Replicant Compression Coding in Besov Spaces

We present here a new proof of the theorem of Birman and Solomyak on the metric entropy of the unit ball of a Besov space B π,q on a regular domain of R . The result is: if s − d(1/π − 1/p)+ > 0, then the Kolmogorov metric entropy satisfies H( ) ∼ −d/s. This proof takes advantage of the representation of such spaces on wavelet type bases and extends the result to more general spaces. The lower bound is a consequence of very simple probabilistic exponential inequalities. To prove the upper bound, we provide a new universal coding based on a thresholding-quantizing procedure using replication. Mathematics Subject Classification. 41A25, 41A46, 65F99, 65N12, 65N55. Received April 9, 2001. Revised January 20, 2003. Introduction The evaluation of the entropy of the balls in Besov spaces is a very important point in modern nonparametric statistics. Entropy is a measure of the complexity of a parameter space which early proved to be especially appropriate for approximation –see Lorentz [22]– but also for statistical estimation. LeCam was the first to obtain general bounds depending on the metric structure of the parameter space for the risk of estimators –see LeCam [20] and [21]–. After this seminal work, Assouad [1] and Birgé [2] pointed out the link between the metric structure and the minimax rates. Nowadays, entropy or construction of -nets is a very common approach to obtain lower bounds in minimax evaluations. Moreover, entropy is also a very powerful tool to derive exponential bounds for the supremum of processes, in particular empirical processes. This has especially important consequences: for instance in many situations, the rate of convergence of the classical MLE or LSE directly follows from entropy calculations. An important tool in modern nonparametric estimation is penalized methods. Here, again exponential bounds on the supremum of the empirical process on a class of functions is a key argument to calibrate the penalties –see for instance, van de Geer [25], for a review–. On the other hand, Besov spaces among other spaces of regularity appear to be especially suited for approximation and statistical applications: for instance, balls of Besov spaces appear to be maximal sets for linear approximation methods (see for instance Nikolskii [23]). They also appear to be maximal sets –i.e. the set of all functions such that a given procedure has a risk bounded by a specified rate of convergence– for general