Abstract: An exact and easily computable expression for the mean integrated squared error (MISE) for the kernel estimator of a general normal mixture density, is given for Gaussian kernels of arbitrary order. This provides a powerful new way of understanding density estimation which complements the usual tools of simulation and asymptotic analysis. The family of normal mixture densities is very flexible and the formulae derived allow simple exact analysis for a wide variety of density shapes. A number of applications of this method giving important new insights into kernel density estimation are presented. Among these is the discovery that the usual asymptotic approximations to the MISE can be quite inaccurate, especially when the underlying density contains substantial fine structure and also strong evidence that the practical importance of higher order kernels is surprisingly small for moderate sample sizes. 1. Introduction. Substantial research has been devoted to kernel density estimation. This is because it provides a simple, yet appealing, context in which to study problems and issues that arise in all types of nonparametric curve estimation. This includes regression, spectral density and hazard estimation, and also a variety of other estimators, including histograms, splines and orthogonal series. Three important and useful tools for understanding the behavior of nonparametric curve estimators are asymptotic analysis, simulation and numerical calculation of error criteria. Each of these methods provides many useful insights into the complicated structure present in the study of curve estimation. However, each has its limitations as well. rhe strength of asymptotic analysis is that it frequently allows simultaneous study of many different specific examples, through general results applying to entire classes of settings. The weakness of asymptotics is that they only describe behavior in the limit. This is still very useful in many situations because the asymptotics describe the actual situation quite well. However, it is less useful when the asymptotics have not yet kicked in (that is, in studying situations where the asymptotically dominant effect has not taken over yet). Perhaps the biggest drawback to asymptotics is that it is very difficult to determine in a given situation which of these possibilities is occurring.

Abstract: We investigate some of the possibilities for improvement of univariate and multivariate kernel density estimates by varying the window over the domain of estimation, pointwise and globally. Two general approaches are to vary the window width by the point of estimation and by point of the sample observation. The first possibility is shown to be of little efficacy in one variable. In particular, nearest-neighbor estimators in all versions perform poorly in one and two dimensions, but begin to be useful in three or more variables. The second possibility is more promising. We give some general properties and then focus on the popular Abramson estimator. We show that in many practical situations, such as normal data, a nonlocality phenomenon limits the commonly applied version of the Abramson estimator to bias of $O(\lbrack h / \log h\rbrack^2)$ instead of the hoped for $O(h^4)$.

Abstract: Kernel estimators for d dimensional data are usually parametrized by either a single smoothing parameter, or d smoothing parameters corresponding to each of the coordinate directions. A generalization of each of these parameterizations is to use a d× d matrix which allows smoothing in arbitrary directions. We demonstrate that, at this level of generality, the usual error approximations and their numerical minimization can be done quite simply using matrix algebra. The minimization formulas have the practical importance that they can be applied to data-driven selection of the smoothing parameters using a ”plug-in approach. Particular attention is paid to the special case of kernel estimation of multivariate normal mixture densities where it is shown that the numerical evaluation and minimization of both asymptotic and exact mean integrated squared error can be set up in a matrix algebraic formulation which requires no numerical integration. This provides a flexible family of multivariate smoothing problems...

Abstract: This research is concerned with modeling the gray-level probability density function of the image and then using it to select optimal threshold values. Image histogram modeling by kernel density estimation is considered. The main goal is to model the histogram properly. A number of analytic techniques for threshold selection are discussed and empirically evaluated on test as well as real-world scenes. >

Abstract: We present a new nonparametric density estimate based on normalized tensor B–Splines. We show under the expected conditions that the non- parametric density estimate converges in mean square error and integrated mean square error. Results of simulations are also presented.

Abstract: Summary Using a global window width kernel estimator to estimate an approximately symmetric probability density with high kurtosis usually leads to poor estimation because good estimation of the peak of the distribution leads to unsatisfactory estimation of the tails and vice versa. The technique proposed corrects for kurtosis via a transformation of the data before using a global window width kernel estimator. The transformation depends on a “generalised smoothing parameter” consisting of two real-valued parameters and a window width parameter which can be selected either by a simple graphical method or, for a completely data-driven implementation, by minimising an estimate of mean integrated squared error. Examples of real and simulated data demonstrate the effectiveness of this approach, which appears suitable for a wide range of symmetric, unimodal densities. Its performance is similar to ordinary kernel estimation in situations where the latter is effective, e.g. Gaussian densities. For densities like the Cauchy where ordinary kernel estimation is not satisfactory, our methodology offers a substantial improvement.

