arXiv subjects

Alexander Goldenshluger

Publications and source records attributed to Alexander Goldenshluger.

14 recordsLinked to original sources

arXiv2026

Optimal single threshold stopping rules and sharp prophet inequalities

This paper considers a finite horizon optimal stopping problem for a sequence of independent and identically distributed random variables, where the objective is to design stopping rules that attempt to select the random variable with the highest value in the sequence. The performance of any stopping rule may be benchmarked relative to the selection of a ``prophet" that has perfect foreknowledge of the largest value. Such comparisons are typically stated in the form of ``prophet inequalities." In this paper we develop a game-theoretic characterization that supports a principled approach for deriving sharp non-asymptotic prophet inequalities for single threshold stopping rules. We demonstrate that sharp constants in the ratio- and difference-type prophet inequalities are determined by the optimal values of infinite two-person zero-sum game on the unit square with particular payoff kernels, while the the solutions to the game provide optimal stopping rules and least favorable distributions. Among other things, this formulation also allows a systematic way to tackle restricted classes of distributions. The proposed framework leads to a numerically efficient algorithmic paradigm that allows computing sharp constants in prophet inequalities with any prescribed level of accuracy.

math.PR↗

arXiv2021

Density Deconvolution with Non-Standard Error Distributions: Rates of Convergence and Adaptive Estimation

It is a typical standard assumption in the density deconvolution problem that the characteristic function of the measurement error distribution is non-zero on the real line. While this condition is assumed in the majority of existing works on the topic, there are many problem instances of interest where it is violated. In this paper we focus on non--standard settings where the characteristic function of the measurement errors has zeros, and study how zeros multiplicity affects the estimation accuracy. For a prototypical problem of this type we demonstrate that the best achievable estimation accuracy is determined by the multiplicity of zeros, the rate of decay of the error characteristic function, as well as by the smoothness and the tail behavior of the estimated density. We derive lower bounds on the minimax risk and develop optimal in the minimax sense estimators. In addition, we consider the problem of adaptive estimation and propose a data-driven estimator that automatically adapts to unknown smoothness and tail behavior of the density to be estimated.

math.ST↗

arXiv2020

Minimax estimation of norms of a probability density: I. Lower bounds

The paper deals with the problem of nonparametric estimating the $L_p$--norm, $p\in (1,\infty)$, of a probability density on $R^d$, $d\geq 1$ from independent observations. The unknown density %to be estimated is assumed to belong to a ball in the anisotropic Nikolskii's space. We adopt the minimax approach, and derive lower bounds on the minimax risk. In particular, we demonstrate that accuracy of estimation procedures essentially depends on whether $p$ is integer or not. Moreover, we develop a general technique for derivation of lower bounds on the minimax risk in the problems of estimating nonlinear functionals. The proposed technique is applicable for a broad class of nonlinear functionals, and it is used for derivation of the lower bounds in the~$L_p$--norm estimation.

math.ST↗

arXiv2020

Minimax estimation of norms of a probability density: II. Rate-optimal estimation procedures

In this paper we develop rate--optimal estimation procedures in the problem of estimating the $L_p$--norm, $p\in (0, \infty)$ of a probability density from independent observations. The density is assumed to be defined on $R^d$, $d\geq 1$ and to belong to a ball in the anisotropic Nikolskii space. We adopt the minimax approach and construct rate--optimal estimators in the case of integer $p\geq 2$. We demonstrate that, depending on parameters of Nikolskii's class and the norm index $p$, the risk asymptotics ranges from inconsistency to $\sqrt{n}$--estimation. The results in this paper complement the minimax lower bounds derived in the companion paper \cite{gl20}.

math.ST↗

arXiv2020

Density deconvolution under general assumptions on the distribution of measurement errors

In this paper we study the problem of density deconvolution under general assumptions on the measurement error distribution. Typically deconvolution estimators are constructed using Fourier transform techniques, and it is assumed that the characteristic function of the measurement errors does not have zeros on the real line. This assumption is rather strong and is not fulfilled in many cases of interest. In this paper we develop a methodology for constructing optimal density deconvolution estimators in the general setting that covers vanishing and non--vanishing characteristic functions of the measurement errors. We derive upper bounds on the risk of the proposed estimators and provide sufficient conditions under which zeros of the corresponding characteristic function have no effect on estimation accuracy. Moreover, we show that the derived conditions are also necessary in some specific problem instances.

math.ST↗

arXiv2020

A Unified Approach for Solving Sequential Selection Problems

In this paper we develop a unified approach for solving a wide class of sequential selection problems. This class includes, but is not limited to, selection problems with no-information, rank-dependent rewards, and considers both fixed as well as random problem horizons. The proposed framework is based on a reduction of the original selection problem to one of optimal stopping for a sequence of judiciously constructed independent random variables. We demonstrate that our approach allows exact and efficient computation of optimal policies and various performance metrics thereof for a variety of sequential selection problems, several of which have not been solved to date.

