the considered U-statistic is degenerate, we discard all the combinations with c 0, in view of 7. On the other hand, following Lemma., we have that { Ψ h h j,k X, ε... Ψ j,k h X h, ε h Cn h l= c l B dj{ h l= l c l ι,...,ι m Σ h E = Cn h l= c mh l B dj m l= c l. 8 Furthermore, mh = m l= lc l > l= c l imlies that the exonent in the last term of 8 is ositive, so that ι,...,ι m Σ h E { Ψ h h j,k X, ε... Ψ j,k h X h, ε h. Cn mh, because B dj n. Finally, using the Stirling aroximation and Eq. 0 we have that m ] m E [{ Θj Θ j C n m h β j,k h n mh as claimed. C C h= h= n m n mh B jd h= n / k Bjd h mh h β j,k k {Θ j h m, The roof of Lemma 3.5 is the counterart of the roof of Lemma 6 in Kerkyacharian et al. [30] in the needlet framework. Proof of Lemma 3.5. Following Kerkyacharian et al. [30] and the results ob- m 3

33 tained in the revious roof, we have that K n j ˆΘ j 0 Θ j 0 = 0 + n + n 0 υ Σ 0 h= β 0 h j,k ι υ,ι Σ h Ψ h with the convention Ψ 0 j,k X ι, ε ι = ; ι υ,ι Σ 0 ι υ,ι Σ h Ψ h j,k X ι, ε ι j,k X ι, ε ι, ι υ,ι Σ h Ψ h j,k X ι, ε ι Ψ h j,k X ι, ε ι = 0 for h < 0. ι υ,ι Σ h Reversing the order of integration and alying an analogous rocedure to the one used in the roof of Proosition 3.4, we achieve the claimed result. Proosition 3.6 is roved by using the general roerties of the needlets. Proof of Proosition 3.6. It is easy to see that { E Θ j = K j n [ n E {ψ j,k X i Y i ] + n as claimed. Bd j n i= K j M + σε ψj,k L S d + Θ j i= n E{ψ j,k X i Y i Acknowledgement. The author wishes to thank M. Konstantinou, A. Ortiz, A. Renzi and N. Turchi for recious discussions and hints. Furthermore, the author wishes to acknowledge the Associate Editor, the referees and the Editorin-Chief for the insightful remarks and suggestions which led to a substantial imrovement of this work., References [] Baldi, P., Kerkyacharian, G., Marinucci, D., Picard, D., 009a. Adative density estimation for directional data using needlets. Ann. Statist. 37, [] Baldi, P., Kerkyacharian, G., Marinucci, D., Picard, D., 009b. Asymtotics for sherical needlets. Ann. Statist. 37, [3] Brown, L.D., Cai, T.T., Zhou, H.H., 00. Nonarametric regression in exonential families. Ann. Statist. 38,

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