Showing papers on "Multivariate kernel density estimation published in 1976"
TL;DR: The kernel method of density estimation from continuous to multivariate binary spaces is described, finding its simple nonparametric nature together with its consistency properties make it an attractive tool in discrimination problems, with some advantages over already proposed parametric counterparts.
Abstract: SUMMARY An extension of the kernel method of density estimation from continuous to multivariate binary spaces is described. Its simple nonparametric nature together with its consistency properties make it an attractive tool in discrimination problems, with some advantages over already proposed parametric counterparts. The method is illustrated by an application to a particular medical diagnostic problem. Simple extensions of the method to categorical data and to data of mixed binary and continuous form are indicated.
658 citations
TL;DR: In this article, an Introduction to the Implementation and Theory of Nonparametric Density Estimation (NDE) is presented. The American Statistician: Vol. 30, No. 3, pp. 105-112.
Abstract: (1976). An Introduction to the Implementation and Theory of Nonparametric Density Estimation. The American Statistician: Vol. 30, No. 3, pp. 105-112.
102 citations
TL;DR: In this article, four new methods for classification of multivariate binary data are presented, based on an orthogonal expansion of the density in terms of discrete Fourier series, and the performance of these methods in 11 populations of various structures was measured in terms for mean error of misclassification and was compared to three well-known methods.
Abstract: Four new methods for classification of multivariate binary data are presented, based on an orthogonal expansion of the density in terms of discrete Fourier series. The performance of these methods in 11 populations of various structures was measured in terms of mean error of misclassification and was compared to three well-known methods. Also, performance in density estimation was measured for the appropriate methods. In general, the new methods seem to be superior for classification as well as for density estimation.
60 citations
TL;DR: In this paper, a theory of estimation of a regression function by the Parzen kernel-type density estimators is developed in the following points: 1) convergence of the estimators to the regression function at a continuous point.
Abstract: In this paper a theory of estimation of a regression function by the Parzen kernel-type density estimators is developed in the following points: 1) convergence of the estimators to the regression function at a continuous point, 2) convergence of the mean square error at a continuous point, and 3) the speed of the convergence in 2).
48 citations
Dissertation•
1 Apr 1976
TL;DR: Two nonparametric probability density estimators are considered and a discrete maximum penalized likelihood estimator is proposed and is shown to be consistent in the mean square error.
Abstract: Two nonparametric probability density estimators are considered. The first is the kernel estimator. The problem of choosing the kernel scaling factor based solely on a random sample is addressed. An interactive mode is discussed and an algorithm proposed to choose the scaling factor automatically. The second nonparametric probability estimate uses penalty function techniques with the maximum likelihood criterion. A discrete maximum penalized likelihood estimator is proposed and is shown to be consistent in the mean square error. A numerical implementation technique for the discrete solution is discussed and examples displayed. An extensive simulation study compares the integrated mean square error of the discrete and kernel estimators. The robustness of the discrete estimator is demonstrated graphically.
32 citations
1 Jul 1976
TL;DR: The multivariate Gaussian process with the same variance/covariance structure as the multivariate kernel density estimator in Euclidean space of dimension d is considered in this article, and weak and strong bounds are placed on the asymptotic behaviour of the maximum of the process over a multidimensional interval which is allowed to increase as the sample size increases.
Abstract: The multivariate Gaussian process with the same variance/covariance structure as the multivariate kernel density estimator in Euclidean space of dimension d is considered. An exact result is obtained for the limit in probability of the maximum of the normalized process. In addition weak and strong bounds are placed on the asymptotic behaviour of the maximum of the process over a multidimensional interval which is allowed to increase as the sample size increases. All the bounds obtained on the process areOnly the uniform continuity of the underlying density is assumed; the conditions on the kernel are also mild.
17 citations
TL;DR: In this article, the existence of optimal estimators for the unknown density function, based on a finite number of independent observations, is studied, where the statistical problem given is invariant with respect to a certain transformation group, and the invariant density estimators form an essentially complete class of decision functions.
Abstract: This paper is concerned with the existence of optimal estimators for the unknown density function, based on a finite number of independent observations. If the statistical problem given is invariant with respect to a certain transformation group, then the invariant density estimators form an essentially complete class of decision functions. If, in particular, the sample space is an Euclidean space and if the densities in question are with respect to Lebesgue measure, an optimal density estimator exists, which is symmetric in the observations and invariant under isometric transformations. If a sufficient sub-σ-algebra ℭ exists, under additional conditions, only ℭ-measurable density estimators are to be considered.
3 citations
TL;DR: In this article, optimal weighting matrices are used for matrix density estimators, and the result is a new class of density estimator, the collection of matrix density estimation estimators.
Abstract: The accuracy of orthogonal series types of density estimators can be conveniently measured in terms of their Mean Integrated Squared Error, or MISE. Further reduction In MISE is achieved by introducing certain weighting factors into the estimators. In this paper we consider optimal weighting matrices, and the result is a new class of density estimators, the collection of matrix density estimators.
2 citations
1 Apr 1976
TL;DR: In this paper, a theory of multivariate life table analysis is developed and estimators for various multivariate interval failure rates are developed for multivariate extensions of the usual estimator in the univariate case.
Abstract: : A theory of multivariate life table analysis is developed. Estimators are developed for various multivariate interval failure rates. These estimators are multivariate extensions of the usual estimator in the univariate case. Under appropriate conditions, the estimators are shown to be asymptotically normal and asymptotically independent. The key tool used is a theorem of Sethuraman (1963) 'Some limit distributions connected with fixed interval analysis', Sankhya, Series A, 25, 395-398.
1 citations
TL;DR: Parzen estimators are often used for nonparametric estimation of probability density functions and a problem-dependent criterion for its value is proposed and illustrated by some examples.
Abstract: Parzen estimators are often used for nonparametric estimation of probability density functions. The smoothness of such an estimation is controlled by the smoothing parameter. A problem-dependent criterion for its value is proposed and illustrated by some examples. Especially in multimodal situations, this criterion led to good results.