Asymptotic distribution of bandwidth selectors in kernel regression estimation

Gasser, Kneip and Köhler (1991) proposed a fast and flexible procedure for automatic bandwidth selection in kernel regression estimation. This article describes this method and additionally derives the joint asymptotic normal distribution of this bandwidth selector with the realizationwise optimal bandwidth.     

Local influence on bandwidth estimation for kernel smoothing

This paper examines the problem of assessing local influence on the optimal bandwidth estimation in kernel smoothing based on cross validation. The bandwidth for kernel smoothing plays an important role in the model fitting and is often estimated using the cross-validation criterion. Following the argument of the second-order approach to local influence suggested by Wu and Luo (1993), we develop a new diagnostic statistic to examine the local influence of the observations on the estimation of the optimal bandwidth, where the perturbation may belong to one of three schemes. These are the response perturbation, the perturbation in the explanatory variable, and the case-weight

perturbation. The proposed diagnostic is nonparametric and is capable of identifying influential observations with strong influence on the bandwidth estimation. An example is presented to illustrate the application of the proposed diagnostic, and the usefulness of the nonparametric approach is illustrated in comparison with some other approaches to the assessment of local influence     

An optimal local bandwidth selector for kernel density estimation

The problem of selecting the bandwidth for optimal kernel density estimation at a point is considered. A class of local bandwidth selectors which minimize smoothed bootstrap estimates of mean-squared error in density estimation is introduced. It is proved that the bandwidth selectors in the class achieve optimal relative rates of convergence, dependent upon the local smoothness of the target density. Practical implementation of the bandwidth selection methodology is discussed. The use of Gaussian-based kernels to facilitate computation of the smoothed bootstrap estimate of mean-squared error is proposed. The performance of the bandwidth selectors is investigated empirically.     

A hybrid bandwidth selection methodology for kernel density estimation

A bandwidth selection method that combines the concept of least-squares cross-validation and the plug-in approach is being introduced in connection with kernel density estimation. A simulation study reveals that this hybrid methodology outperforms some commonly used bandwidth selection rules. It is shown that the proposed approach can also be readily employed in the context of variable kernel density estimation. We conclude with two illustrative examples.     

Subsampling-extrapolation bandwidth selection in bivariate kernel density estimation

This paper focuses on bivariate kernel density estimation that bridges the gap between univariate and multivariate applications. We propose a subsampling-extrapolation bandwidth matrix selector that improves the reliability of the conventional cross-validation method. The proposed procedure combines a U-statistic expression of the mean integrated squared error and asymptotic theory, and can be used in both cases of diagonal bandwidth matrix and unconstrained bandwidth matrix. In the subsampling stage, one takes advantage of the reduced variability of estimating the bandwidth matrix at a smaller subsample size m (m < n); in the extrapolation stage, a simple linear extrapolation is used to remove the incurred bias. Simulation studies reveal that the proposed method reduces the variability of the cross-validation method by about 50% and achieves an expected integrated squared error that is up to 30% smaller than that of the benchmark cross-validation. It shows comparable or improved performance compared to other competitors across six distributions in terms of the expected integrated squared error. We prove that the components of the selected bivariate bandwidth matrix have an asymptotic multivariate normal distribution, and also present the relative rate of convergence of the proposed bandwidth selector.     

A weighted least-squares cross-validation bandwidth selector for kernel density estimation

Since the late 1980s, several methods have been considered in the literature to reduce the sample variability of the least-squares cross-validation bandwidth selector for kernel density estimation. In this article, a weighted version of this classical method is proposed and its asymptotic and finite-sample behavior is studied. The simulation results attest that the weighted cross-validation bandwidth performs quite well, presenting a better finite-sample performance than the standard cross-validation method for “easy-to-estimate” densities, and retaining the good finite-sample performance of the standard cross-validation method for “hard-to-estimate” ones.     

Bandwidth selection for kernel density estimation: a review of fully automatic selectors

On the one hand, kernel density estimation has become a common tool for empirical studies in any research area. This goes hand in hand with the fact that this kind of estimator is now provided by many software packages. On the other hand, since about three decades the discussion on bandwidth selection has been going on. Although a good part of the discussion is about nonparametric regression, this parameter choice is by no means less problematic for density estimation. This becomes obvious when reading empirical studies in which practitioners have made use of kernel densities. New contributions typically provide simulations only to show that the own selector outperforms some of the existing methods. We review existing methods and compare them on a set of designs that exhibit few bumps and exponentially falling tails. We concentrate on small and moderate sample sizes because for large ones the differences between consistent methods are often negligible, at least for practitioners. As a byproduct we find that a mixture of simple plug-in and cross-validation methods produces bandwidths with a quite stable performance.     

The influence function of the optimal bandwidth for kernel density estimation

Theories about the bandwidth of kernel density estimation have been well established by many statisticians. However, the influence function of the bandwidth has not been well investigated. The influence function of the optimal bandwidth that minimizes the mean integrated square error is derived and the asymptotic property of the bandwidth selectors based on the influence function is provided.     

Adaptive normal reference bandwidth based on quantile for kernel density estimation

Bandwidth selection is an important problem of kernel density estimation. Traditional simple and quick bandwidth selectors usually oversmooth the density estimate. Existing sophisticated selectors usually have computational difficulties and occasionally do not exist. Besides, they may not be robust against outliers in the sample data, and some are highly variable, tending to undersmooth the density. In this paper, a highly robust simple and quick bandwidth selector is proposed, which adapts to different types of densities.     

Quasi-universal bandwidth selection for kernel density estimators

Let f?_n, h denote the kernel density estimate based on a sample of size n drawn from an unknown density f. Using techniques from L₂ projection density estimators, the author shows how to construct a data-driven estimator f?_n, h which satisfies This paper is inspired by work of Stone (1984), Devroye and Lugosi (1996) and Birge and Massart (1997).     

