Nonparametric kernel regression estimation near endpoints

When kernel regression is used to produce a smooth estimate of a curve over a finite interval, boundary problems detract from the global performance of the estimator. A new kernel is derived to reduce this boundary problem. A generalized jackknife combination of two unsatisfactory kernels produces the desired result. One motivation for adopting a jackknife combination is that they are simple to construct and evaluate. Furthermore, as in other settings, the bias reduction property need not cause an inordinate increase in variability. The convergence rate with the new boundary kernel is the same as for the non-boundary. To illustrate the general approach, a new second-order boundary kernel, which is continuously linked to the Epanechnikov (1969, Theory Probab. Appl. 14, 153–158) kernel, is produced. The asymptotic mean square efficiencies relative to smooth optimal kernels due to Gasser and Müller (1984, Scand. J. Statist. 11, 171–185), Müller (1991, Biometrika 78, 521–530) and Müller and Wang (1994, Biometrics 50, 61–76) indicate that the new kernel is also competitive in this sense.     

Asymptotic Unbiased Kernel Estimator for Line Transect Sampling

In this article, we propose a new estimator for the density of objects using line transect data. The proposed estimator combines the nonparametric kernel estimator with parametric detection function: the exponential or the half normal detection function to estimate the density of objects. The selection of the detection function depends on the testing of the shoulder condition assumption. If the shoulder condition is true then the half-normal detection function is introduced together with the kernel estimator. Otherwise, the negative exponential is combined with the kernel estimator. Under these assumptions, the proposed estimator is asymptotically unbiased and it is strongly consistent estimator for the density of objects using line transect data. The simulation results indicate that the proposed estimator is very successful in taking the advantage of the parametric detection function available.     

Multivariate boundary kernels and a continuous least squares principle

Whereas there are many references on univariate boundary kernels, the construction of boundary kernels for multivariate density and curve estimation has not been investigated in detail. The use of multivariate boundary kernels ensures global consistency of multivariate kernel estimates as measured by the integrated mean-squared error or sup-norm deviation for functions with compact support. We develop a class of boundary kernels which work for any support, regardless of the complexity of its boundary. Our construction yields a boundary kernel for each point in the boundary region where the function is to be estimated. These boundary kernels provide a natural continuation of non-negative kernels used in the interior onto the boundary. They are obtained as solutions of the same kernel-generating variational problem which also produces the kernel function used in the interior as its solution. We discuss the numerical implementation of the proposed boundary kernels and their relationship to locally weighted least squares. Along the way we establish a continuous least squares principle and a continuous analogue of the Gauss–Markov theorem.     

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.     

Uniformly most powerful unbiased test for shoulder condition in point transect sampling

Point transect sampling is a well-known methodology for estimating wildlife population density. In this context, the usual approach is to assume a model for the detection function. Thus, the estimate depends on the shape of the detection function. In particular, the estimation is influenced by the so-called shoulder condition, which guarantees that detection is nearly certain at small distances from the observer. For instance, the half-normal model satisfies this condition, whereas the negative exponential model does not. Testing whether the shoulder condition is consistent with data is a crucial issue. In this paper we propose the uniformly most powerful unbiased test for the shoulder condition in the exponential mixture model of the half-normal and the negative exponential. Critical values of the proposed test are calculated for large samples by means of asymptotic distribution theory and for small samples via Monte Carlo simulations. Finally, a case study is presented.     

Deconvolution boundary kernel method in nonparametric density estimation

This paper considers the nonparametric deconvolution problem when the true density function is left (or right) truncated. We propose to remove the boundary effect of the conventional deconvolution density estimator by using a special class of kernels: the deconvolution boundary kernels. Methods for constructing such kernels are provided. The mean squared error properties, including the rates of convergence, are investigated for supersmooth and ordinary smooth errors. Numerical simulations show that the deconvolution boundary kernel estimator successfully removes the boundary effects of the conventional deconvolution density estimator.     

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.     

A family of kernels and their associated deconvolving kernels for normally distributed measurement errors

A family of kernels (with the sinc kernel as the simplest member) is introduced for which the associated deconvolving kernels (assuming normally distributed measurement errors) can be represented by relatively simple analytic functions. For this family, deconvolving kernel density estimation is not more sophisticated than ordinary kernel density estimation. Application examples suggest that it may be advantageous to overestimate the measurement error, because the resulting deconvolving kernels can partially compensate for the blurring inherent to the density estimation itself. A corollary of this proposition is that, even without error, it may be rational to use deconvolving rather than ordinary kernels.     

