Abstract: We study the estimation of a regression function by the kernel method. Under mild conditions on the “window”, the “bandwidth” and the underlying distribution of the bivariate observations {(X i , Y i)}, we obtain the weak and strong uniform convergence rates on a bounded interval. These results parallel those of Silverman (1978) on density estimation and extend those of Schuster and Yakowitz (1979) and Collomb (1979) on regression estimation.

Abstract: By representing the location and scale parameters of an absolutely continuous distribution as functionals of the usually unknown probability density function, it is possible to provide estimates of these parameters in terms of estimates of the unknown functionals.

Abstract: We consider the estimation of the 90 and 95 percentiles of a normal distribution and also the construction of one-sided 90% and 95% -normal ranges. Three methods are proposed -the sample percentile method, and two based on kernel estimates of the density function using Fryer's method and the leaving-one-out method for choosing a smoothing parameter. A simulation study compares the methods in terms of bias, variance and mean square error of the population percentile estimates and of the eovers of the consequent normal ranges.

Abstract: The rates at which integrated mean square and mean squre errors of nonparametric density estimation by orthogonal series method for sequences of strictly stationary strong mixing random variables are obtained. These rates are better than those known to hold for the independent case and they are shown to hold for Markov processes. In fact our results when specialized to the independent case are improvements over previously known results of Schwartz (1967,Ann. Math. Statist.,38, 1262–1265). An extension of the results to estimation of the bivariate density is also given.

Abstract: A crucial problem in kernel density estimates of a probability density function is the selection of the bandwidth. The aim of this study is to propose a procedure for selecting both fixed and variable bandwidths. The present study also addresses the question of how different variable bandwidth kernel estimators perform in comparison with each other and to the fixed type of bandwidth estimators. The appropriate algorithms for implementation of the proposed method are given along with a numerical simulation.The numerical results serve as a guide to determine which bandwidth selection method is most appropriate for a given type of estimator over a vide class of probability density functions, Also, we obtain a numerical comparison of the different types of kernel estimators under various types of bandwidths.

Abstract: Spatial point patterns which possess a natural origin are considered. Two ways of estimating the marginal radial and angular probability density functions associated with the stochastic process underlying such a pattern are presented. These methods are based on one and two-dimensional kernel functions respectively. The angular density estimate can be used to detect angular trend and to test for angular uniformity within a particular sector about the origin. The two methods of estimation produce essentially the same results. That based on the one-dimensional kernel is recommended because it is computationally simpler.

Abstract: L p notion of the weak, mean, and strong consistency of the kernel method of multivariate density estimation is proposed and studied. The results expand, unify, or generalize most known results in the literature. Rates of convergence in mean and strongL p-consistencies are presented.

Abstract: For nonparametric estimates of multivariate density function based on trigonometric series expansions general conditions for weak and strong pointwise consistency were proved. Moreover the rate of conver­gence has been studied.

Abstract: : The problem addressed concerns the estimation of a p-dimensional multivariate density, given only a set of n observation vectors, together with information that the density function is likely to be reasonably smooth. A solution is proposed which employs up to n + 1/2 p(p+1) smoothing parameters, all of which may be estimated by their posterior means. This avoids the well-known difficulties, associated with even one-dimensional kernel estimators, of estimating the bandwidth or smoothing parameter by a mathematical procedure. The posterior mean value function, unconditional upon the smoothing parameters, turns out to be a data-based mixture of multivariate t-distributions. The corresponding estimate of the sampling covariance matrix may be viewed as a shrinkage estimator of the Bayes-Stein type. The results involve some finite series which may be evaluated by straightforward simulation procedure. (Author)

Abstract: : A technique for modeling bivariate data that is based on the theory of orthogonal expansions in a separable Hilbert space is examined. A new nonparametric density estimation procedure is developed using an information criterion and is shown to be equivalent to least squares estimation of a density when the criterion function is computed with respect to the empirical distribution function. Computer programs are presented that implement the procedure for the univariate and bivariate cases. Examples utilizing these programs are given and comparisons made to existing density estimation techniques. (Author)

Abstract: : A method of estimation for the parameters of the multivariate normal distribution based on the characteristic function (density) and its sample counterpart is given. These M-estimators are dependent on a user-specified parameter. The response of the parameter estimates and observation weights to variation of this user-specified parameter allows a sensitivity analysis of the data and the model considered as a single entity. The estimators have desirable robustness properties, are easy to compute and use, are relatively efficient at the multivariate normal and are useful in identifying potential outliers and problems with the statistical assumptions or the data. The method is extended to include multivariate experimental designs with attention restricted to the two-way cross classification. Several illustrations are provided. (Author)

Abstract: Based on sample values of a one-dimensional random variable a nonparametric maximum likelihood estimate for the unknown probability density is introduced as the solution of an optimization problem in an appropriate Hilbert space. This solution turns out to be a polynomial spline function, and a complete characterization is given using recent results on the differentiability of the optimal value of a parametrized family of optimization problems. An important feature of this estimate is that its support interval results in a quite natural way from the formulation of the problem and is not fixed in advance. The estimator is shown to have a certain consistency property for a special class of density functions. Numerical results will be given in a subsequent paper.

Abstract: We consider the problem of estimating a probability density from observations from that density which are further contaminated by random errors. We propose a method of estimation using spline functions, discuss the numerical implementation of the method, and prove its consistency. The problem is motivated by the analysis of DNA content obtained by microfluorometry, and an example of such an analysis is included.

Abstract: In a decreasing sequence of intervals centered on the true mode the normalized kernel estimate of the density converges weakly to a nonstationary Gaussian random process. The expected value of this process is a parabola through the origin. The covariance function of this process depends on the smoothness of the kernel. When the kernel is mean-square differentiable the location of the maximum of this process has a normal distribution. When the kernel is discontinuous the location of the maximum has a distribution related to a solution of the heat equation.

Abstract: In this paper we derive a law of the logarithm for the maximal deviation between a kernel density estimator and the true underlying density function. Extensions to higher derivatives are included. The results are applied to get optimal window-widths with respect to almost sure uniform convergence.