Discrete associated kernels method and extensions

Discrete kernel estimation of a probability mass function (p.m.f.), often mentioned in the literature, has been far less investigated in comparison with continuous kernel estimation of a probability density function (p.d.f.). In this paper, we are concerned with a general methodology of discrete kernels for smoothing a p.m.f. f. We give a basic of mathematical tools for further investigations. First, we point out a generalizable notion of discrete associated kernel which is defined at each point of the support of f and built from any parametric discrete probability distribution. Then, some properties of the corresponding estimators are shown, in particular pointwise and global (asymptotical) properties. Other discrete kernels are constructed from usual discrete probability distributions such as Poisson, binomial and negative binomial. For small samples sizes, underdispersed discrete kernel estimators are more interesting than the empirical estimator; thus, an importance of discrete kernels is illustrated. The choice of smoothing bandwidth is classically investigated according to cross-validation and, novelly, to excess of zeros methods. Finally, a unification way of this method concerning the general probability function is discussed.     

Cross-validation Bandwidth Matrices for Multivariate Kernel Density Estimation

Abstract.  The performance of multivariate kernel density estimates depends crucially on the choice of bandwidth matrix, but progress towards developing good bandwidth matrix selectors has been relatively slow. In particular, previous studies of cross-validation (CV) methods have been restricted to biased and unbiased CV selection of diagonal bandwidth matrices. However, for certain types of target density the use of full (i.e. unconstrained) bandwidth matrices offers the potential for significantly improved density estimation. In this paper, we generalize earlier work from diagonal to full bandwidth matrices, and develop a smooth cross-validation (SCV) methodology for multivariate data. We consider optimization of the SCV technique with respect to a pilot bandwidth matrix. All the CV methods are studied using asymptotic analysis, simulation experiments and real data analysis. The results suggest that SCV for full bandwidth matrices is the most reliable of the CV methods. We also observe that experience from the univariate setting can sometimes be a misleading guide for understanding bandwidth selection in the multivariate case.     

Central limit theorem for the variable bandwidth kernel density estimators

In this paper we study the ideal variable bandwidth kernel density estimator introduced by McKay (1993a, b) and Jones et al. (1994) and the plug-in practical version of the variable bandwidth kernel estimator with two sequences of bandwidths as in Giné and Sang (2013). Based on the bias and variance analysis of the ideal and plug-in variable bandwidth kernel density estimators, we study the central limit theorems for each of them. The simulation study confirms the central limit theorem and demonstrates the advantage of the plug-in variable bandwidth kernel method over the classical kernel method.     

Density estimation using asymmetric kernels and Bayes bandwidths with censored data

We propose a modification to the regular kernel density estimation method that use asymmetric kernels to circumvent the spill over problem for densities with positive support. First a pivoting method is introduced for placement of the data relative to the kernel function. This yields a strongly consistent density estimator that integrates to one for each fixed bandwidth in contrast to most density estimators based on asymmetric kernels proposed in the literature. Then a data-driven Bayesian local bandwidth selection method is presented and lognormal, gamma, Weibull and inverse Gaussian kernels are discussed as useful special cases. Simulation results and a real-data example illustrate the advantages of the new methodology.     

Minimax properties of beta kernel estimators

In this paper, we are interested in the study of beta kernel density estimators from an asymptotic minimax point of view. These estimators allows to estimate density functions with support in [0,1]. It is well-known that beta kernel estimators are - on the contrary of classical kernel estimators - “free of boundary effect” and thus are very useful in practice. The goal of this paper is to prove that there is a price to pay: for very regular density functions or for certain losses, these estimators are not minimax. Nevertheless they are minimax for classical regularities such as regularity of order two or less than two, supposed commonly in the practice and for some classical losses.     

Comparison of Kernel Density Estimators with Assumption on Number of Modes

A data-driven bandwidth choice for a kernel density estimator called critical bandwidth is investigated. This procedure allows the estimation to have as many modes as assumed for the density to estimate. Both Gaussian and uniform kernels are considered. For the Gaussian kernel, asymptotic results are given. For the uniform kernel, an argument against these properties is mentioned. These theoretical results are illustrated with a simulation study that compares the kernel estimators that rely on critical bandwidth with another one that uses a plug-in method to select its bandwidth. An estimator that consists in estimates of density contour clusters and takes assumptions on number of modes into account is also considered. Finally, the methodology is illustrated using environment monitoring data.     

A Unified View of Nonparametric Trend-Cycle Predictors Via Reproducing Kernel Hilbert Spaces

We provide a common approach for studying several nonparametric estimators used for smoothing functional time series data. Linear filters based on different building assumptions are transformed into kernel functions via reproducing kernel Hilbert spaces. For each estimator, we identify a density function or second order kernel, from which a hierarchy of higher order estimators is derived. These are shown to give excellent representations for the currently applied symmetric filters. In particular, we derive equivalent kernels of smoothing splines in Sobolev and polynomial spaces. The asymmetric weights are obtained by adapting the kernel functions to the length of the various filters, and a theoretical and empirical comparison is made with the classical estimators used in real time analysis. The former are shown to be superior in terms of signal passing, noise suppression and speed of convergence to the symmetric filter.     

