Results 1  10
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24
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 ..."
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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 ..."
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Cited by 17 (1 self)
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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.
Asymptotics and optimal bandwidth selection for highest density region estimation
 Annals of Statistics
"... We study kernel estimation of highest density regions (HDR). Our main contributions are twofold. Firstly, we derive a uniforminbandwidth asymptotic approximation to a risk that is appropriate for HDR estimation. This approximation is then used to derive a bandwidth selection rule for HDR estimat ..."
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Cited by 11 (1 self)
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We study kernel estimation of highest density regions (HDR). Our main contributions are twofold. Firstly, we derive a uniforminbandwidth asymptotic approximation to a risk that is appropriate for HDR estimation. This approximation is then used to derive a bandwidth selection rule for HDR estimation possessing attractive asymptotic properties. We also present the results of numerical studies that illustrate the benefits of our theory and methodology.
PLUGIN ESTIMATION OF LEVEL SETS IN A NONCOMPACT SETTING WITH APPLICATIONS IN MULTIVARIATE RISK THEORY
, 2011
"... This paper deals with the problem of estimating the level sets L(c) = {F(x) ≥ c}, with c ∈ (0,1), of an unknown distribution function F on R 2 +. A plugin approach is followed. That is, given a consistent estimator Fn of F, we estimate L(c) by Ln(c) = {Fn(x) ≥ c}. In our setting, noncompactnes ..."
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Cited by 8 (4 self)
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This paper deals with the problem of estimating the level sets L(c) = {F(x) ≥ c}, with c ∈ (0,1), of an unknown distribution function F on R 2 +. A plugin approach is followed. That is, given a consistent estimator Fn of F, we estimate L(c) by Ln(c) = {Fn(x) ≥ c}. In our setting, noncompactness property is a priori required for the level sets to estimate. We state consistency results with respect to the Hausdorff distance and the volume of the symmetric difference. Our results are motivated by applications in multivariate risk theory. In particular we propose a new bivariate version of the Conditional Tail Expectation by conditioning the twodimensional random vector to be in the level set L(c). We also present simulated and real examples which illustrate our theoretical results.
Clusters and water flows: a novel approach to modal clustering through Morse theory
, 2014
"... The problem of finding groups in data (cluster analysis) has been extensively studied by researchers from the fields of Statistics and Computer Science, among others. However, despite its popularity it is widely recognized that the investigation of some theoretical aspects of clustering has been re ..."
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Cited by 5 (2 self)
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The problem of finding groups in data (cluster analysis) has been extensively studied by researchers from the fields of Statistics and Computer Science, among others. However, despite its popularity it is widely recognized that the investigation of some theoretical aspects of clustering has been relatively sparse. One of the main reasons for this lack of theoretical results is surely the fact that, unlike the situation with other statistical problems as regression or classification, for some of the cluster methodologies it is quite difficult to specify a population goal to which the databased clustering algorithms should try to get close. This paper aims to provide some insight into the theoretical foundations of the usual nonparametric approach to clustering, which understands clusters as regions of high density, by presenting an explicit formulation for the ideal population clustering.
cluster
, 2004
"... A technique for conducting point pattern analysis of ..."
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Cited by 3 (0 self)
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A technique for conducting point pattern analysis of
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 ..."
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Cited by 1 (0 self)
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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.
support and
, 2012
"... Using the k−nearest neighbor restricted Delaunay polyhedron to estimate the density ..."
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Using the k−nearest neighbor restricted Delaunay polyhedron to estimate the density
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"... Nonparametric estimation of regression level sets using kernel plugin estimator ..."
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Nonparametric estimation of regression level sets using kernel plugin estimator
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 ..."
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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.