Block‐threshold‐adapted Estimators via a Maxiset Approach

Abstract

International audienceWe study the maxiset performance of a large collection of block thresholding wavelet estimators, namely the horizontal block thresholding family. We provide sufficient conditions on the choices of rates and threshold values to ensure that the involved adaptive estimators obtain large maxisets. Moreover, we prove that any estimator of such a family reconstructs the Besov balls with a near-minimax optimal rate that can be faster than the one of any separable thresholding estimator. Then, we identify, in particular cases, the best estimator of such a family, that is, the one associated with the largest maxiset. As a particularity of this paper, we propose a refined approach that models method-dependent threshold values. By a series of simulation studies, we confirm the good performance of the best estimator by comparing it with the other members of its family

Combining thresholding rules: a new way to improve the performance of wavelet estimators

“…We recall that estimators induced by these rules are, respectively, the Blockshrink estimator studied by Cai (1997) and Autin et al (2011b) and the Hard Tree estimator studied by Autin (2004Autin ( , 2008a and Autin et al (2011a). Our numerical results clearly illustrate the need to use the combination of the previous estimators, called the Block Tree estimator, rather than the Blockshrink estimator or the Hard Tree estimator, since this Block Tree estimator behaves well over all the 12 functions considered here.…”

“…This approach consists of determining the maxiset of a thresholding procedure that is the maximal functional space for which the quadratic risk of the procedure reaches a given rate of convergence. As previously discussed in Cohen, De Vore, Kerkyacharian, and Picard (2001b), Picard (2000, 2002), Autin (2004Autin ( , 2008a, Autin, Le Pennec, Loubes, and Rivoirard (2010), Autin, Freyermuth, and von Sachs (2011a) and Autin, Freyermuth, and von Sachs (2011b), this approach can be successful at differentiating between minimax-equivalent procedures whenever their maxisets are nested. Without such embeddings, the comparison would be impossible.…”

“…As examples, we cite estimators that rely on vertical-block thresholding rules (see Cohen et al 2001b;Autin 2008a,b;Autin et al 2011a) or horizontal-block thresholding rules (see, among others, Cai 1997Cai , 1999Cai , 2008Hall, Kerkycharian, and Picard 1998a,b;Cai and Silverman 2001;Cai and Zhou 2009;Autin et al 2011b). When looking at these procedures, their maxisets contain those of procedures based on elitist rules, including Hard and Soft thresholding estimators, and also many Bayes procedures (see Autin, Picard, and Rivoirard 2006).…”

“…The (μ B , m) thresholding estimator was studied by Cai (1997) and Autin et al (2011b). It relies on a rule which keeps empirical wavelet coefficients if the l 2 -norm of the empirical wavelet coefficients from their block is larger than the threshold value mt .…”

“…Blocks are non-overlapping with common size ln( −1 ) = − ln(F −1 (t )) , where F −1 is the inverse function of F : −→ F( ) := t and x denotes the smallest integer bigger than or equal to x. A precise description is given in Cai (1997) and Autin et al (2011b) in particular for handling boundaries. More generally, for wise choices of μ, the resulting estimators show good theoretical and practical performance.…”