|This paper studies minimax aspects of nonparametric classi cation. We rst study minimax estimation of the conditional probability of a class label, given the feature variable. This function, say f; is assumed to be in a general nonparametric class. We show the minimax rate of convergence under square L2 loss is determined by the massiveness of the class as measured by metric entropy. The secon...

Since Stein’s 1956 seminal paper, shrinkage has played a fundamental role in both parametric and nonparametric inference. This article discusses minimaxity and adaptive minimaxity in nonparametric function estimation. Three interrelated problems, function estimation under global integrated squared error, estimation under pointwise squared error, and nonparametric confidence intervals, are consi...

Mallows has conjectured that among distributions which are Gaussian but for occasional contamination by additive noise, the one having least Fisher information has (two-sided) geometric contamination. A very similar problem arises in estimation of a non-negative vector parameter in Gaussian white noise when it is known also that most, i.e. (1 − ǫ), components are zero. We provide a partial asym...

Driven by a wide range of applications in high-dimensional data analysis, there has been significant recent interest in the estimation of large covariance matrices. In this paper, we consider optimal estimation of a covariance matrix as well as its inverse over several commonly used parameter spaces under the matrix l1 norm. Both minimax lower and upper bounds are derived. The lower bounds are ...

We consider the minimax rate of testing (or estimation) of non-linear functionals defined on semiparametric models. Existing methods appear not capable of determining a lower bound on the minimax rate of testing (or estimation) for certain functionals of interest. In particular, if the semiparametric model is indexed by several infinite-dimensional parameters. To cover these examples we extend ...

General results on adaptive function estimation are obtained with respect to a collection of estimation strategies for both density estimation and nonparametric regression under square L 2 loss. It is shown that without knowing which strategy in a given countable collection works best for the underlying function, a single strategy can be constructed by mixing the proposed ones so that it is ada...

the problem of density estimation on R from an independent sample X1, ...XN with common density f is concerned. The behavior of the minimax Lp-risk, 1 ≤ p ≤ ∞, is studied when f belongs to a Hölder class of regularity s on the real line. The lower bound for the minimax risk is provided. We show that the linear estimator is not efficient in this setting and construct a wavelet adaptive estimator...

The nonparametric minimax estimation of an analytic density at a given point, under random censorship, is considered. Although the problem of estimating density is known to be irregular in a certain sense, we make some connections relating this problem to the problem of estimating smooth functionals. Under condition that the censoring is not too severe, we establish the exact limiting behavior ...

Function estimation over the Besov spaces under pointwise r (1 ≤ r < ∞) risks is considered. Minimax rates of convergence are derived using a constrained risk inequality and wavelets. Adaptation under pointwise risks is also considered. Sharp lower bounds on the cost of adaptation are obtained and are shown to be attainable by a wavelet estimator. The results demonstrate important differences b...

The nonparametric problem of estimating a variance based on a sample of size n from a univariate distribution which has a known bounded range but is otherwise arbitrary is treated. For squared error loss, a certain linear function of the sample variance is seen to be minimax for each n from 2 through 13, except n = 4. For squared error loss weighted by the reciprocal of the variance, a constant...