Precision matrix is of significant importance in a wide range of applications in multivariate analysis. This paper considers adaptive minimax estimation of sparse precision matrices in the high dimensional setting. Optimal rates of convergence are established for a range of matrix norm losses. A fully data driven estimator based on adaptive constrained `1 minimization is proposed and its rate o...

In this paper, we study the problem of pointwise estimation of a multivariate function. We develop a general pointwise estimation procedure that is based on selection of estimators from a large parameterized collection. An upper bound on the pointwise risk is established and it is shown that the proposed selection procedure specialized for different collections of estimators leads to minimax an...

We find the minimax rate of convergence in Hausdorff distance for estimating a manifold M of dimension d embedded in R given a noisy sample from the manifold. Under certain conditions, we show that the optimal rate of convergence is n−2/(2+d). Thus, the minimax rate depends only on the dimension of the manifold, not on the dimension of the space in which M is embedded.

This paper addresses an estimation problem of an additive functional of φ, which is defined as θ(P ;φ) = ∑ k i=1 φ(pi), given n i.i.d. random samples drawn from a discrete distribution P = (p1, ..., pk) with alphabet size k. We have revealed in the previous paper [1] that the minimax optimal rate of this problem is characterized by the divergence speed of the fourth derivative of φ in a range o...

A fundamental quantity in statistical decision theory is the notion of the minimax risk as5 sociated with an estimation problem. It is based on a saddlepoint problem, in which nature plays the 6 role of adversary in choosing the underlying problem instance, and the statistician seeks an estimator 7 with good properties uniformly over a class of problem instances. We argue that in many modern 8 ...

Minimax parameter estimation aims at characterizing the set of all values of the parameter vector that minimize the largest absolute deviation between the experimental data and the corresponding model outputs. It is well known, however, to be extremely sensitive to outliers in the data resulting, e.g., of sensor failures. In this paper, a new method is proposed to robustify minimax estimation b...

The problems of predictive density estimation with Kullback-Leibler loss, optimal universal data compression for MDL model selection, and the choice of priors for Bayes factors in model selection are interrelated. Research in recent years has identified procedures which are minimax for risk in predictive density estimation and for redundancy in universal data compression. Here, after reviewing ...

New formulas are given for the minimax linear risk in estimating a linear functional of an unknown object from indirect data contaminated with random Gaussian noise. The formulas cover a variety of loss functions, and do not require the symmetry of the convex a priori class. It is shown that affine minimax rules are within a few percent of minimax even among nonlinear rules, for a variety of lo...

We study the problem of matrix estimation and matrix completion under a general framework. This framework includes several important models as special cases such as the gaussian mixture model, mixed membership model, bi-clustering model and dictionary learning. We consider the optimal convergence rates in a minimax sense for estimation of the signal matrix under the Frobenius norm and under the...