Adaptive smoothing in associated kernel discrete functions estimation using Bayesian approach

This paper demonstrates that cross-validation (CV) and Bayesian adaptive bandwidth selection can be applied in the estimation of associated kernel discrete functions. This idea is originally proposed by Brewer [A Bayesian model for local smoothing in kernel density estimation, Stat. Comput. 10 (2000), pp. 299–309] to derive variable bandwidths in adaptive kernel density estimation. Our approach considers the adaptive binomial kernel estimator and treats the variable bandwidths as parameters with beta prior distribution. The best variable bandwidth selector is estimated by the posterior mean in the Bayesian sense under squared error loss. Monte Carlo simulations are conducted to examine the performance of the proposed Bayesian adaptive approach in comparison with the performance of the Asymptotic mean integrated squared error estimator and CV technique for selecting a global (fixed) bandwidth proposed in Kokonendji and Senga Kiessé [Discrete associated kernels method and extensions, Stat. Methodol. 8 (2011), pp. 497–516]. The Bayesian adaptive bandwidth estimator performs better than the global bandwidth, in particular for small and moderate sample sizes.     

Multivariate generalized Birnbaum—Saunders kernel density estimators

In this article, we first propose the classical multivariate generalized Birnbaum–Saunders kernel estimator for probability density function estimation in the context of multivariate non negative data. Then, we apply two multiplicative bias correction (MBC) techniques for multivariate kernel density estimator. Some properties (bias, variance, and mean integrated squared error) of the corresponding estimators are also investigated. Finally, the performances of the classical and MBC estimators based on family of generalized Birnbaum–Saunders kernels are illustrated by a simulation study.     

BIAS REDUCTION AND ELIMINATION WITH KERNEL ESTIMATORS

A great deal of research has focused on improving the bias properties of kernel estimators. One proposal involves removing the restriction of non-negativity on the kernel to construct “higher-order” kernels that eliminate additional terms in the Taylor's series expansion of the bias. This paper considers an alternative that uses a local approach to bandwidth selection to not only reduce the bias, but to eliminate it entirely. These so-called “zero-bias bandwidths” are shown to exist for univariate and multivariate kernel density estimation as well as kernel regression. Implications of the existence of such bandwidths are discussed. An estimation strategy is presented, and the extent of the reduction or elimination of bias in practice is studied through simulation and example.     

On finite sample properties of nonparametric discrete asymmetric kernel estimators

The discrete kernel method was developed to estimate count data distributions, distinguishing discrete associated kernels based on their asymptotic behaviour. This study investigates the class of discrete asymmetric kernels and their resulting non-consistent estimators, but this theoretical drawback of the estimators is balanced by some interesting features in small/medium samples. The role of modal probability and variance of discrete asymmetric kernels is highlighted to help better understand the performance of these kernels, in particular how the binomial kernel outperforms other asymmetric kernels. The performance of discrete asymmetric kernel estimators of probability mass functions is illustrated using simulations, in addition to applications to real data sets.     

Beta-Bernstein Smoothing for Regression Curves with Compact Support

ABSTRACT. The problem of boundary bias is associated with kernel estimation for regression curves with compact support. This paper proposes a simple and uni(r)ed approach for remedying boundary bias in non-parametric regression, without dividing the compact support into interior and boundary areas and without applying explicitly different smoothing treatments separately. The approach uses the beta family of density functions as kernels. The shapes of the kernels vary according to the position where the curve estimate is made. Theyare symmetric at the middle of the support interval, and become more and more asymmetric nearer the boundary points. The kernels never put any weight outside the data support interval, and thus avoid boundary bias. The method is a generalization of classical Bernstein polynomials, one of the earliest methods of statistical smoothing. The proposed estimator has optimal mean integrated squared error at an order of magnitude n ^−4/5, equivalent to that of standard kernel estimators when the curve has an unbounded support.     

Long run variance estimation and robust regression testing using sharp origin kernels with no truncation

A new family of kernels is suggested for use in long run variance (LRV) estimation and robust regression testing. The kernels are constructed by taking powers of the Bartlett kernel and are intended to be used with no truncation (or bandwidth) parameter. As the power parameter (ρ)$(\rho )$ increases, the kernels become very sharp at the origin and increasingly downweight values away from the origin, thereby achieving effects similar to a bandwidth parameter. Sharp origin kernels can be used in regression testing in much the same way as conventional kernels with no truncation, as suggested in the work of Kiefer and Vogelsang [2002a, Heteroskedasticity-autocorrelation robust testing using bandwidth equal to sample size. Econometric Theory 18, 1350–1366, 2002b, Heteroskedasticity-autocorrelation robust standard errors using the Bartlett kernel without truncation, Econometrica 70, 2093–2095] Analysis and simulations indicate that sharp origin kernels lead to tests with improved size properties relative to conventional tests and better power properties than other tests using Bartlett and other conventional kernels without truncation.     

