New Multivariate Product Density Estimators

Devroye, Luc; Krzyżak, Adam

doi:10.1006/jmva.2001.2021

Cited by 30 publications

(11 citation statements)

References 20 publications

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“…For example, using an L p norm on the d-vectors of differences between ranks, one can show that the classical k-NN regression function estimate is universally consistent in the sense of Stone [37]. This was observed by Olshen [30], and shown by Devroye [8] (see also [15,16,10,2] for related works). Rules based upon statistically equivalent blocks (see, e.g., [1,34,13,14], and [9, Section 21.4]) are other important examples of regression methods invariant with respect to monotone transformations of the coordinate axes.…”

Section: Introductionmentioning

confidence: 61%

An affine invariant k-nearest neighbor regression estimate

Biau

Devroye

Dujmović

et al. 2012

Journal of Multivariate Analysis

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Section: Introductionmentioning

confidence: 61%

An affine invariant k-nearest neighbor regression estimate

Biau

Devroye

Dujmović

et al. 2012

Journal of Multivariate Analysis

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“…4. The k -nearest neighbor distribution density estimator (NNDE), which is based on the distance to the nearest k observation position evaluation to the tested random dimension and has been examined by Devroye and Krzyżak [9];…”

Section: The Density Estimation Algorithms Analysedmentioning

confidence: 99%

“…is the k-th nearest x neighbor in the whole sample of observations {X(t), t = 1, ..., n}. k is the nearest neighbor density estimator depending on the coordinate system, as the distances vary after their changes x − X(t) , 1 ≤ t ≤ n. In order to avoid this dependence, the following changes are introduced [9]:…”

Section: The Density Estimation Algorithms Analysedmentioning

confidence: 99%

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Accuracy of Nonparametric Density Estimation for Univariate Gaussian Mixture Models: A Comparative Study

Arnastauskaitė

Ruzgas

2020

Mathematical Modelling and Analysis

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Flexible and reliable probability density estimation is fundamental in unsupervised learning and classification. Finite Gaussian mixture models are commonly used for this purpose. However, the parametric form of the distribution is not always known. In this case, non-parametric density estimation methods are used. Usually, these methods become computationally demanding as the number of components increases. In this paper, a comparative study of accuracy of some nonparametric density estimators is made by means of simulation. The following approaches have been considered: an adaptive bandwidth kernel estimator, a projection pursuit estimator, a logspline estimator, and a k-nearest neighbor estimator. It was concluded that data clustering as a pre-processing step improves the estimation of mixture densities. However, in case data does not have clearly defined clusters, the pre-preprocessing step does not give that much of advantage. The application of density estimators is illustrated using municipal solid waste data collected in Kaunas (Lithuania). The data distribution is similar (i.e., with kurtotic unimodal density) to the benchmark distribution introduced by Marron and Wand. Based on the homogeneity tests it can be concluded that distributions of the municipal solid waste fractions in Kutaisi (Georgia), Saint-Petersburg (Russia), and Boryspil (Ukraine) are statistically indifferent compared to the distribution of waste fractions in Kaunas. The distribution of waste data collected in Kaunas (Lithuania) follows the general observations introduced by Marron and Wand (i.e., has one mode and certain kurtosis).

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“…Note from Lemma 1.1 and (1.7) and (1.8) thatĥ N (x) bears the same relationship to B (N ) andF N that h N (x) does to B (N ) and F . One can show thatĥ N (X) → h(X) in various senses as N grows without bound; see, for example, [8], Theorem 12.7 of [1] and [4,5,10,11,14]. A particularly strong notion of convergence, but one that matters for applications, is unconditional almost sure convergence, where "unconditional" is meant with respect to the learning sample and test case.…”

Section: Definitions and Notationmentioning

confidence: 99%

Tree-structured regression and the differentiation of integrals

Olshen¹

2007

Ann. Statist.

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This paper provides answers to questions regarding the almost sure limiting behavior of rooted, binary tree-structured rules for regression. Examples show that questions raised by Gordon and Olshen in 1984 have negative answers. For these examples of regression functions and sequences of their associated binary tree-structured approximations, for all regression functions except those in a set of the first category, almost sure consistency fails dramatically on events of full probability. One consequence is that almost sure consistency of binary tree-structured rules such as CART requires conditions beyond requiring that (1) the regression function be in ${\mathcal {L}}^1$, (2) partitions of a Euclidean feature space be into polytopes with sides parallel to coordinate axes, (3) the mesh of the partitions becomes arbitrarily fine almost surely and (4) the empirical learning sample content of each polytope be ``large enough.'' The material in this paper includes the solution to a problem raised by Dudley in discussions. The main results have a corollary regarding the lack of almost sure consistency of certain Bayes-risk consistent rules for classification.Comment: Published at http://dx.doi.org/10.1214/009053606000001000 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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New Multivariate Product Density Estimators

Cited by 30 publications

References 20 publications

An affine invariant k-nearest neighbor regression estimate

An affine invariant k-nearest neighbor regression estimate

Accuracy of Nonparametric Density Estimation for Univariate Gaussian Mixture Models: A Comparative Study

Tree-structured regression and the differentiation of integrals

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