Abstract: Nonparametric density estimation requires the specification of smoothing parameters. The demands of statistical objectivity make it highly desirable to base the choice on properties of the data set. In this article we introduce some biased cross-validation criteria for selection of smoothing parameters for kernel and histogram density estimators, closely related to one investigated in Scott and Factor (1981). These criteria are obtained by estimating L 2 norms of derivatives of the unknown density and provide slightly biased estimates of the average squared L 2 error or mean integrated squared error. These criteria are roughly the analog of Wahba's (1981) generalized cross-validation procedure for orthogonal series density estimators. We present the relationship of the biased cross-validation procedure to the least squares cross-validation procedure, which provides unbiased estimates of the mean integrated squared error. Both methods are shown to be based on U statistics. We compare the two metho...

Abstract: SUMMARY We study two natural classes of kernel density estimators for use with spherical data. Members of both classes have already been used in practice. The classes have an element in common, but for the most part they are disjoint. However, all members of the first class are asymptotically equivalent to one another, and to a single element of the second class. In this sense the second class 'contains' the first. It includes some estimators which out-perform all those in the first class, if loss is measured in either squared-error or Kullback-Leibler senses. Explicit formulae are given for bias, variance and loss, and large-sample properties of these quantities are described. Numerical illustrations are presented.

Abstract: 1. Historical Background 2. Some Approaches to Nonparametric Density Estimation 3. Maximum Likelihood Density Estimation 4. Maximum Penalized Likelihood Density Estimation 5. Discrete Maximum Penalized Likelihood Estimation 6. Nonparametric Density of Estimation in Higher Dimensions 7. Nonparametric Regression and Intensity Function Estimation 8. Model Building and Speculative Data Analysis Appendix I. An Introduction to Mathematical Optimization Theory Appendix II. Numerical Solution of Constrained Optimization Problems Appendix III. Optimization Algorithms for Noisy Problems Appendix IV. A Brief Primer in Simulation Index.

Abstract: In the setting of nonparametric multivariate density estimation, theorems are established which allow a comparison of the Kullback-Leibler and the least-squares cross-validation methods of smoothing parameter selection The family of delta sequence estimators (including kemel, orthogonal series, histogram and histospline estimators) is considered These theorems also show that either type of cross validation can be used to compare different estimators (eg, kernel versus orthogonal series) 1 Introduction Consider the problem of trying to estimate a d-dimensional probability density function, f(x), using a random sample, Xl,, Xn, from f Most proposed estimators of f depend on a "smoothing parameter," say X E DR +, whose selection is crucial to the performance of the estimator In this paper, for the large class of delta sequence estimators, theorems are obtained which allow comparison of two smoothing parameter selectors which are known to be asymptotically optimal An important consequence of these results is that either smoothing parameter selector may be used for a data based comparison of two density estimators, for example, kernel versus orthogonal series Another attractive feature of these results is that they are set in a quite general framework, special cases of which provide simpler proofs of several recent asymptotic optimality results In Sections 2 and 3 the family of delta sequence estimators and the smoothing parameter selectors are given The theorems are stated in Section 4, with some remarks in Section 5 The rest of the paper consists of proofs

Abstract: Annual flood risk is usually estimated by fitting an a priori assumed probability distribution funetion to the observed annual extremepeak series (parametrie method). The main shorteomings of such a proeedure are: the seleetion of a distribution, reliability of parameters (espeeially for skewed data with short record length), inability to analyze multimodal distributions resulting from flooding due to snowmelt versus thunderstorm aetivity, and treatment of outliers.

Abstract: We note that the well-known FABIAN'S approach to stochastic approximation of minima of smooth regressiion functions carries over to kernel estimation of smooth densities in a straightforward manner. Both finitely many times differentiable and analytic densities are considered.

Abstract: Sufficient conditions are obtained for asymptotically unbiased and consistent estimators of prior probability density.

Abstract: A class of Maximum Penalized Likelihood Estimators (MPLE) of the density function is constructed. The density estimator enables the solution of other nonparametric estimation problems: (1) density estimation for censored data, (2) intensity function estimation for inhomogeneous Poisson processes, (3) multivarate regression, (4) estimation of hazard-rate functions. The flexibility of the penalty function permits the construction of splines with improved performance at the peaks and valleys of the density curves. This construction leads to a large-scale, constrained optimization problem. The dimension of this problem and the denseness of the Hessian matrix make Newton's method inconvenient to use. Instead, a truncated-Newton method is suggested. This method is better able to handle the size and exploit the idiosyncracies of the optimization problem. Graphs of estimators are given.

Abstract: : A new family of Fourier-based estimators of the parameters of the multivariate Gaussian distribution is presented. The estimators are equivalent to parametric density estimators. Three distinct estimators arise, each of which is robust and reduces to the maximum likelihood estimator as a special case. By varying the window width of a parametric density estimator, a set of diagnostics which are useful in problems of outlier detection and clustering are obtained. An example, using a trivariate data set, is given. Keywords: Covariance matrices.

Abstract: The kernel density estimators can be considered as averages of (usually) symmetric probability density functions which are centered at the sample data points Consequently, the appropriate function-space setting for these kernel density estimators is of considerable interest and has been discussed in the literature In this paper, the space of real-valued continuous functions which go to zero at ±∞, Co (R), with the supremum norm, is proposed for these considerations Laws of large numbers are developed for Co (R) which have direct application in establishing the uniform strong consistency of the kernel density estimators Moreover, under mild conditions on the kernel functions, it can be shown that no proper subspace of Co (R) will suffice for these considerations

Abstract: We study the asymptotic behavior of the errors in the Parzen estimators of normal and Cauchy distribution densities for different kernels.

Abstract: . A linear stationary and invertible process yt models the second-order properties of T observations on a discrete time series, up to finitely many unknown parameters θ. Two estimators of the residuals or innovations ɛt of yt are presented, based on a θ estimator which is root-T consistent with respect to a wide class of ɛt distributions, such as a Gaussian estimator. One sets unobserved yt equal to their mean, the other treats yt as a circulant and may be best computed via two passes of the fast Fourier transform. The convergence of both estimators to ɛt is investigated. We apply the estimated ɛt to estimate the probability density function of ɛt. Kernel density estimators are shown to converge uniformly in probability to the true density. A new sub-class of linear time series models is motivated.

Abstract: Let f be a nonparametric probability density function on the real line. For estimating f(x) at a given point x we consider a recursive estimator fn (x) consisting of kernel functions. For given α (0 0 we define a certain class of stopping rules Nd and take as a 2d–width confidence interval for f(x). In this paper we derive the rate at which the probability converges to α as d tends to zero.