Abstract: Recognition and extraction of features in a nonparametric density estimate are highly dependent on correct calibration. The data-driven choice of bandwidth h in kernel density estimation is a difficult one that is compounded by the fact that the globally optimal h is not generally optimal for all values of x. In recognition of this fact a new type of graphical tool, the mode tree, is proposed. The basic mode tree plot relates the locations of modes in density estimates with the bandwidths of those estimates. Additional information can be included on the plot indicating factors such as the size of modes, how modes split, and the locations of antimodes and bumps. The use of a mode tree in adaptive multimodality investigations is proposed, and an example is given to show the value in using a normal kernel, as opposed to the biweight or other kernels, in such investigations. Examples of such investigations are provided for Ahrens's chondrite data and van Winkle's Hidalgo stamp data. Finally, the biva...

Abstract: In this article, a class of penalized likelihood probability density estimators is proposed and studied. The true log density is assumed to be a member of a reproducing kernel Hilbert space on a finite domain, not necessarily univariate, and the estimator is defined as the unique unconstrained minimizer of a penalized log likelihood functional in such a space. Under mild conditions, the existence of the estimator and the rate of convergence of the estimator in terms of the symmetrized Kullback-Leibler distance are established. To make the procedure applicable, a semiparametric approximation of the estimator is presented, which sits in an adaptive finite dimensional function space and hence can be computed in principle. The theory is developed in a generic setup and the proofs are largely elementary. Algorithms are yet to follow.

Abstract: Kernel density estimation methods have recently been introduced as viable and flexible alternatives to parametric methods for flood frequency estimation. Key properties of such estimators are reviewed in this paper. Attention is focused on the selection of the kernel function and the bandwidth. These are the parameters of the method. Existing techniques for kernel and bandwidth selection are applied to three situations: Gaussian data, skewed data (three-parameter gamma), and mixture data. The intent was to investigate issues relevant to parameter estimation as well as to the likely performance of these methods with the small sample sizes typical in hydrology. Bandwidths chosen by minimizing a performance criterion related to the distribution function lead to much smaller mean square errors of tail probabilities than those chosen by cross-validation methods designed for density estimation. However, this can lead to estimates that degenerate to the empirical distribution function, and hence to an unusable flood frequency curve. Variable bandwidths with heavy tailed kernels appear to do best. Kernel estimators are increasingly more competitive in terms of mean square error of estimate as the underlying distribution gets more complex.

Abstract: Publisher Summary Nonparametric function estimation has been one of the most active fields of statistical research in the 1980s. In nonparametric inference (e.g., rank tests) and in robust statistics, methods have been developed that are less sensitive to a priori assumptions, the assumption of normal errors being an example. Nonparametric function estimation follows a similar line of thought. Nonparametric density estimation is closely related to this tradition: it provides an estimate of the distributional pattern without assuming an underlying parametric family of distributions. Regression estimation becomes nonparametric when the requirement of an a priori specified-parametric functional model for the dependence structure is relaxed. Many of these methods fall under the general heading “smoothing methods,” but reach by intent and technique beyond some purely heuristic smoothing method. In data analysis, nonparametric function estimation is intended as an exploratory tool and today's general availability of graphical techniques adds to its usefulness. The field of nonparametric function estimation comprises the types of functions such as hazard rate functions, spectral densities, and intensities. The chapter discusses the concept of regression estimation, problems that are formally almost the same for density estimation, and the development of algorithms and software for regression.

Abstract: We consider the problem of nonparametric regression when there are d explanatory variables which lie in a compact set . Perhaps the most common example is two explanatory variables which lie in the unit square. We adapt kernel-type smoothers suitable for estimating a function of one variable to the estimation of a function of two or more variables. Kernel estimators are attractive because their explicit representation as a weighted local average makes them easy to implement and to understand. Especially in two dimensions, the ease and interpretability of these smoothers is an argument for their use. In addition, the theoretical tractability of kernel smoothers makes them attractive as components of more complicated estimation schemes; see for example Staniswalis (1989a), or Goldstein and Messer (1992). The radially symmetric kernel estimators we study here present a conceptually appealing approach to two issues which arise for kernel estimators in higher dimensions: first, what is the optimal shape of the...

Abstract: Using the well known kernel method of estimation for multivariate probability densities due to Rosenblatt (1956), Parzen (1962) and Cacoullos (1966), testing the goodness of fit in the one sample, two sample, and testing symmetry cases are discussed. Test statistics presented here are based on the L2-norms . The estimates presented here for δ i i=l,2, 3 are modifications on those discussed by Bickel and Rosenblatt (1973), Rosenblatt (1975) and Hall (1984), and are asymptotically normal under the null and also under the alternatives with conditions much simpler than assumed in literature.

