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.     

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.     

On the Local Polynomial Estimators of the Counting Process Intensity Function and its Derivatives

Abstract. We consider the properties of the local polynomial estimators of a counting process intensity function and its derivatives. By expressing the local polynomial estimators in a kernel smoothing form via effective kernels, we show that the bias and variance of the estimators at boundary points are of the same magnitude as at interior points and therefore the local polynomial estimators in the context of intensity estimation also enjoy the automatic boundary correction property as they do in other contexts such as regression. The asymptotically optimal bandwidths and optimal kernel functions are obtained through the asymptotic expressions of the mean square error of the estimators. For practical purpose, we suggest an effective and easy‐to‐calculate data‐driven bandwidth selector. Simulation studies are carried out to assess the performance of the local polynomial estimators and the proposed bandwidth selector. The estimators and the bandwidth selector are applied to estimate the rate of aftershocks of the Sichuan earthquake and the rate of the Personal Emergency Link calls in Hong Kong.     

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.     

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.     

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.     

Kernel spline regression

The authors propose «kernel spline regression,» a method of combining spline regression and kernel smoothing by replacing the polynomial approximation for local polynomial kernel regression with the spline basis. The new approach retains the local weighting scheme and the use of a bandwidth to control the size of local neighborhood. The authors compute the bias and variance of the kernel linear spline estimator, which they compare with local linear regression. They show that kernel spline estimators can succeed in capturing the main features of the underlying curve more effectively than local polynomial regression when the curvature changes rapidly. They also show through simulation that kernel spline regression often performs better than ordinary spline regression and local polynomial regression.     

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.     

Kernel estimation in transect sampiing withoyt the shoulder condition

We consider the estiinution of wildlife population density based on line transect data. Nonparametric kernel method is employed, without the usual assumption that the detection curve has a shoulder at distance zero, with the help of a special class of kernels called boundary kernels. Asymptotic distribution results are included. It is pointed out that the boundery kernel of Zhang and Karunamuni (1998) (see also Müller and Wang (1994)) performs better (for asmyptotic mean square error consideration) than that of the boundary kernel of M¨ller (1991). But both of these kernels are clearly superior to the half-nonnal and one-sided Epanechnikov kernel when the shoulder condition fails to hold. In practice, however, for small to moderate sample sizes, caution should be exercised in using bounrlary kernels in that the density estimate might become negative. A Monte Carlo study is also presented, comparing the performance of four kernels applied to detection data, with and without the shoulder condition, Two bundary kernels for deriatives are also included for the point transect case.     

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.     

Simple Transformation Techniques for Improved Non-parametric Regression

We propose and investigate two new methods for achieving less bias in non- parametric regression. We show that the new methods have bias of order h ⁴, where h is a smoothing parameter, in contrast to the basic kernel estimator's order h ². The methods are conceptually very simple. At the first stage, perform an ordinary non-parametric regression on { x_i , Y_i } to obtain m^ ( x_i ) (we use local linear fitting). In the first method, at the second stage, repeat the non-parametric regression but on the transformed dataset { m^ ( x_i , Y_i )}, taking the estimator at x to be this second stage estimator at m^ ( x ). In the second, and more appealing, method, again perform non-parametric regression on { m^ ( x_i , Y_i )}, but this time make the kernel weights depend on the original x scale rather than using the m^ ( x ) scale. We concentrate more of our effort in this paper on the latter because of its advantages over the former. Our emphasis is largely theoretical, but we also show that the latter method has practical potential through some simulated examples.     

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.     

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.     

Using pseudometrics in kernel density estimation

Common kernel density estimators (KDE) are generalised, which involve that assumptions on the kernel of the distribution can be given. Instead of using metrics as input to the kernels, the new estimators use parameterisable pseudometrics. In general, the volumes of the balls in pseudometric spaces are dependent on both the radius and the location of the centre. To enable constant smoothing, the volumes of the balls need to be calculated and analytical expressions are preferred for computational reasons. Two suitable parametric families of pseudometrics are identified. One of them has common KDE as special cases. In a few experiments, the proposed estimators show increased statistical power when proper assumptions are made. As a consequence, this paper describes an approach, where partial knowledge about the distribution can be used effectively. Furthermore, it is suggested that the new estimators are adequate for statistical learning algorithms such as regression and classification.     

A locally adaptive transformation method of boundary correction in kernel density estimation

Kernel smoothing methods are widely used in many research areas in statistics. However, kernel estimators suffer from boundary effects when the support of the function to be estimated has finite endpoints. Boundary effects seriously affect the overall performance of the estimator. In this article, we propose a new method of boundary correction for univariate kernel density estimation. Our technique is based on a data transformation that depends on the point of estimation. The proposed method possesses desirable properties such as local adaptivity and non-negativity. Furthermore, unlike many other transformation methods available, the proposed estimator is easy to implement. In a Monte Carlo study, the accuracy of the proposed estimator is numerically analyzed and compared with the existing methods of boundary correction. We find that it performs well for most shapes of densities. The theory behind the new methodology, along with the bias and variance of the proposed estimator, are presented. Results of a data analysis are also given.     

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.     

Using recursive algorithms for the efficient identification of smoothing spline ANOVA models

In this paper we present a unified discussion of different approaches to the identification of smoothing spline analysis of variance (ANOVA) models: (i) the “classical” approach (in the line of Wahba in Spline Models for Observational Data, 1990; Gu in Smoothing Spline ANOVA Models, 2002; Storlie et al. in Stat. Sin., 2011) and (ii) the State-Dependent Regression (SDR) approach of Young in Nonlinear Dynamics and Statistics (2001). The latter is a nonparametric approach which is very similar to smoothing splines and kernel regression methods, but based on recursive filtering and smoothing estimation (the Kalman filter combined with fixed interval smoothing). We will show that SDR can be effectively combined with the “classical” approach to obtain a more accurate and efficient estimation of smoothing spline ANOVA models to be applied for emulation purposes. We will also show that such an approach can compare favorably with kriging.     

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.     

A boundary kernel for local polynomial regression

We define and compute a boundary kernel for local polynomial regression, We prove that the new kernel provides improvement over the existing kernels, Simulations show the improvement in finite samples.