The estimation of derivatives of a nonparametric regression function when the data are correlated

A kernel estimator of a derivative of arbitrary order of a nonparametric average population curve is considered for a correlated-errors model with balanced replicate measurements at each design point. Asymptotic expansions of the mean squared error are derived for two classes of correlation functions in the model. Consistency, choice of smoothing parameter, and rates of convergence are examined for the important special cases of estimating the first and second derivatives.     

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.     

Nonparametric regression estimation at design poles and zeros

ABSTRACT In most treatments of nonparametric regression, it is assumed that the marginal density of the explanatory variables is strictly bounded away from zero and infinity. This note investigates the pointwise asymptotics for nonparametric regression when this assumption fails, that is, the marginal density of the explanatory variable has either an isolated zero or a pole at the point of interest.     

Functions of Concordance and Dependence with Related Measures

Some functions that serve as building blocks for construction of a wider range of modes of concordance and dependence are pointed. We probe into interplays of such modes. From the standpoint of their conformity to stochastic dominance ordering of distributions within a Fréchet class, all such derived modes display some parallelism under certain conditions. We finally suggest a novel numeric measure of dependence that covers similar existing measures in literature.     

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.     

C93. The relevance of imaginary alternatives

Some practical approaches to the problem of choosing parameters which control the smoothness of kernel-based density estimators are investigated. Fixed and variable kernels are considered, and particularly simple approaches are investigated in the latter case. The performances of a wide range of estimators are compared in a simulation study.     

混合核函数对支持向量机分类性能的改进

通过对核矩阵的计算和研究,从理论上对常用的核函数进行了评估.在此基础上,通过实验仿真证实了通过优选后的核函数所组成的混合核函数对支持向量机分类性能的改善,为核函数的选择提供了参考.     

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.     

The Henderson Smoother in Reproducing Kernel Hilbert Space

The Henderson smoother has been traditionally applied for trend-cycle estimation in the context of nonparametric seasonal adjustment software officially adopted by statistical agencies. This study introduces a Henderson third-order kernel representation by means of the reproducing kernel Hilbert space (RKHS) methodology. Two density functions and corresponding orthonormal polynomials have been calculated. Both are shown to give excellent representations for short- and medium-length filters. Theoretical and empirical comparisons of the Henderson third-order kernel asymmetric filters are made with the classical ones. The former are shown to be superior in terms of signal passing, noise suppression, and revision size.     

Practical performance of local likelihood for circular density estimation

ABSTRACT

Local likelihood has been mainly developed from an asymptotic point of view, with little attention to finite sample size issues. The present paper provides simulation evidence of how likelihood density estimation practically performs from two points of view. First, we explore the impact of the normalization step of the final estimate, second we show the effectiveness of higher order fits in identifying modes present in the population when small sample sizes are available. We refer to circular data, nevertheless it is easily seen that our findings straightforwardly extend to the Euclidean setting, where they appear to be somehow new.