Bivariate uniform deconvolution

We construct a density estimator in the bivariate uniform deconvolution model. For this model, we derive four inversion formulas to express the bivariate density that we want to estimate in terms of the bivariate density of the observations. By substituting a kernel density estimator of the density of the observations, we then obtain four different estimators. Next we construct an asymptotically optimal convex combination of these four estimators. Expansions for the bias, variance, as well as asymptotic normality are derived. Some simulated examples are presented.     

Data‐driven choice of the smoothing parametrization for kernel density estimators

There are several levels of sophistication when specifying the bandwidth matrix H to be used in a multivariate kernel density estimator, including H to be a positive multiple of the identity matrix, a diagonal matrix with positive elements or, in its most general form, a symmetric positive‐definite matrix. In this paper, the author proposes a data‐based method for choosing the smoothing parametrization to be used in the kernel density estimator. The procedure is fully illustrated by a simulation study and some real data examples. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Nonparametric density estimation for multivariate bounded data

We propose a new nonparametric estimator for the density function of multivariate bounded data. As frequently observed in practice, the variables may be partially bounded (e.g. nonnegative) or completely bounded (e.g. in the unit interval). In addition, the variables may have a point mass. We reduce the conditions on the underlying density to a minimum by proposing a nonparametric approach. By using a gamma, a beta, or a local linear kernel (also called boundary kernels), in a product kernel, the suggested estimator becomes simple in implementation and robust to the well known boundary bias problem. We investigate the mean integrated squared error properties, including the rate of convergence, uniform strong consistency and asymptotic normality. We establish consistency of the least squares cross-validation method to select optimal bandwidth parameters. A detailed simulation study investigates the performance of the estimators. Applications using lottery and corporate finance data are provided.     

Considerations for optimal nonparametric regression under a generalizederror structure

Nonparametric smoothing, such as kernel or spline estimation, has been examined extensively under the assumption of uncorrelated errors. This paper addresses the effects of potential correlation on consistency and other asymptotic properties in a repeated-measures model, using directly optimized linear smoothers of the replicate means. Unrestricted optimal weights, with respect to squared error loss, are used to confirm a lack of consistency for all linear estimators in an autocorrelated errors model. The results indicate kernel methods that work well for an uncorrelated errors model may not have the ability to perform satisfactorily when correlation is introduced, due to an asymmetry in the optimal weights, which disappears for an uncorrelated errors model. These would include data-driven bandwidth selection methods, adjustments of the bandwidth to accommodate correlation, higher-order kernels, and related bias reduction techniques. The analytic results suggest alternative approaches, not considered here in detail, which have shown merit.     

Gains from joint cross validation bandwidth selection for derivatives of conditional multidimensional densities

This paper studies bandwidth selection for kernel estimation of derivatives of multidimensional conditional densities, a non-parametric realm unexplored in the literature. This paper extends Baird [Cross validation bandwidth selection for derivatives of multidimensional densities. RAND Working Paper series, WR-1060; 2014] in its examination of conditional multivariate densities, derives and presents criteria for arbitrary kernel order and density dimension, shows consistency of the estimators, and investigates a minimization criterion which jointly estimates numerator and denominator bandwidths. I conduct a Monte Carlo simulation study for various orders of kernels in the Gaussian family and compare the new cross validation criterion with those implied by Baird [Cross validation bandwidth selection for derivatives of multidimensional densities. RAND Working Paper series, WR-1060; 2014]. The paper finds that higher order kernels become increasingly important as the dimension of the distribution increases. I find that the cross validation criterion developed in this paper that jointly estimates the derivative of the joint density (numerator) and the marginal density (denominator) does orders of magnitude better than criteria that estimate the bandwidths separately. I further find that using the infinite order Dirichlet kernel tends to have the best results.     

Bandwidth selection procedures tor kernel density estimates

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.     

A Remedy for Kernel Estimation Under Random Design

Two common kernel-based methods for non-parametric regression estimation suffer from well-known drawbacks when the design is random. The Gasser-Müller estimator is inadmissible due to its high variance while the Nadaraya-Watson estimator has zero asymptotic efficiency because of poor bias behavior. Under asymptotic consideration, the local linear estimator avoids these two drawbacks of kernel estimators and achieves minimax optimality. However, when based on compact support kernels its finite sample behavior is disappointing because sudden kinks may show up in the estimate.

This paper proposes a modification of the kernel estimator, called the binned convolution estimator leading to a fast O(n) method. Provided the design density is continously differentiable and the conditional fourth moments exist the binned convolution estimator has asymptotic properties identical with those of the local linear estimator.