Results 1  10
of
11
Asymptotic normality of plugin level set estimates
 Annals of Applied Probability
, 2009
"... We establish the asymptotic normality of the Gmeasure of the symmetric difference between the level set and a plugintype estimator of it formed by replacing the density in the definition of the level set by a kernel density estimator. Our proof will highlight the efficacy of Poissonization method ..."
Abstract

Cited by 24 (2 self)
 Add to MetaCart
We establish the asymptotic normality of the Gmeasure of the symmetric difference between the level set and a plugintype estimator of it formed by replacing the density in the definition of the level set by a kernel density estimator. Our proof will highlight the efficacy of Poissonization methods in the treatment of large sample theory problems of this kind.
Exact Rates in Density Support Estimation
"... Let f be an unknown multivariate probability density with compact support Sf. Given n independent observations X1,...,Xn drawn from f, this paper is devoted to the study of the estimator Ŝn of Sf defined as unions of balls centered at the Xi and of common radius rn. measure the proximity between S ..."
Abstract

Cited by 17 (1 self)
 Add to MetaCart
Let f be an unknown multivariate probability density with compact support Sf. Given n independent observations X1,...,Xn drawn from f, this paper is devoted to the study of the estimator Ŝn of Sf defined as unions of balls centered at the Xi and of common radius rn. measure the proximity between Ŝn and Sf, we employ a general criterion dg, based on some function g, which encompasses many statistical situations of interest. Under mild assumptions on the sequence (rn) and some analytic conditions on f and g, the exact rates of convergence of dg(Ŝn, Sf) are obtained using tools from Riemannian geometry. The conditions on the radius sequence are found to be sharp and consequences of the results are discussed from a statistical perspective.
Asymptotic normality in density support estimation
 Electron. J. Probab
"... c i E l e c t r o n J o u r n a l o f ..."
Complexity Penalized Support Estimation
, 2004
"... We consider the estimation of the support of a probability density function with iid observations. The estimator to be considered is a minimizer of a complexity penalized excess mass criterion. We present a fast algorithm for the construction of the estimator. The estimator is able to estimate s ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
We consider the estimation of the support of a probability density function with iid observations. The estimator to be considered is a minimizer of a complexity penalized excess mass criterion. We present a fast algorithm for the construction of the estimator. The estimator is able to estimate supports which consists of disconnected regions. We will prove that the estimator achieves minimax rates of convergence up to a logarithmic factor simultaneously over a scale of Hölder smoothness classes for the boundary of the support. The proof assumes a sharp boundary for the support.
Adaptation to lowest density regions with application to support recovery
, 2014
"... Adaptation to lowest density regions with ..."
Confidence Regions for Level Sets
, 2012
"... This paper discusses a universal approach to the construction of confidence regions for level sets {h(x) ≥ 0} ⊂Rd of a function h of interest. The proposed construction is based on a plugin estimate of the level sets using an appropriate estimate hn of h. The approach provides finite sample upper ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
This paper discusses a universal approach to the construction of confidence regions for level sets {h(x) ≥ 0} ⊂Rd of a function h of interest. The proposed construction is based on a plugin estimate of the level sets using an appropriate estimate hn of h. The approach provides finite sample upper and lower confidence limits. This leads to generic conditions under which the constructed confidence regions achieve a prescribed coverage level asymptotically. The construction requires an estimate of quantiles of the distribution of sup∆n hn(x) − h(x)  for appropriate sets ∆n ⊂ R d. In contrast to related work from the literature, the existence of a weak limit for an appropriately normalized process {hn(x),x ∈ D} is not required. This adds significantly to the challenge of deriving asymptotic results for the corresponding coverage level. Our approach is exemplified in the case of a density level set utilizing a kernel density estimator and a bootstrap procedure.
Asymptotic Normality in Density Support Estimation
"... Let X1,..., Xn be n independent observations drawn from a multivariate probability density f with compact support Sf. This paper is devoted to the study of the estimator ˆ Sn of Sf defined as the union of balls centered at the Xi and with common radius rn. Using tools from Riemannian geometry, and u ..."
Abstract
 Add to MetaCart
(Show Context)
Let X1,..., Xn be n independent observations drawn from a multivariate probability density f with compact support Sf. This paper is devoted to the study of the estimator ˆ Sn of Sf defined as the union of balls centered at the Xi and with common radius rn. Using tools from Riemannian geometry, and under mild assumptions on f and the sequence (rn), we prove a central limit theorem for λ(Sn∆Sf), where λ denotes the Lebesgue measure on R d and ∆ the symmetric difference operation.
Estimation of density level sets with a given probability content
, 2012
"... Given a random vector X valued in Rd with density f and an arbitrary probability number p ∈ (0; 1), we consider the estimation of the upper level set {f ≥ t(p)} of f corresponding to probability content p, that is, such that the probability that X belongs to {f ≥ t(p)} is equal to p. Based on an i.i ..."
Abstract
 Add to MetaCart
Given a random vector X valued in Rd with density f and an arbitrary probability number p ∈ (0; 1), we consider the estimation of the upper level set {f ≥ t(p)} of f corresponding to probability content p, that is, such that the probability that X belongs to {f ≥ t(p)} is equal to p. Based on an i.i.d. random sample X1,..., Xn drawn from f, we define the plugin level set estimate {f̂n ≥ t(p)n}, where t(p)n is a random threshold depending on the sample and f̂n is a nonparametric kernel density estimate based on the same sample. We establish the exact convergence rate of the Lebesgue measure of the symmetric difference between the estimated and actual level sets.
Summary
"... We present algorithms for finding the level set tree of a multivariate density estimate. That is, we find the separated components of level sets of the estimate for a series of levels, gather information on the separated components, such as volume and barycenter, and present the information togethe ..."
Abstract
 Add to MetaCart
We present algorithms for finding the level set tree of a multivariate density estimate. That is, we find the separated components of level sets of the estimate for a series of levels, gather information on the separated components, such as volume and barycenter, and present the information together with the tree structure of the separated components. The algorithm proceeds by first building a binary tree which partitions the support of the density estimate, followed by bottomup travels of this tree during which we join those parts of the level sets which touch each other. As a byproduct we present an algorithm for evaluating a kernel estimate on a large multidimensional grid. Since we find the barycenters of the separated components of the level sets also for high levels, our method finds the locations of local extremes of the estimate.
Author manuscript, published in "Electronic Journal of Probability (2009) 26172635" Asymptotic Normality in Density Support Estimation
, 2009
"... Let X1,..., Xn be n independent observations drawn from a multivariate probability density f with compact support Sf. This paper is devoted to the study of the estimator ˆ Sn of Sf defined as unions of balls centered at the Xi and of common radius rn. Using tools from Riemannian geometry, and under ..."
Abstract
 Add to MetaCart
(Show Context)
Let X1,..., Xn be n independent observations drawn from a multivariate probability density f with compact support Sf. This paper is devoted to the study of the estimator ˆ Sn of Sf defined as unions of balls centered at the Xi and of common radius rn. Using tools from Riemannian geometry, and under mild assumptions on f and the sequence (rn), we prove a central limit theorem for λ(Sn∆Sf), where λ denotes the Lebesgue measure on R d and ∆ the symmetric difference operation.