Orthonormal Series and Approximation.- Density Estimation for Small Samples.- Nonparametric Regression Time Analysis of Multivariate Functions Filtering Asymptotics.- Nonseries Methods.

Consider a problem of recovery smooth function (signal, image) f/spl isin//spl Fscr//spl isin/L/sub 2/([0, 1]/sup d/) passed through an unknown filter and then contaminated by noise. A typical model discussed in the paper is described stochastic differential equation dY/sub f//sup /spl epsi//(t)=(Hf)(t)dt+/spl epsi/dW(t), t/spl isin/[0, d/, epsi/>0 where H linear operator modeling W Brownian motion (sheet) The aim to recover f with asymptotically (as epsi//spl rarr/0) minimax mean integrated...

Abstract The problem is to estimate the probability density of a random variable contaminated by an independent measurement error. I explore one worst-case scenario when characteristic function this error decreases exponentially and thus optimal estimators converge only with logarithmic rate. particular example such any normal, Cauchy, or another stable variable. For setting circular data, suggest asymptotically efficient data-driven estimator that adaptive both smoothness estimated...

ABSTRACT The bivariate mean residual life (BMRL) is a key characteristic of pair lifetimes given that they exceed values. Its nonparametric estimation well known for the case directly observed pairs lifetimes. For right‐censored observations, typical in survival analysis, methodology estimating BMRL univariate right‐censoring when two are censored by same censoring variable. paper considers general case, and proposes data‐driven based on conditional function which interest its own....

Multiwavelets are relative newcomers into the world of wavelets. Thus, it has not been a surprise that used methods denoising modified universal thresholding procedures developed for uniwavelets. On other hand, specific multiwavelet discrete transform is typical errors identically distributed and correlated, whereas theory based on assumption independent normal errors. we suggest an alternative procedure Efromovich-Pinsker algorithm. We show this optimal over wide class noise distributions....

Regression problems are traditionally analyzed via univariate characteristics like the regression function, scale function and marginal density of errors. These useful informative whenever association between predictor response is relatively simple. More detailed information about can be provided by conditional given predictor. For first time in literature, this article develops theory minimax estimation for settings with fixed random designs predictors, bounded unbounded responses a vast...

The concept of biased data is well known and its practical applications range from social sciences biology to economics quality control. These observations arise when a sampling procedure chooses an observation with probability that depends on the value observation. This interesting because it favors some neglects others. It biasing does not change rates nonparametric density estimation, but no results are available about sharp constants. article presents asymptotic minimax estimation. In...

Estimation of the density regression errors is a fundamental issue in analysis and it typically explored via parametric approach. This article uses nonparametric approach with mean integrated squared error (MISE) criterion. It solves long-standing problem, formulated two decades ago by Mark Pinsker, about estimation setting accuracy an oracle that knows underlying errors. The solution implies that, under mild assumption on differentiability design function, MISE data-driven estimator attains...

An orthogonal series estimator of the conditional density a response given vector continuous and ordinal/nominal categorical predictors is suggested. The based on writing as sum projections all possible subspaces reduced dimensionality then estimating each projection via shrinkage procedure. procedure uses universal thresholding dyadic-blockwise for low high frequencies, respectively. data-driven, adaptive to underlying smoothness density, attains minimax rate mean integrated squared error...

Nonparametric regression with predictors missing at random (MAR), where the probability of depends only on observed variables, is considered. Univariate predictor primary case interest. A new adaptive orthogonal series estimator developed. Large sample theory shows that rate-minimax and it also sharp-minimax whenever are completely (MCAR). Furthermore, confidence bands, estimation nuisance functions, including conditional observing predictor, design density scale, multiple Numerical study a...

A problem of estimating the integral a squared regression function and its derivatives has been addressed in number papers. For case heteroscedastic model where smoothness underlying function, design density, variance errors are known, asymptotically sharp minimax lower bound estimator were found Pastuchova & Khasminski (1989). However, there apparently no results on either rate optimal or adaptive, data‐driven, estimation when neither degree nor scale distribution known. After brief...

Abstract The paper considers, for the first time in literature, sharp minimax design of predictors and sequential estimation regression functions a classical heteroscedastic nonparametric regression. suggested methodology controlled experiments with fixed-size samples is based on minimization coefficient difficulty an underlying model, which defined as factor changing sample size that makes problem comparable homoscedastic unit-variance errors uniform design. It established optimal density...

We consider the problem of sharp-optimal estimation a response function $f(x)$ in random design nonparametric regression under general model where pair observations $(Y, X)$ has joint density $p(y, x) = p(y|f(x)) \pi(x)$. wish to estimate with optimal minimax mean integrated squared error convergence as sample size tends $\infty$. Traditional regularity assumptions on conditional $p(y| \theta)$ assumed for parameter $\theta$ are sufficient risk well existence best constant and rate...

Minimax mean-squared error estimates of quadratic functionals smooth functions have been constructed for a variety smoothness classes. In contrast to many nonparametric function estimation problems there are both regular and irregular cases. the cases minimax converges at rate proportional inverse sample size, whereas in case much slower rates rule. We investigate problem adaptive functional when degree underlying is not known. It shown that estimators cannot achieve convergence...

The theory of adaptive estimation and oracle inequalities for the case Gaussian-shift--finite-interval experiments has made significant progress in recent years. In particular, sharp-minimax estimators exact exponential-type have been suggested a vast set functions including analytic Sobolev with any positive index as well Efromovich--Pinsker Stein blockwise-shrinkage estimators. Is it possible to obtain similar results more interesting applied problem density and/or dual characteristic...

The relationship between speciation times and the corresponding of gene divergence is interest in phylogenetic inference as a means understanding past evolutionary dynamics populations estimating timing events. It has long been recognized that might substantially pre-date Although distribution difference these previously studied for case two populations, this not explicitly computed larger species phylogenies. Here we derive simple method computing trees arbitrary size. A two-stage procedure...

Abstract The main paradigm of the modern wavelet theory spatial adaptation formulated by Donoho and Johnstone is that there a divergence between linear minimax heuristic guiding algorithm development leads to necessity using strongly nonlinear adaptive thresholded methods. On other hand, it well known estimates are best whenever an estimated function smooth. Is possible suggest quasi-linear estimate, adding estimate minimal number terms on finest scales, offers advantages at same time...

Do a random censorship and/or order restrictions (e.g., nonnegativity, monotonicity, convexity) affect estimation of smooth density under mean integrated squared error (MISE)? Under mild assumptions, the known asymptotic results, which are concerned only with rates, answer "no." This answer, especially for censored data, contradicts practical experience and statistical intuition. So what can be said about constants MISE convergence? It is shown that asymptotically (a) does constant, this...