Remarks on Some Nonparametric Estimates of a Density Function
01 Sep 1956-Annals of Mathematical Statistics (Institute of Mathematical Statistics)-Vol. 27, Iss: 3, pp 832-837
TL;DR: In this article, some aspects of the estimation of the density function of a univariate probability distribution are discussed, and the asymptotic mean square error of a particular class of estimates is evaluated.
Abstract: This note discusses some aspects of the estimation of the density function of a univariate probability distribution. All estimates of the density function satisfying relatively mild conditions are shown to be biased. The asymptotic mean square error of a particular class of estimates is evaluated.
Citations
More filters
TL;DR: In this article, the theory and application of Smoothed particle hydrodynamics (SPH) since its inception in 1977 are discussed, focusing on the strengths and weaknesses, the analogy with particle dynamics and the numerous areas where SPH has been successfully applied.
Abstract: In this review the theory and application of Smoothed particle hydrodynamics (SPH) since its inception in 1977 are discussed. Emphasis is placed on the strengths and weaknesses, the analogy with particle dynamics and the numerous areas where SPH has been successfully applied.
4,070 citations
TL;DR: Kernel methods are of flexible form and can be used where simple parametric models are found to be inappropriate or difficult to specify and give alternative approaches to the Anderson (1982) Fourier transform methods.
Abstract: In this paper kernel methods for the nonparametric estimation of the utilization distribution from a random sample of locational observations made on an animal in its home range are described. They are of flexible form, thus can be used where simple parametric models are found to be inappropriate or difficult to specify. Two examples are given to illustrate the fixed and adaptive kernel approaches in data analysis and to compare the methods. Various choices for the smoothing parameter used in kernel methods are discussed. Since kernel methods give alternative approaches to the Anderson (1982) Fourier transform methods, some comparisons are made.
3,949 citations
Book•
01 Jan 1996
TL;DR: The Bayes Error and Vapnik-Chervonenkis theory are applied as guide for empirical classifier selection on the basis of explicit specification and explicit enforcement of the maximum likelihood principle.
Abstract: Preface * Introduction * The Bayes Error * Inequalities and alternatedistance measures * Linear discrimination * Nearest neighbor rules *Consistency * Slow rates of convergence Error estimation * The regularhistogram rule * Kernel rules Consistency of the k-nearest neighborrule * Vapnik-Chervonenkis theory * Combinatorial aspects of Vapnik-Chervonenkis theory * Lower bounds for empirical classifier selection* The maximum likelihood principle * Parametric classification *Generalized linear discrimination * Complexity regularization *Condensed and edited nearest neighbor rules * Tree classifiers * Data-dependent partitioning * Splitting the data * The resubstitutionestimate * Deleted estimates of the error probability * Automatickernel rules * Automatic nearest neighbor rules * Hypercubes anddiscrete spaces * Epsilon entropy and totally bounded sets * Uniformlaws of large numbers * Neural networks * Other error estimates *Feature extraction * Appendix * Notation * References * Index
3,598 citations
TL;DR: Applications of gradient estimation to pattern recognition are presented using clustering and intrinsic dimensionality problems, with the ultimate goal of providing further understanding of these problems in terms of density gradients.
Abstract: Nonparametric density gradient estimation using a generalized kernel approach is investigated. Conditions on the kernel functions are derived to guarantee asymptotic unbiasedness, consistency, and uniform consistency of the estimates. The results are generalized to obtain a simple mcan-shift estimate that can be extended in a k -nearest-neighbor approach. Applications of gradient estimation to pattern recognition are presented using clustering and intrinsic dimensionality problems, with the ultimate goal of providing further understanding of these problems in terms of density gradients.
3,125 citations
References
More filters
Book•
01 Jan 1950
7 citations
"Remarks on Some Nonparametric Estim..." refers background in this paper
...Moreover, (2) is an unbiased estimate of F(b) — F(a), since E f SiviXw-tXJdy** f ES(y;X1} • • -7Xn) dy = f f(y) dy = F(b) - F(a)...
[...]
...But then (2) fsiviXw-tXJdy is a symmetric estimate of F(b) - F(a) - [ f(y) dy....
[...]