math.PR↗

arXiv2018

Nonparametric density estimation from observations with multiplicative measurement errors

In this paper we study the problem of pointwise density estimation from observations with multiplicative measurement errors. We elucidate the main feature of this problem: the influence of the estimation point on the estimation accuracy. In particular, we show that, depending on whether this point is separated away from zero or not, there are two different regimes in terms of the rates of convergence of the minimax risk. In both regimes we develop kernel--type density estimators and prove upper bounds on their maximal risk over suitable nonparametric classes of densities. We show that the proposed estimators are rate--optimal by establishing matching lower bounds on the minimax risk. Finally we test our estimation procedures on simulated data.

stat.ME↗

arXiv2012

Bandwidth selection in kernel density estimation: Oracle inequalities and adaptive minimax optimality

We address the problem of density estimation with $\mathbb{L}_s$-loss by selection of kernel estimators. We develop a selection procedure and derive corresponding $\mathbb{L}_s$-risk oracle inequalities. It is shown that the proposed selection rule leads to the estimator being minimax adaptive over a scale of the anisotropic Nikol'skii classes. The main technical tools used in our derivations are uniform bounds on the $\mathbb{L}_s$-norms of empirical processes developed recently by Goldenshluger and Lepski [Ann. Probab. (2011), to appear].

math.ST↗

arXiv2012

Uniform bounds for norms of sums of independent random functions

In this paper, we develop a general machinery for finding explicit uniform probability and moment bounds on sub-additive positive functionals of random processes. Using the developed general technique, we derive uniform bounds on the ${\mathbb{L}}_s$-norms of empirical and regression-type processes. Usefulness of the obtained results is illustrated by application to the processes appearing in kernel density estimation and in nonparametric estimation of regression functions.

math.PR↗

arXiv2009

Woodroofe's one-armed bandit problem revisited

We consider the one-armed bandit problem of Woodroofe [J. Amer. Statist. Assoc. 74 (1979) 799--806], which involves sequential sampling from two populations: one whose characteristics are known, and one which depends on an unknown parameter and incorporates a covariate. The goal is to maximize cumulative expected reward. We study this problem in a minimax setting, and develop rate-optimal polices that involve suitable modifications of the myopic rule. It is shown that the regret, as well as the rate of sampling from the inferior population, can be finite or grow at various rates with the time horizon of the problem, depending on "local" properties of the covariate distribution. Proofs rely on martingale methods and information theoretic arguments.

math.PR↗

arXiv2009

A universal procedure for aggregating estimators

In this paper we study the aggregation problem that can be formulated as follows. Assume that we have a family of estimators $\mathcal{F}$ built on the basis of available observations. The goal is to construct a new estimator whose risk is as close as possible to that of the best estimator in the family. We propose a general aggregation scheme that is universal in the following sense: it applies for families of arbitrary estimators and a wide variety of models and global risk measures. The procedure is based on comparison of empirical estimates of certain linear functionals with estimates induced by the family $\mathcal{F}$. We derive oracle inequalities and show that they are unimprovable in some sense. Numerical results demonstrate good practical behavior of the procedure.

math.ST↗

arXiv2008

Universal pointwise selection rule in multivariate function estimation

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 and adaptive minimax estimators in various settings.

math.ST↗

arXiv2006

Recovering convex boundaries from blurred and noisy observations

We consider the problem of estimating convex boundaries from blurred and noisy observations. In our model, the convolution of an intensity function $f$ is observed with additive Gaussian white noise. The function $f$ is assumed to have convex support $G$ whose boundary is to be recovered. Rather than directly estimating the intensity function, we develop a procedure which is based on estimating the support function of the set $G$. This approach is closely related to the method of geometric hyperplane probing, a well-known technique in computer vision applications. We establish bounds that reveal how the estimation accuracy depends on the ill-posedness of the convolution operator and the behavior of the intensity function near the boundary.

math.ST↗

arXiv2005

The Hough transform estimator

This article pursues a statistical study of the Hough transform, the celebrated computer vision algorithm used to detect the presence of lines in a noisy image. We first study asymptotic properties of the Hough transform estimator, whose objective is to find the line that ``best'' fits a set of planar points. In particular, we establish strong consistency and rates of convergence, and characterize the limiting distribution of the Hough transform estimator. While the convergence rates are seen to be slower than those found in some standard regression methods, the Hough transform estimator is shown to be more robust as measured by its breakdown point. We next study the Hough transform in the context of the problem of detecting multiple lines. This is addressed via the framework of excess mass functionals and modality testing. Throughout, several numerical examples help illustrate various properties of the estimator. Relations between the Hough transform and more mainstream statistical paradigms and methods are discussed as well.

math.ST↗