Mark G. Low

Optimal adaptive estimation of a quadratic functional

Minimax mean-squared error estimates of quadratic functionals of smooth functions have been constructed for a variety of smoothness classes. In contrast to many nonparametric function estimation problems there are both regular and irregular cases. In the regular cases the minimax mean-squared error converges at a rate proportional to the inverse of the sample size, whereas in the irregular case much slower rates are the rule. We investigate the problem of adaptive estimation of a quadratic functional of a smooth function when the degree of smoothness of the underlying function is not known. It is shown that estimators cannot achieve the minimax rates of convergence simultaneously over two parameter spaces when at least one of these spaces corresponds to the irregular case. A lower bound for the mean squared error is given which shows that any adaptive estimator which is rate optimal for the regular case must lose a logarithmic factor in the irregular case. On the other hand, we give a rather simple adaptive estimator which is sharp for the regular case and attains this lower bound in the irregular case. Moreover, we explicitly describe a subset of functions where our adaptive estimator loses the logarithmic factor and show that this subset is relatively small.

Estimation and Confidence Sets for Sparse Normal Mixtures

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.

Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift

This paper establishes the global asymptotic equivalence between a Poisson process with variable intensity and white noise with drift under sharp smoothness conditions on the unknown function. This equivalence is also extended to density estimation models by Poissonization. The asymptotic equivalences are established by constructing explicit equivalence mappings. The impact of such asymptotic equivalence results is that an investigation in one of these nonparametric models automatically yields asymptotically analogous results in the other models.

Testing composite hypotheses, Hermite polynomials and optimal estimation of a nonsmooth functional

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 ∑ |θ i | from an observation Y ∼ N(θ, I n ). 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.

Adaptive confidence balls

Adaptive confidence balls are constructed for individual resolution levels as well as the entire mean vector in a multiresolution framework. Finite sample lower bounds are given for the minimum expected squared radius for confidence balls with a prespecified confidence level. The confidence balls are centered on adaptive estimators based on special local block thresholding rules. The radius is derived from an analysis of the loss of this adaptive estimator. In addition adaptive honest confidence balls are constructed which have guaranteed coverage probability over all of R N and expected squared radius adapting over a maximum range of Besov bodies.

Nonquadratic Estimators of a Quadratic Functional

Estimation of a quadratic functional over parameter spaces that are not quadratically convex is considered. It is shown, in contrast to the theory for quadratically convex parameter spaces, that optimal quadratic rules are often rate suboptimal. In such cases minimax rate optimal procedures are constructed based on local thresholding. These nonquadratic procedures are sometimes fully efficient even when optimal quadratic rules have slow rates of convergence. Moreover, it is shown that when estimating a quadratic functional nonquadratic procedures may exhibit different elbow phenomena than quadratic procedures.

An adaptation theory for nonparametric confidence intervals

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.

On adaptive estimation of linear functionals

Adaptive estimation of linear functionals over a collection of parameter spaces is considered. A between-class modulus of continuity, a geometric quantity, is shown to be instrumental in characterizing the degree of adaptability over two parameter spaces in the same way that the usual modulus of continuity captures the minimax difficulty of estimation over a single para, meter space. A general construction of optimally adaptive estimators based on an ordered modulus of continuity is given. The results are complemented by several illustrative examples.

Minimax estimation of linear functionals over nonconvex parameter spaces

The minimax theory for estimating linear functionals is extended to the case of a finite union of convex parameter spaces. Upper and lower bounds for the minimax risk can still be described in terms of a modulus of continuity. However in contrast to the theory for convex parameter spaces rate optimal procedures are often required to be nonlinear. A construction of such nonlinear procedures is given. The results developed in this paper have important applications to the theory of adaptation.

Adaptive estimates of linear functionals

SummaryGiven a collection of nested closed, convex symmetric sets and a linear functional, we find estimates which are within a logarithm term of being simultaneously asymptotically minimax. Moreover, these estimates can be constructed so that the loss of this logarithm term only occurs on a small subset of functions. These estimates are quasi-optimal since there do not exist estimators which do not lose a logarithm term on some part of the parameter spaces.