SUMMARY Much recent effort has sought asymptotically minimax methods for recovering infinite dimensional objects—curves, densities, spectral densities, images—from noisy data. A now rich and complex body of work develops nearly or exactly minimax estimators for an array of interesting problems. Unfortunately, the results have rarely moved into practice, for a variety of reasons—among them being similarity to known methods, computational intractability and lack of spatial adaptivity. We discuss a method for curve estimation based on n noisy data: translate the empirical wavelet coefficients towards the origin by an amount √(2 log n)σ/√n. The proposal differs from those in current use, is computationally practical and is spatially adaptive; it thus avoids several of the previous objections. Further, the method is nearly minimax both for a wide variety of loss functions—pointwise error, global error measured in Lp‐norms, pointwise and global error in estimation of derivatives—and for a wide range of smoothness classes, including standard Holder and Sobolev classes, and bounded variation. This is a much broader near optimality than anything previously proposed: we draw loose parallels with near optimality in robustness and also with the broad near eigenfunction properties of wavelets themselves. Finally, the theory underlying the method is interesting, as it exploits a correspondence between statistical questions and questions of optimal recovery and information‐based complexity.

No abstract

No abstract

We investigate invariant random fields on the sphere using a new type of spherical wavelets, called needlets. These are compactly supported in frequency and enjoy excellent localization properties in real space, with quasiexponentially decaying tails. We show that, for random fields on the sphere, the needlet coefficients are asymptotically uncorrelated for any fixed angular distance. This property is used to derive CLT and functional CLT convergence results for polynomial functionals of the needlet coefficients: here the asymptotic theory is considered in the high-frequency sense. Our proposals emerge from strong empirical motivations, especially in connection with the analysis of cosmological data sets