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树结构响应的非参数回归模型AI 解读

Developments in science and technology over the last two decades has motivated the study of complex data objects. In this article, we consider the topological properties of a population of tree-structured objects. Our interest centers on modeling the relationship between a tree-structured response and other covariates. For tree-structured objects, this poses serious challenges since most regression methods rely on linear operations in Euclidean space. We generalize the…

对称正定矩阵的局部多项式回归AI 解读

Summary Local polynomial regression has received extensive attention for the non-parametric estimation of regression functions when both the response and the covariate are in Euclidean space. However, little has been done when the response is in a Riemannian manifold. We develop an intrinsic local polynomial regression estimate for the analysis of symmetric positive definite matrices as responses that lie in a Riemannian manifold with covariate in Euclidean space. The…

加权距离加权判别及其渐近性质AI 解读

While Distance Weighted Discrimination (DWD) is an appealing approach to classification in high dimensions, it was designed for balanced datasets. In the case of unequal costs, biased sampling, or unbalanced data, there are major improvements available, using appropriately weighted versions of DWD (wDWD). A major contribution of this paper is the development of optimal weighting schemes for various nonstandard classification problems. In addition, we discuss several…

高维小样本情形下主成分分析的一致性AI 解读

Principal Component Analysis (PCA) is an important tool of dimension reduction especially when the dimension (or the number of variables) is very high. Asymptotic studies where the sample size is fixed, and the dimension grows [i.e., High Dimension, Low Sample Size (HDLSS)] are becoming increasingly relevant. We investigate the asymptotic behavior of the Principal Component (PC) directions. HDLSS asymptotics are used to study consistency, strong inconsistency and subspace…

高维小样本的几何表示在较弱条件下成立AI 解读

High-dimension, low-small-sample size datasets have different geometrical properties from those of traditional low-dimensional data. In their asymptotic study regarding increasing dimensionality with a fixed sample size, Hall et al. (2005) showed that each data vector is approximately located on the vertices of a regular simplex in a high-dimensional space. A perhaps unappealing aspect of their result is the underlying assumption which requires the variables, viewed as a…

曲线估计的尺度空间视角AI 解读

Scale space theory from computer vision leads to an interesting and novel approach to nonparametric curve estimation. The family of smooth curve estimates indexed by the smoothing parameter can be represented as a surface called the scale space surface. The smoothing parameter here plays the same role as that played by the scale of resolution in a visual system. In this paper, we study in detail various features of that surface from a statistical viewpoint. Weak convergence…

图像处理中的边缘保持平滑器AI 解读

Classical smoothers have limited usefulness in image processing, because sharp “edges” tend to be blurred. There is a literature on edge-preserving smoothers, but these all require moderately large “smooth stretches.” Here we discuss an approach to this problem called “sigma filtering” and propose an improvement based on running M estimation. Both computational and theoretical aspects are developed. For image processing, the methods have a niche between standard filtering…

自动边界修正AI 解读

Many popular curve estimators based on smoothing have difficulties caused by boundary effects. These effects are visually disturbing in practice and can play a dominant role in theoretical analysis. Local polynomial regression smoothers are known to correct boundary effects automatically. Some analogs are implemented for density estimation and the resulting estimators also achieve automatic boundary corrections. In both settings of density and regression estimation, we…

非参数风险率估计中平滑参数的 Bootstrap 选择AI 解读

An asymptotic representation of the mean weighted integrated squared error for the kernel-based estimator of the hazard rate in the presence of right-censored samples is obtained for different bootstrap resampling methods. As a consequence, a new bandwidth selector based on the bootstrap is introduced. Very satisfactory simulations results are obtained in comparison to the cross-validation selector for different models, using WARPed (i.e., binned) versions of the estimators.

摘要暂缺

密度估计带宽选择简述AI 解读

There has been major progress in recent years in data-based bandwidth selection for kernel density estimation. Some “second generation” methods, including plug-in and smoothed bootstrap techniques, have been developed that are far superior to well-known “first generation” methods, such as rules of thumb, least squares cross-validation, and biased cross-validation. We recommend a “solve-the-equation” plug-in bandwidth selector as being most reliable in terms of overall…

定性平滑的视觉误差准则AI 解读

An important gap, between the classical mathematical theory and the practice and implementation of nonparametric curve estimation, is due to the fact that the usual norms on function spaces measure something different from what the eye can see visually in a graphical presentation. Mathematical error criteria that more closely follow “visual impression” are developed and analyzed from both graphical and mathematical viewpoints. Examples from wavelet regression and kernel…