On multivariate associated kernels to estimate general density functions

Multivariate associated kernel estimators, which depend on both target point and bandwidth matrix, are appropriate for distributions with partially or totally bounded supports and generalize the classical ones such as the Gaussian. Previous studies on multivariate associated kernels have been restricted to products of univariate associated kernels, also considered having diagonal bandwidth matrices. However, it has been shown in classical cases that, for certain forms of target density such as multimodal ones, the use of full bandwidth matrices offers the potential for significantly improved density estimation. In this paper, general associated kernel estimators with correlation structure are introduced. Asymptotic properties of these estimators are presented; in particular, the boundary bias is investigated. Generalized bivariate beta kernels are handled in more details. The associated kernel with a correlation structure is built with a variant of the mode-dispersion method and two families of bandwidth matrices are discussed using the least squared cross validation method. Simulation studies are done. In the particular situation of bivariate beta kernels, a very good performance of associated kernel estimators with correlation structure is observed compared to the diagonal case. Finally, an illustration on a real dataset of paired rates in a framework of political elections is presented.     

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.     

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.     

Optimal kernels when estimating non-smooth densities

The derivation of new kernel functions for the kernel estimator of an unknown density function is given. These kernels are shown to be optimal in some sense when the underlying density f is continuous but its derivative f′ is not, and consequently a solu tion is presented for an unsolved problem which was stated by van Eeden (1985). Other attractive features of these kernels are also discussed and a number of graphs are listed.     

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.     

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.     

Kernel density estimation on the torus

Kernel density estimation for multivariate, circular data has been formulated only when the sample space is the sphere, but theory for the torus would also be useful. For data lying on a d-dimensional torus (d?1), we discuss kernel estimation of a density, its mixed partial derivatives, and their squared functionals. We introduce a specific class of product kernels whose order is suitably defined in such a way to obtain L₂-risk formulas whose structure can be compared to their Euclidean counterparts. Our kernels are based on circular densities; however, we also discuss smaller bias estimation involving negative kernels which are functions of circular densities. Practical rules for selecting the smoothing degree, based on cross-validation, bootstrap and plug-in ideas are derived. Moreover, we provide specific results on the use of kernels based on the von Mises density. Finally, real-data examples and simulation studies illustrate the findings.     

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.     

A new non parametric estimator for Pdf based on inverse gamma distribution

ABSTRACT

The non parametric approach is considered to estimate probability density function (Pdf) which is supported on(0, ∞). This approach is the inverse gamma kernel. We show that it has same properties as gamma, reciprocal inverse Gaussian, and inverse Gaussian kernels such that it is free of the boundary bias, non negative, and it achieves the optimal rate of convergence for the mean integrated squared error. Also some properties of the estimator were established such as bias and variance. Comparison of the bandwidth selection methods for inverse gamma kernel estimation of Pdf is done.     

On boundary correction in kernel density estimation

It is well known now that kernel density estimators are not consistent when estimating a density near the finite end points of the support of the density to be estimated. This is due to boundary effects that occur in nonparametric curve estimation problems. A number of proposals have been made in the kernel density estimation context with some success. As of yet there appears to be no single dominating solution that corrects the boundary problem for all shapes of densities. In this paper, we propose a new general method of boundary correction for univariate kernel density estimation. The proposed method generates a class of boundary corrected estimators. They all possess desirable properties such as local adaptivity and non-negativity. In simulation, it is observed that the proposed method perform quite well when compared with other existing methods available in the literature for most shapes of densities, showing a very important robustness property of the method. The theory behind the new approach and the bias and variance of the proposed estimators are given. Results of a data analysis are also given.     

Computationally efficient classes of higher-order kernel functions

Classes of higher-order kernels for estimation of a probability density are constructed by iterating the twicing procedure. Given a kernel K of order l, we build a family of kernels K_m of orders l(m + 1) with the attractive property that their Fourier transforms are simply 1 — {1 —$(.)}^m+1, where ? is the Fourier transform of K. These families of higher-order kernels are well suited when the fast Fourier transform is used to speed up the calculation of the kernel estimate or the least-squares cross-validation procedure for selection of the window width. We also compare the theoretical performance of the optimal polynomial-based kernels with that of the iterative twicing kernels constructed from some popular second-order kernels.     

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.