An evaluation of likelihood-based bandwidth selectors for spatial and spatiotemporal kernel estimates

Spatial point pattern data sets are commonplace in a variety of different research disciplines. The use of kernel methods to smooth such data is a flexible way to explore spatial trends and make inference about underlying processes without, or perhaps prior to, the design and fitting of more intricate semiparametric or parametric models to quantify specific effects. The long-standing issue of ‘optimal’ data-driven bandwidth selection is complicated in these settings by issues such as high heterogeneity in observed patterns and the need to consider edge correction factors. We scrutinize bandwidth selectors built on leave-one-out cross-validation approximation to likelihood functions. A key outcome relates to previously unconsidered adaptive smoothing regimens for spatiotemporal density and multitype conditional probability surface estimation, whereby we propose a novel simultaneous pilot-global selection strategy. Motivated by applications in epidemiology, the results of both simulated and real-world analyses suggest this strategy to be largely preferable to classical fixed-bandwidth estimation for such data.     

Nonparametric density deconvolution by weighted kernel estimators

Nonparametric density estimation in the presence of measurement error is considered. The usual kernel deconvolution estimator seeks to account for the contamination in the data by employing a modified kernel. In this paper a new approach based on a weighted kernel density estimator is proposed. Theoretical motivation is provided by the existence of a weight vector that perfectly counteracts the bias in density estimation without generating an excessive increase in variance. In practice a data driven method of weight selection is required. Our strategy is to minimize the discrepancy between a standard kernel estimate from the contaminated data on the one hand, and the convolution of the weighted deconvolution estimate with the measurement error density on the other hand. We consider a direct implementation of this approach, in which the weights are optimized subject to sum and non-negativity constraints, and a regularized version in which the objective function includes a ridge-type penalty. Numerical tests suggest that the weighted kernel estimation can lead to tangible improvements in performance over the usual kernel deconvolution estimator. Furthermore, weighted kernel estimates are free from the problem of negative estimation in the tails that can occur when using modified kernels. The weighted kernel approach generalizes to the case of multivariate deconvolution density estimation in a very straightforward manner.     

Consistent deconvolution in density estimation

Suppose we have n observations from X = Y + Z, where Z is a noise component with known distribution, and Y has an unknown density f. When the characteristic function of Z is nonzero almost everywhere, we show that it is possible to construct a density estimate f_n such that for all f, Iim_n| |=0.     

Variable kernel density estimation

This paper considers the problem of selecting optimal bandwidths for variable (sample‐point adaptive) kernel density estimation. A data‐driven variable bandwidth selector is proposed, based on the idea of approximating the log‐bandwidth function by a cubic spline. This cubic spline is optimized with respect to a cross‐validation criterion. The proposed method can be interpreted as a selector for either integrated squared error (ISE) or mean integrated squared error (MISE) optimal bandwidths. This leads to reflection upon some of the differences between ISE and MISE as error criteria for variable kernel estimation. Results from simulation studies indicate that the proposed method outperforms a fixed kernel estimator (in terms of ISE) when the target density has a combination of sharp modes and regions of smooth undulation. Moreover, some detailed data analyses suggest that the gains in ISE may understate the improvements in visual appeal obtained using the proposed variable kernel estimator. These numerical studies also show that the proposed estimator outperforms existing variable kernel density estimators implemented using piecewise constant bandwidth functions.     

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.     

Simple boundary correction for kernel density estimation

If a probability density function has bounded support, kernel density estimates often overspill the boundaries and are consequently especially biased at and near these edges. In this paper, we consider the alleviation of this boundary problem. A simple unified framework is provided which covers a number of straightforward methods and allows for their comparison: generalized jackknifing generates a variety of simple boundary kernel formulae. A well-known method of Rice (1984) is a special case. A popular linear correction method is another: it has close connections with the boundary properties of local linear fitting (Fan and Gijbels, 1992). Links with the optimal boundary kernels of Müller (1991) are investigated. Novel boundary kernels involving kernel derivatives and generalized reflection arise too. In comparisons, various generalized jackknifing methods perform rather similarly, so this, together with its existing popularity, make linear correction as good a method as any. In an as yet unsuccessful attempt to improve on generalized jackknifing, a variety of alternative approaches is considered. A further contribution is to consider generalized jackknife boundary correction for density derivative estimation. En route to all this, a natural analogue of local polynomial regression for density estimation is defined and discussed.     

Shape constrained kernel density estimation

In this paper, a method for estimating monotone, convex and log-concave densities is proposed. The estimation procedure consists of an unconstrained kernel estimator which is modified in a second step with respect to the desired shape constraint by using monotone rearrangements. It is shown that the resulting estimate is a density itself and shares the asymptotic properties of the unconstrained estimate. A short simulation study shows the finite sample behavior.     

Nonparametric localized bandwidth selection for Kernel density estimation

As conventional cross-validation bandwidth selection methods do not work properly in the situation where the data are serially dependent time series, alternative bandwidth selection methods are necessary. In recent years, Bayesian-based methods for global bandwidth selection have been studied. Our experience shows that a global bandwidth is however less suitable than a localized bandwidth in kernel density estimation based on serially dependent time series data. Nonetheless, a di?cult issue is how we can consistently estimate a localized bandwidth. This paper presents a nonparametric localized bandwidth estimator, for which we establish a completely new asymptotic theory. Applications of this new bandwidth estimator to the kernel density estimation of Eurodollar deposit rate and the S&P 500 daily return demonstrate the effectiveness and competitiveness of the proposed localized bandwidth.     

On kernel density derivative estimation

Estimators of derivatives of a density function based on polynomial multiples of kernels are compared with those based on differentiated kernels.