28,387 research outputs found
Conditional density estimation in a regression setting
Regression problems are traditionally analyzed via univariate characteristics
like the regression function, scale function and marginal density of regression
errors. These characteristics are useful and informative whenever the
association between the predictor and the response is relatively simple. More
detailed information about the association can be provided by the conditional
density of the response given the predictor. For the first time in the
literature, this article develops the theory of minimax estimation of the
conditional density for regression settings with fixed and random designs of
predictors, bounded and unbounded responses and a vast set of anisotropic
classes of conditional densities. The study of fixed design regression is of
special interest and novelty because the known literature is devoted to the
case of random predictors. For the aforementioned models, the paper suggests a
universal adaptive estimator which (i) matches performance of an oracle that
knows both an underlying model and an estimated conditional density; (ii) is
sharp minimax over a vast class of anisotropic conditional densities; (iii) is
at least rate minimax when the response is independent of the predictor and
thus a bivariate conditional density becomes a univariate density; (iv) is
adaptive to an underlying design (fixed or random) of predictors.Comment: Published in at http://dx.doi.org/10.1214/009053607000000253 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On clustering procedures and nonparametric mixture estimation
This paper deals with nonparametric estimation of conditional den-sities in
mixture models in the case when additional covariates are available. The
proposed approach consists of performing a prelim-inary clustering algorithm on
the additional covariates to guess the mixture component of each observation.
Conditional densities of the mixture model are then estimated using kernel
density estimates ap-plied separately to each cluster. We investigate the
expected L 1 -error of the resulting estimates and derive optimal rates of
convergence over classical nonparametric density classes provided the
clustering method is accurate. Performances of clustering algorithms are
measured by the maximal misclassification error. We obtain upper bounds of this
quantity for a single linkage hierarchical clustering algorithm. Lastly,
applications of the proposed method to mixture models involving elec-tricity
distribution data and simulated data are presented
General empirical Bayes wavelet methods and exactly adaptive minimax estimation
In many statistical problems, stochastic signals can be represented as a
sequence of noisy wavelet coefficients. In this paper, we develop general
empirical Bayes methods for the estimation of true signal. Our estimators
approximate certain oracle separable rules and achieve adaptation to ideal
risks and exact minimax risks in broad collections of classes of signals. In
particular, our estimators are uniformly adaptive to the minimum risk of
separable estimators and the exact minimax risks simultaneously in Besov balls
of all smoothness and shape indices, and they are uniformly superefficient in
convergence rates in all compact sets in Besov spaces with a finite secondary
shape parameter. Furthermore, in classes nested between Besov balls of the same
smoothness index, our estimators dominate threshold and James-Stein estimators
within an infinitesimal fraction of the minimax risks. More general block
empirical Bayes estimators are developed. Both white noise with drift and
nonparametric regression are considered.Comment: Published at http://dx.doi.org/10.1214/009053604000000995 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Decompounding on compact Lie groups
Noncommutative harmonic analysis is used to solve a nonparametric estimation
problem stated in terms of compound Poisson processes on compact Lie groups.
This problem of decompounding is a generalization of a similar classical
problem. The proposed solution is based on a char- acteristic function method.
The treated problem is important to recent models of the physical inverse
problem of multiple scattering.Comment: 26 pages, 3 figures, 25 reference
Variational Analysis of Constrained M-Estimators
We propose a unified framework for establishing existence of nonparametric
M-estimators, computing the corresponding estimates, and proving their strong
consistency when the class of functions is exceptionally rich. In particular,
the framework addresses situations where the class of functions is complex
involving information and assumptions about shape, pointwise bounds, location
of modes, height at modes, location of level-sets, values of moments, size of
subgradients, continuity, distance to a "prior" function, multivariate total
positivity, and any combination of the above. The class might be engineered to
perform well in a specific setting even in the presence of little data. The
framework views the class of functions as a subset of a particular metric space
of upper semicontinuous functions under the Attouch-Wets distance. In addition
to allowing a systematic treatment of numerous M-estimators, the framework
yields consistency of plug-in estimators of modes of densities, maximizers of
regression functions, level-sets of classifiers, and related quantities, and
also enables computation by means of approximating parametric classes. We
establish consistency through a one-sided law of large numbers, here extended
to sieves, that relaxes assumptions of uniform laws, while ensuring global
approximations even under model misspecification
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