Abstract: An adaptive maximum likelihood estimator based on the estimation of the log-density by B-splines is introduced. A data-driven method of selecting the smoothing paramieter involved in the consequent density estimation is demonstrated. A Monte Carlo study is conducted to evaluate the small sample performance of the estimator in a location and a regression problem. The adaptive estimator is seen to compare favorably to some standard estimates. We show that the estimator is asymptotically efficient. 1. Introduction. The problem of adaptive estimation was introduced by Stein (1956). One wishes to estimate a Euclidean parameter 0 in the presence of an infinite-dimensional shape parameter G (usually the density). An adaptive estimate performs asymptotically as well (in the sense that the limiting distributions are the same) with G unknown as any estimate which utilizes knowledge of G. Note that the term adaptive estimation has been used elsewhere in the literature in the lesser sense of adapting to the data in some way. The estimates considered here are adaptive in a much stronger sense. An adaptive estimator of the center of symmetry of an unknown distribution was constructed by Stone (1975). Bickel (1982) dealt with the multiple regression problem and simplified Stein's conditions for the circumstances under which adaptive estimation is possible. However, all the aforementioned work pertains to large sample behavior. Problems arise when one tries to apply these procedures in a practical small sample situation. The adaptive estimates proposed all depend on nonparametric density estimation and specifically the use of kernel density estimation. Essentially, one replaces the true density used in a one-step maximum likelihood estimate by an estimate of that density. A problem which pervades nonparametric density estimation is the choice of the smoothing parameters. Much work has been done on this subject and various schemes for the optimal choice of smoothing parameters have been proposed, mostly for the mean integrated square error criteria. Unfortunately, this is of little help in selecting the optimal smoothing parameters for the estimation of the parameters in the location or regression problem since these methods may give a good estimate of the density, but that is secondary to the problem at hand. We need to estimate the score, not the density, and our criterion is to minimize the MSE, say, of the estimate of 0. In fact, experimentation reveals that the optimal choice of smoothing parameters as regards the estimation of the location parameter tends to produce an under

Abstract: : This paper presents a class of non-parametric density estimators on a low dimensional space. The support of these estimators is defined by the convex hull of the set of observations. A random sample from the set of observations is used to tessellate the interior of the convex hull. The attribution of empirical probability mass to the tiles resulting from the tessellation produces a density estimate. With a set of appropriate linear constraints on the attribution of mass, the estimator is shown to be a conditional maximum likelihood estimator. Repeating this procedure, and averaging these density estimates within tiles, produces a bootstrap estimate of the density function. The results of this resampling and density estimation process are presented in graphic form.

Abstract: An approach for removing boundary bias in nonparametric density esti-mation is considered. The technique is based on suitable finite-dimensional projections in Hilbert space. Applications to boundary bias removal with kernel and trigonometric series estimators are presented.

Abstract: In some experiments, only values smaller than all previous ones are observed, such as destructive stress testing and industrial quality control experiments. Here, for such record-breaking data, kernel density estimation is considered. For a single record-breaking sample, consistent estimation is not possible except in the extreme tails of the distribution. Hence, replication is required, and for m such independent record-breaking samples, the kernel density estimator is shown to be strongly consistent and asymptotically normal as m → ∞. Also, some computer simulation results and examples are presented.

Abstract: Estimators of derivatives of a density function based on differences of the empirical distribution function (Maltz 1974) are identified as derivatives of kernel density estimators using particular kernel functions. Properties of this family of kernels are investigated.

Abstract: Estimation of the spectral density S(f) of a stationary random process can be viewed as a nonparametric statistical estimation problem. A nonparametric approach based on a wavelet representation for the logarithm of the unknown S(f) is introduced. This approach offers the ability to capture significant components of S(f) at different resolution levels by application of a significance test, and guarantees nonnegativity of the spectral density estimator. >

Abstract: In this paper robustness properties are studied for kernel density estimators. A plug-in and least squares cross-validation bandwidth selector are considered. In an asymptotic analysis and in a simulation study it is shown that the robustness of kernel density estimates depends strongly on the chosen bandwidth selector.

Abstract: Importance‐sampling technique has been used in recent years in conjunction with Monte Carlo simulation method to evaluate the reliability of structural systems. Since the efficiency of the importance‐sampling method depends primarily on the choice of the importance‐sampling density, the use of the kernel method to estimate the optimal importance‐sampling density is proposed. This method deviates from the current practice of prescribing the importance‐sampling density from a given parametric family of density functions. Instead, the data obtained from an initial Monte Carlo run are utilized to determine the required importance‐sampling density. The kernel method yields unbiased estimates of the probability of failure. Two measures are developed to quantify the efficiency of the kernel method relative to the basic Monte Carlo method. The first measure, called the marginal efficiency, is used as an indicator of the effectiveness of the kernel method, whereas the second measure, the overall efficiency, define...

Abstract: Many data based methods for choosing the bandwidth of a kernel density estimator depend on unknown constants associated with auxiliary bandwidths which arise at the functional estimation stages. These constants are typically replaced by the corresponding constants for some reference distribution, or they are estimated.In this paper, it is argued that at least some stage of estimation for the constants is preferable to simply using the Normal reference without any estimation at all.Furthermore, it is seen that it is not enough to have only consistent estimates of the needed quantities.Sufficient rates of convergence for estimation of these constants to have negligible asymptotic effect are given.However, it is also noted that higher stages of estimation do not always give an improvement.

Abstract: The paper contains exponential inequalities for dependent random variables. As a measure of dependence we use φand ρ-mixing coefficients, the last one being based on the maximal coefficient of correlation. These results allow us to study the problem of uniform strong convergence for the kernel estimators of a density and for a kernel predictor for stochastic processes. Our uniform consistency theorems extend some known results