Abstract: Two types of unsupervised learning techniques for nonparametric multivariate density estimation are discussed, where no assumption is made about the data being drawn from any of known parametric families of distribution. The first type is based on a robust kernel method which uses locally tuned radial basis (Gaussian) functions. The second type is based on an exploratory projection pursuit technique which uses orthogonal polynomial approximation to 1-D density along several projections from multidimensional data. Performance evaluations using training data from mixture Gaussian and mixture Cauchy densities are presented. >

Abstract: The problem of estimating the integral of the square of a probability density function is considered, It is shown that under some regularity conditions the kernel estimate of this functional is asymptotically normally distributed. An expression for the smoothing parameter that minimizes the mean square error of the estimate is derived. Results of simulation studies are included.AMS (1980) Subject Classification: Primary 62G07 Secondary 60FOS.

Abstract: Summary The performance of kernel density estimation, in terms of mean integrated squared error, is investigated in the opposite of the usual situation, namely when the bandwidth is large. This affords noteworthy insights including the special role taken by the normal density function as kernel and a tie-in with ‘semiparametric’ approaches to density estimation.

Abstract: The use of kernel density estimate and some probabilistic argument leads to a new and improved estimate of density. Results of a simulation study based on small and moderately large samples confirm the better performance in regard to bias of the new estimate as compared to kernel density estimate and another estimate proposed before. The paper also presents the estimate obtained by the three methods for two real and one artificial data set.

Abstract: We introduce a universal functional Q(f) which measures the difficulty a given density ƒ poses to tne standard Kernel density estimate if one uses me optimal smoothing factor. The functional is well-defined (but possibly infinite) for all densities, regardless of their smoothness or tail properties. It is proportional to the limit of n2/5Eƒ| - ƒ| where ƒnis the optimal kernel estimate. This paper settles some questions legft unanswered in Devroye and Gyorfi (1985) and Hall and Wall (1988) 1991 Mathematics subject classifications:Primary 62 G07 Secondary:62G05, 62 F12, 60F25

Abstract: Probabilistic neural networks (PNN) build internal density representations based on the kernel or Parzen estimator and use Bayesian decision theory in order to build up arbitrarily complex decision boundaries. As in the classical kernel estimator, the training is performed in a single pass of the data and asymptotic convergence is guaranteed. One important factor affecting convergence is the kernel width. Theory only provides an optimal width in the case of normally distributed data. This problem becomes acute in multivariate cases. In this paper we present an asymptotically optimal method of setting kernel widths for multivariate Gaussian kernels based on the theory of filtered kernel estimators and show how this can be realized as a filtered kernel PNN architecture. Performance comparisons are made with competing methods.© (1993) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Abstract: The merits and limitations of parametric and nonparametric methods and the value of historical floods and palaeoflood information are reviewed and discussed A mixture density estimation procedure based on the Gumbel (EV1) distribution kernel is introduced and a modified maximum likelihood criterion is developed for estimation of model parameters Using the recorded data and pre-gauging floods in China and a limited number of simulation experiments, the flood quantiles estimated by the proposed model are compared with those estimated by parametric and nonparametric methods It is found that the mixture density estimation method can fit real data points more closely than can its parametric counterparts, and that it is competitive with the other considered candidates

Abstract: An improvement to the classic adaptive kernel estimator has been made by incorporating first order dynamics in a neural network framework that results in a fully self-consistent probability density function (pdf) estimate. The dynamics give rise to nonlinear interactions between the kernel parameters, resulting in a self-consistent pdf estimate. This is in contrast to the adaptive kernel estimator which is a simple three step procedure. Adaptive kernel estimates have asymptotic convergence rates of O(h/sup 4/) if the errors involved in the pilot estimate can be ignored. This is compared to standard kernel estimators which converge as O(h/sup 2/). By using a fully self-consistent method, this approach is also able to approach the theoretical O(h/sup 4/) convergence rate while providing smoother estimates of the distribution tails than the adaptive kernel estimator. A one-dimensional application to the estimation of a log-normal distribution is included as an example.

Abstract: EDRS PRICE MFO1 /PCO1 Plus Postage. DESCRIPTORS *Analysis of Variance; Computer Simulation; Estimation (Mathematics); Monte Carlo Methods; *Multivariate Analysis; Probability; *Robustness (Statistics); *Sample Size IDENTIFIERS Biomedical Computer Programs; Hotellings t; Likelihood Ratio Tests; Pill is Trace; Roys Largest Root; Statistical Analysis System; Statistical Package for the Social Sciences; *Type I Errors

Abstract: Representation and Geometry of Multivariate Data. Nonparametric Estimation Criteria. Histograms: Theory and Practice. Frequency Polygons. Averaged Shifted Histograms. Kernel Density Estimators. The Curse of Dimensionality and Dimension Reduction. Nonparametric Regression and Additive Models. Special Topics. Appendices. Indexes.