The principal result is that, under conditions, to any nonparametric regression problem there corresponds an asymptotically equivalent sequence of white noise with drift problems, and conversely. This asymptotic equivalence is in a global and uniform sense. Any normalized risk function attainable in one problem is asymptotically attainable in the other, with the difference in normalized risks converging to zero uniformly over the entire parameter space. The results are constructive. A recipe is provided for producing these asymptotically equivalent procedures. Some implications and generalizations of the principal result are also discussed. University of PennsylvaniaThe principal result is that, under conditions, to any nonparametric regression problem there corresponds an asymptotically equivalent sequence of white noise with drift problems, and conversely. This asymptotic equivalence is in a global and uniform sense. Any normalized risk function attainable in one problem is asymptotically attainable in the other, with the difference in normalized risks converging to zero uniformly over the entire parameter space. The results are constructive. A recipe is provided for producing these asymptotically equivalent procedures. Some implications and generalizations of the principal result are also discussed.

For high dimensional statistical models, researchers have begun to focus on situations which can be described as having relatively few moderately large coefficients. Such situations lead to some very subtle statistical problems. In particular, Ingster and Donoho and Jin have considered a sparse normal means testing problem, in which they described the precise demarcation or detection boundary. Meinshausen and Rice have shown that it is even possible to estimate consistently the fraction of nonzero coordinates on a subset of the detectable region, but leave unanswered the question of exactly in which parts of the detectable region consistent estimation is possible. In the present paper we develop a new approach for estimating the fraction of nonzero means for problems where the nonzero means are moderately large. We show that the detection region described by Ingster and Donoho and Jin turns out to be the region where it is possible to consistently estimate the expected fraction of nonzero coordinates. This theory is developed further and minimax rates of convergence are derived. A procedure is constructed which attains the optimal rate of convergence in this setting. Furthermore, the procedure also provides an honest lower bound for confidence intervals while minimizing the expected length of such an interval. Simulations are used to enable comparison with the work of Meinshausen and Rice, where a procedure is given but where rates of convergence have not been discussed. Extensions to more general Gaussian mixture models are also given.Comment: Published in at http://dx.doi.org/10.1214/009053607000000334 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A general lower bound is developed for the minimax risk when estimating an arbitrary functional. The bound is based on testing two composite hypotheses and is shown to be effective in estimating the nonsmooth functional 1 n |θi| from an observation Y ∼ N (θ, In). This problem exhibits some features that are significantly different from those that occur in estimating conventional smooth functionals. This is a setting where standard techniques fail to yield sharp results.A sharp minimax lower bound is established by applying the general lower bound technique based on testing two composite hypotheses. A key step is the construction of two special priors and bounding the chi-square distance between two normal mixtures. An estimator is constructed using approximation theory and Hermite polynomials and is shown to be asymptotically sharp minimax when the means are bounded by a given value M . It is shown that the minimax risk equals β 2 * M 2 ( log log n log n ) 2 asymptotically, where β * is the Bernstein constant. The general techniques and results developed in the present paper can also be used to solve other related problems.

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A nonparametric adaptation theory is developed for the construction of confidence intervals for linear functionals. A between class modulus of continuity captures the expected length of adaptive confidence intervals. Sharp lower bounds are given for the expected length and an ordered modulus of continuity is used to construct adaptive confidence procedures which are within a constant factor of the lower bounds. In addition, minimax theory over nonconvex parameter spaces is developed.