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.     

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.     

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.     

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.     

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.     

New compactly supported spatiotemporal covariance functions from SPDEs

Potential theory and Dirichlet’s priciple constitute the basic elements of the well-known classical theory of Markov processes and Dirichlet forms. This paper presents new classes of fractional spatiotemporal covariance models, based on the theory of non-local Dirichlet forms, characterizing the fundamental solution, Green kernel, of Dirichlet boundary value problems for fractional pseudodifferential operators. The elements of the associated Gaussian random field family have compactly supported non-separable spatiotemporal covariance kernels admitting a parametric representation. Indeed, such covariance kernels are not self-similar but can display local self-similarity, interpolating regular and fractal local behavior in space and time. The associated local fractional exponents are estimated from the empirical log-wavelet variogram. Numerical examples are simulated for illustrating the properties of the space–time covariance model class introduced.     

Smoothed categorical regression based on direct kernel estimates

A nonparametric method is considered which yields smoothed estimates of the response probabilities when the response variable is categorical. The method is based on Lauder's (1983) direct kernel estimates which are extended to allow for ordinal kernels. Thus one can make use of the ordinal scale of the response variable. A class of predictive loss functions is introduced on which the cross-validatory choice of smoothing parameters is based. Plots of the smoothed response probabilities may be used to uncover the form of covariate effects     

Gaussian-based kernels

We derive a class of higher-order kernels for estimation of densities and their derivatives, which can be viewed as an extension of the second-order Gaussian kernel. These kernels have some attractive properties such as smoothness, manageable convolution formulae, and Fourier transforms. One important application is the higher-order extension of exact calculations of the mean integrated squared error. The proposed kernels also have the advantage of simplifying computations of common window-width selection algorithms such as least-squares cross-validation. Efficiency calculations indicate that the Gaussian-based kernels perform almost as well as the optimal polynomial kernels when die order of the derivative being estimated is low.     

Detection of a change point with local polynomial fits for the random design case

Regression functions may have a change or discontinuity point in the ν th derivative function at an unknown location. This paper considers a method of estimating the location and the jump size of the change point based on the local polynomial fits with one‐sided kernels when the design points are random. It shows that the estimator of the location of the change point achieves the rate n^?1/(2ν+1) when ν is even. On the other hand, when ν is odd, it converges faster than the rate n^?1/(2ν+1) due to a property of one‐sided kernels. Computer simulation demonstrates the improved performance of the method over the existing ones.     

Learning rates of multi-kernel regression by orthogonal greedy algorithm

We investigate the problem of regression from multiple reproducing kernel Hilbert spaces by means of orthogonal greedy algorithm. The greedy algorithm is appealing as it uses a small portion of candidate kernels to represent the approximation of regression function, and can greatly reduce the computational burden of traditional multi-kernel learning. Satisfied learning rates are obtained based on the Rademacher chaos complexity and data dependent hypothesis spaces.     

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 classifier for multi-dimensional datasets based on Bayesian multiple kernel grouping learning

This paper proposes an algorithm for the classification of multi-dimensional datasets based on the conjugate Bayesian Multiple Kernel Grouping Learning (BMKGL). Using conjugate Bayesian framework improves the computation efficiency. Multiple kernels instead of a single kernel avoid the kernel selection problem which is also a computationally expensive work. Through grouping parameter learning, BMKGL can simultaneously integrate information from different dimensions and find the dimensions which contribute more to the variations of the outcome for the purpose of interpretable property. Meanwhile, BMKGL can select the most suitable combination of kernels for different dimensions so as to extract the most appropriate measure for each dimension and improve the accuracy of classification results. The simulation results illustrate that our learning process has better performance in prediction results and stability compared to some popular classifiers, such as k-nearest neighbours algorithm, support vector machine algorithm and naive Bayes classifier. BMKGL also outperforms previous methods in terms of accuracy and interpretation for the heart disease and EEG datasets.     

Testing for a finite mixture model with two components

Summary.  We consider a finite mixture model with k components and a kernel distribution from a general one-parameter family. The problem of testing the hypothesis k =2 versus k 3 is studied. There has been no general statistical testing procedure for this problem. We propose a modified likelihood ratio statistic where under the null and the alternative hypotheses the estimates of the parameters are obtained from a modified likelihood function. It is shown that estimators of the support points are consistent. The asymptotic null distribution of the modified likelihood ratio test proposed is derived and found to be relatively simple and easily applied. Simulation studies for the asymptotic modified likelihood ratio test based on finite mixture models with normal, binomial and Poisson kernels suggest that the test proposed performs well. Simulation studies are also conducted for a bootstrap method with normal kernels. An example involving foetal movement data from a medical study illustrates the testing procedure.     

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.     

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.     

具有核化函数的部分线性模型及其应用

本文基于再生核希尔伯特空间中的再生核，将核技巧与高斯-赛责尔迭代算法相结合，提出了具有核化函数的部分线性模型（PLMKF）及其算法收敛性条件等相关内容，具体包括：（1）基于OLS的PLMKF；（2）基于岭估计的PLMKF；（3）基于GLS的PLMKF；（4）基于多核学习的PLMKF。它们构成了PLMKF家族，具有一定的相互转化关系。在数值模拟中，本文验证了各个算法的有效性，比较了基于OLS与GLS、单核与多核的PLMKF模拟结果。实际应用中，在大幅外推情景下，PLMKF仍保持了良好的泛化能力，预测精度高于PLM、GAM和SVR。     

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.     

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.     

Bayesian adaptive bandwidth selector for multivariate discrete kernel estimator

We treat a non parametric estimator for joint probability mass function, based on multivariate discrete associated kernels which are appropriated for multivariate count data of small and moderate sample sizes. Bayesian adaptive estimation of the vector of bandwidths using the quadratic and entropy loss functions is considered. Exact formulas for the posterior distribution and the vector of bandwidths are obtained. Numerical studies indicate that the performance of our approach is better, comparing with other bandwidth selection techniques using integrated squared error as criterion. Some applications are made on real data sets.     

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.