改进的可变窗宽核密度估计AI 解读

Variable window width kernel density estimators, with the width varying proportionally to the square root of the density, have been thought to have superior asymptotic properties. The rate of convergence has been claimed to be as good as those typical for higher-order kernels, which makes the variable width estimators more attractive because no adjustment is needed to handle the negativity usually entailed by the latter. However, in a recent paper, Terrell and Scott show…

通过变换降低核密度估计的边界偏差AI 解读

SUMMARY We consider kernel estimation of a univariate density whose support is a compact interval. If the density is non-zero at either boundary, then the usual kernel estimator can be seriously biased. ‘Reflection’ at a boundary removes some bias, but unless the first derivative of the density is 0 at the boundary the estimator with reflection can still be much more severely biased at the boundary than in the interior. We propose to transform the data to a density that has…

精确平均积分平方误差AI 解读

An exact and easily computable expression for the mean integrated squared error (MISE) for the kernel estimator of a general normal mixture density, is given for Gaussian kernels of arbitrary order. This provides a powerful new way of understanding density estimation which complements the usual tools of simulation and asymptotic analysis. The family of normal mixture densities is very flexible and the formulae derived allow simple exact analysis for a wide variety of density…

更接近最优值的回归平滑参数AI 解读

Abstract It is well known that data-driven regression smoothing parameters ħ based on cross-validation and related methods exhibit a slow rate of convergence to their optimum. In an earlier article we showed that this rate can be as slow as n –1/10; that is, for a bandwidth ħ 0 optimizing the averaged squared error, n 1/10 (ħ — ħ 0)/ħ 0 tends to an asymptotic normal distribution. In this article we consider mean averaged squared error optimal bandwidths h 0. This (nonrandom)…

平均导数估计的带宽选择AI 解读

Abstract The average derivative is the expected value of the derivative of a regression function. Kernel methods have been proposed as a means of estimating this quantity. The problem of bandwidth selection for these kernel estimators is addressed here. Asymptotic representations are found for the variance and squared bias. These are compared with each other to find an insightful representation for a bandwidth optimizing terms of lower order than n –1. It is interesting…

接近最优值的回归平滑参数AI 解读

It is well known that data-driven regression smoothing parameters h based on cross-validation and related methods exhibit a slow rate of convergence to their optimum. In an earlier article we showed that this rate can be as slow as n-1/10; that is, for a bandwidth h0 optimizing the averaged squared error, n1/10 (h - h0)/h0 tends to an asymptotic normal distribution. In this article we consider mean averaged squared error optimal bandwidths h0. This (nonrandom) smoothing…

平均导数估计的带宽选择AI 解读

The average derivative is the expected value of the derivative of a regression function. Kernel methods have been proposed as a means of estimating this quantity. The problem of bandwidth selection for these kernel estimators is addressed here. Asymptotic representations are found for the variance and squared bias. These are compared with each other to find an insightful representation for a bandwidth optimizing terms of lower order than n-1. It is interesting that, for…

一种简单的根号 n 带宽选择器AI 解读

The asymptotically best bandwidth selectors for a kernel density estimator currently require the use of either unappealing higher order kernel pilot estimators or related Fourier transform methods. The point of this paper is to present a methodology which allows the fastest possible rate of convergence with the use of only nonnegative kernel estimators at all stages of the selection process. The essential idea is derived through careful study of factorizations of the pilot…

相关误差下两种带宽选择方法的比较AI 解读

For nonparametric regression, in the case of dependent observations, cross-validation is known to be severely affected by dependence. This effect is precisely quantified through a limiting distribution for the cross-validated bandwidth. The performance of two methods, the "leave-$(2l + 1)$-out" version of cross-validation and partitioned cross-validation, which adjust for the effect of dependence on bandwidth selection is investigated. The bandwidths produced by these two…

交叉验证函数中的局部极小值AI 解读

SUMMARY The method of least squares cross-validation for choosing the bandwidth of a kernel density estimator has been the object of considerable research, through both theoretical analysis and simulation studies. The method involves the minimization of a certain function of the bandwidth. One of the less attractive features of this method, which has been observed in simulation studies but has not previously been understood theoretically, is that rather often the…

非参数回归的 Bootstrap 同时误差带AI 解读

Simultaneous error bars are constructed for nonparametric kernel estimates of regression functions. The method is based on the bootstrap, where resampling is done from a suitably estimated residual distribution. The error bars are seen to give asymptotically correct coverage probabilities uniformly over any number of gridpoints. Applications to an economic problem are given and comparison to both pointwise and Bonferroni-type bars is presented through a simulation study.

密度估计中的数据变换AI 解读

For the density estimation problem the global window width kernel density estimator does not perform well when the underlying density has features that require different amounts of smoothing at different locations. In this article we propose to transform the data with the intention that a global window width is more appropriate for the density of the transformed data. The density estimate of the original data is the “back-transform” by change of variables of the global…