Oracle Inequalities for Efromovich–Pinsker Blockwise Estimates

Oracle inequality is a relatively new statistical tool for the analysis of nonparametric adaptive estimates. Oracle is a good pseudo-estimate that is based on both data and an underlying estimated curve. An oracle inequality shows how well an adaptive estimator mimics the oracle for a particular underlying curve. The most advanced oracle inequalities have been recently obtained by Cavalier and Tsybakov (2001) for Stein type blockwise estimates used in filtering a signal from a stationary white Gaussian process. The authors also conjecture that a similar result can be obtained for Efromovich–Pinsker (EP) type blockwise estimators where their approach, based on Stein's formula for risk calculation, does not work. This article proves the conjecture and extends it upon more general models which include not stationary and dependent processes. Other possible extensions, a discussion of practical implications and a numerical study are also presented.     

Analysis of blockwise shrinkage wavelet estimates via lower bounds for no-signal setting

A blockwise shrinkage is a popular procedure of adaptation that has allowed the statisticians to establish an impressive bouquet of asymptotic mathematical results and develop softwares for solving practical problems. Traditionally risks of the estimates are studied via upper bounds that imply sufficient conditions for a blockwise shrinkage procedure to be minimax. This article suggests to analyze the estimates via exact (non-asymptotic) lower bounds established for a no-signal setting. The approach complements the familiar minimax, Bayesian and numerical analysis, it allows to find necessary conditions for a procedure to attain desired rates, and it sheds a new light on popular choices of blocks and thresholds recommended in the literature. Mathematical results are complemented by a numerical study. Supported in part by NSF Grants DMS-9971051 and DMS-0243606.     

Second Order Efficient Estimating a Smooth Distribution Function and its Applications

Methodology and Computing in Applied Probability - Consider a problem of estimation of a cumulative distribution function of a random variable supported on a finite interval, with a circular random...     

Best Fourier Approximation and Application in Efficient Blurred Signal Reconstruction

We expand upon the known results on sharp linear Fourier methods of approximation where the approximation is the best in terms of both rate and constant among all polynomial procedures of approximation. So far these results have been studied due to their mathematical beauty rather than their practical importance. In this paper we show that they are the core mathematics underlying best statistical methods of solving noisy ill-posed problems. In particular, we suggest a procedure for recovery of noisy blurred signals based on samples of small sizes where a traditional statistics concludes that the complexity of such a setting makes the problem not worthy of a further study. Thus, we present a problem where a combination of the classical approximation theory and statistics leads to interesting practical results.     

Wavelet Analysis of Big Data Contaminated by Large Noise in an fMRI Study of Neuroplasticity

Functional magnetic resonance imaging (fMRI) allows researchers to analyze brain activity on a voxel level, but using this ability is complicated by dealing with Big Data and large noise. A traditional remedy is averaging over large parts of brain in combination with more advanced technical innovations in reducing fMRI noise. In this paper a novel statistical approach, based on a wavelet analysis of standard fMRI data, is proposed and its application to an fMRI study of neuron plasticity of 24 healthy adults is presented. The aim of that study was to recognize changes in connectivity between left and right motor cortices (the neuroplasticity) after button clicking training sessions. A conventional method of the data analysis, based on averaging images, has implied that for the group of 24 participants the connectivity increased after the training. The proposed wavelet analysis suggests to analyze pathways between left and right hemispheres on a voxel-to-voxel level and for each participant via estimation of corresponding cross-correlations. This immediately necessitates statistical analysis of large-p-small-n correlation matrices contaminated by large noise. Furthermore, distributions that we are dealing in the analysis are neither Gaussian nor sub-Gaussian but sub-exponential. The paper explains how the problem may be solved and presents results of a dynamic analysis of the ability of a human brain to reorganize itself for 24 healthy adults. Results show that the ability of a brain to reorganize itself varies widely even among healthy individuals, and this observation is important for our understanding of a human brain and treatment of brain diseases.     

Optimal nonparametric estimation of the density of regression errors with finite support

Knowledge of the probability distribution of error in a regression problem plays an important role in verification of an assumed regression model, making inference about predictions, finding optimal regression estimates, suggesting confidence bands and goodness of fit tests as well as in many other issues of the regression analysis. This article is devoted to an optimal estimation of the error probability density in a general heteroscedastic regression model with possibly dependent predictors and regression errors. Neither the design density nor regression function nor scale function is assumed to be known, but they are suppose to be differentiable and an estimated error density is suppose to have a finite support and to be at least twice differentiable. Under this assumption the article proves, for the first time in the literature, that it is possible to estimate the regression error density with the accuracy of an oracle that knows “true” underlying regression errors. Real and simulated examples illustrate importance of the error density estimation as well as the suggested oracle methodology and the method of estimation.     

Hazard rate estimation for left truncated and right censored data

Left truncation and right censoring (LTRC) presents a unique challenge for nonparametric estimation of the hazard rate of a continuous lifetime because consistent estimation over the support of the lifetime is impossible. To understand the problem and make practical recommendations, the paper explores how the LTRC affects a minimal (called sharp) constant of a minimax MISE convergence over a fixed interval. The corresponding theory of sharp minimax estimation of the hazard rate is presented, and it shows how right censoring, left truncation and interval of estimation affect the MISE. Obtained results are also new for classical cases of censoring or truncation and some even for the case of direct observations of the lifetime of interest. The theory allows us to propose a relatively simple data-driven estimator for small samples as well as the methodology of choosing an interval of estimation. The estimation methodology is tested numerically and on real data.     

Oracle inequality for conditional density estimation and an actuarial example

Conditional density estimation in a parametric regression setting, where the problem is to estimate a parametric density of the response given the predictor, is a classical and prominent topic in regression analysis. This article explores this problem in a nonparametric setting where no assumption about shape of an underlying conditional density is made. For the first time in the literature, it is proved that there exists a nonparametric data-driven estimator that matches performance of an oracle which: (i) knows the underlying conditional density, (ii) adapts to an unknown design of predictors, (iii) performs a dimension reduction if the response does not depend on the predictor, (iv) is minimax over a vast set of anisotropic bivariate function classes. All these results are established via an oracle inequality which is on par with ones known in the univariate density estimation literature. Further, the asymptotically optimal estimator is tested on an interesting actuarial example which explores a relationship between credit scoring and premium for basic auto-insurance for 54 undergraduate college students.