Abstract: We show that Kronmal and Tarter's well-known rule for selecting the terms in an orthogonal series density estimator can lead to poor performance and even inconsistency in certain cases. These difficulties arise when the underlying density has a nonmonotone sequence of Fourier coefficients, as is likely to be the case with sharply peaked or multimodal distributions. We suggest a way of overcoming these shortcomings.

Abstract: Comparisons of parametric and nonparametric approaches to discriminant analysis have been reported in which the kernel density method was surprisingly superior to conventional methods such as the linear discriminant function under conditions of normality. The assumptions underlying these comparisons, particularly those of independence, and their implications for product kernel methods are critically examined. A new comparison is made allowing for correlation. It is found that for independent or modestly positively correlated variables, the kernel method is superior to conventional parametric methods unadjusted for the particular correlation structure. With other correlation structures, however, the kernel method behaves erratically and may give poor classification or poor estimation of odds or both depending on the underlying parametric configuration.

Abstract: Nonparametric recursive kernel estimators of a multivariate data generating process (DGP) are presented and their asymptotic biases, variances and distributions are examined. Weak and strong consistency of these estimators are also proved. Remarks on the choice of the kernel function and the bandwidth function are made. Recursive estimates of the marginal and the conditional DGP are deduced from the estimates of the multivariate density. Finally, an application of these estimates to estimation and specification of econometric models is pointed out.

Abstract: Unrestricted nonparametric multivariate density estimation suffers from difficult convergence and computational problems. One way to overcome these problems is to exploit presumed or estimated structure in the density. The isopieth density estimator presumes or estimates the structure of the contours of the density to effectively reduce the dimensionality. The estimator incorporates an order-preserving algorithm to insure that higher isopleths have higher density estimates than lower isopleths. Convergence properties and a simulation are presented. The importance of edge effects is also noted.

Abstract: The nonparametric estimate derived from the Hermite orthogonal system of the functional I=\int f^{2}(x) dx where f is an unknown probability density, is studied. Sufficient conditions for the weak and strong consistency of the estimate are presented, and the rate of convergence is given. In particular, under mild assumptions on f , the rate of mean-square error convergence is O(n^{-1}) , whereas for almost complete convergence it is O((n^{-1} \log n)^{1/2}) . Moreover, several possible applications in the area of nonparametric inference of the estimate are indicated.

Abstract: The derivation of a new class of nonparametric density function estimators, the so-called bootstrap functional estimators (BFE's), is given. These estimators are shown to be strongly consistent under fairly nonrestrictive conditions. Some small-sample properties are discussed and a number of graphs are presented.

Abstract: The standard error of the sample mean for autocorrelated data is directly proportional to the value of the spectral density evaluated at zero frequency of the process being sampled. Thus confidence intervals for the true mean using traditional formulas can be greatly in error.This paper describes both parametric and nonparametric methods of spectral density estimation and illustrates numerically the basic results using a personal computer program called TIMESLAB which acts as a laboratory for studying such problems.The results of the paper indicate that in many situations, adjusting for auto-correlation is easily performed.

Abstract: The relationship is studied between certain estimates of the distribution density, and also between problems of parametric estimation of the density and pointwise estimation of parametric functions.

Abstract: : This paper makes two important contributions to the theory of bandwidth selection for kernel density estimators under right censorship. First, an asymptotic representation of the integrated squared error into easily understood variance and squared bias components is given. Second, it is shown that if the bandwidth is chosen by the data-based method of least squares cross-validation, then it is asymptotically optimal in a compelling sense. A by-product of the first part is an interesting comparison of the two most popular kernel estimators. Keywords: Nonparametric density estimation; Smoothing parameter.