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.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ144 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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.Comment: Published at http://dx.doi.org/10.1214/009053606000000326 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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].Comment: Published in at http://dx.doi.org/10.1214/11-AOS883 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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.Comment: Published in at http://dx.doi.org/10.1214/08-AAP589 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

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.Comment: Published at http://dx.doi.org/10.1214/009053604000000760 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

On adaptive minimax density estimation on RdR^d

We address the problem of adaptive minimax density estimation on \bR^d with \bL_p--loss on the anisotropic Nikol'skii classes. We fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm indexes in definitions of the functional class and of the risk. In particular, we show that there are four different regimes with respect to the behavior of the minimax risk. We develop a single estimator which is (nearly) optimal in orderover the complete scale of the anisotropic Nikol'skii classes. Our estimation procedure is based on a data-driven selection of an estimator from a fixed family of kernel estimators

Optimal change-point estimation from indirect observations

We study nonparametric change-point estimation from indirect noisy observations. Focusing on the white noise convolution model, we consider two classes of functions that are smooth apart from the change-point. We establish lower bounds on the minimax risk in estimating the change-point and develop rate optimal estimation procedures. The results demonstrate that the best achievable rates of convergence are determined both by smoothness of the function away from the change-point and by the degree of ill-posedness of the convolution operator. Optimality is obtained by introducing a new technique that involves, as a key element, detection of zero crossings of an estimate of the properly smoothed second derivative of the underlying function.Comment: Published at http://dx.doi.org/10.1214/009053605000000750 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org