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离散化与插值后的核密度估计AI 解读

The kernel estimate of a probability density function inherits its smoothness properties from the kernel density chosen by the investigator. Nevertheless, for computational (and especially graphical) reasons, exact kernel density estimates are often presented in (piecewise constant) discretized or (piecewise linear) interpolated form, using the exact estimate only at some grid of points. The asymptotic integrated mean squared error properties of these modifications are…

论投影寻踪回归AI 解读

We construct a tractable mathematical model for kernel-based projection pursuit regression approximation. The model permits computation of explicit formulae for bias and variance of estimators. It is shown that the bias of an orientation estimate dominates error about the mean--indeed, the latter is asymptotically negligible in comparison with bias. However, bias and error about the mean are of the same order in the case of projection pursuit curve estimates. Implications of…

离散含噪数据下积分方程正则化解的收敛速率AI 解读

Given data $y_i = (Kg)(u_i) + \varepsilon_i$ where the $\varepsilon$'s are random errors, the $u$'s are known, $g$ is an unknown function in a reproducing kernel space with kernel $r$ and $K$ is a known integral operator, it is shown how to calculate convergence rates for the regularized solution of the equation as the evaluation points $\{u_i\}$ become dense in the interval of interest. These rates are shown to depend on the eigenvalue asymptotics of $KRK^\ast$, where $R$ is…

基于局部中位数的核估计量的渐近性质AI 解读

The desire to make nonparametric regression robust leads to the problem of conditional median function estimation. Under appropriate regularity conditions, a sequence of local median estimators can be chosen to achieve the optimal rate of convergence $n^{-1/(2+d)}$ both pointwise and in the $L^q (1 \leq q < \infty)$ norm restricted to a compact. It can also be chosen to achieve the optimal rate of convergence $(n^{-1} \log n)^{1/(2+d)}$ in the $L^\infty$ norm restricted to a…

线性平滑器与加性模型AI 解读

We study linear smoothers and their use in building nonparametric regression models. In the first part of this paper we examine certain aspects of linear smoothers for scatterplots; examples of these are the running-mean and running-line, kernel and cubic spline smoothers. The eigenvalue and singular value decompositions of the corresponding smoother matrix are used to describe qualitatively a smoother, and several other topics such as the number of degrees of freedom of a…

非参数政策分析AI 解读

This article considers the problem of predicting the mean effect of a change in the distribution of certain policy-related variables on a dependent variable (Y). This is conventionally done using a parametric model. If, however, the conditional expectation of Y, given policy and nonpolicy variables X, is unaltered by the policy intervention, and if the support of X after the policy intervention lies within the support of X before the intervention, then this analysis can be…

摘要暂缺

审计中筛式抽样的效率AI 解读

Dollar-unit sampling is a popular method for sampling accounts by auditors. In this article, we analyze and compare two methods of obtaining dollar-unit samples, simple random sampling and sieve sampling. Two methods of assigning errors to the dollar units in the population are considered. For a variety of study populations based on actual accounting populations, it is found that sieve sampling, with either error assignment method, produces a more precise point estimator…

线性模型的自适应 L 估计AI 解读

Asymptotically efficient (adaptive) estimators for the slope parameters of the linear regression model are constructed based upon the "regression quantile" statistics suggested by Koenker and Bassett. The estimators are natural analogues of the adaptive $L$-estimators of location of Sacks, but employ kernel-density type estimators of the optimal $L$-estimator weight function.

投影近似及其与核方法的对偶关系AI 解读

Projection pursuit regression and kernel regression are methods for estimating a smooth function of several variables from noisy data obtained at scattered sites. Methods based on local averaging can perform poorly in high dimensions (curse of dimensionality). Intuition and examples have suggested that projection based approaches can provide better fits. For what sorts of regression functions is this true? When and by how much do projection methods reduce the curse of…

核估计的局部带宽选择AI 解读

A kernel estimate of a curve that uses an adaptive procedure for local selection of the bandwidth is considered here. A two-step procedure is proposed for estimating the local bandwidth that minimizes the mean squared error (MSE) of a kernel estimator for nonparametric regression. First, a consistent estimate of the exact MSE is constructed. Then the bandwidth that minimizes the estimate of the MSE is calculated. Sufficient conditions under which this bandwidth is…

摘要暂缺

状态空间模型中信号提取、影响分析与交叉验证的快速算法AI 解读

A fast algorithm is developed for computing the conditional mean and variance of the signal given the observations in a signal plus noise model. The resulting recursions can be applied immediately to provide new and efficient formulae for smoothing part or all of the state vector. The ideas of studentized residuals and leverage from regression analysis are generalized to state space models, and the new algorithm is used to compute the various measures. The results are also…

Durbin–Watson检验的三次样条扩展AI 解读

We propose a new test for model misspecification in linear regression, based on a measure of ‘smoothness’ of the regression residuals. This ‘curvature statistic’ is related to a property of cubic splines and leads to a convenient, powerful test for patterns. The distribution of the curvature statistic may be derived from assumptions of normality. For equally spaced data, the calculations are simplified and a table of significance points of the distribution is given.…

核密度估计中平滑参数的Bootstrap选择AI 解读

Cross-validation based on integrated squared error has already been applied to the choice of smoothing parameter in the kernel method of density estimation. In this paper, an alternative resampling plan, based on the bootstrap, is proposed in an attempt to estimate mean integrated squared error. This leads to a further data-based choice of smoothing parameter. The two methods are compared and some simulations and examples demonstrate the relative merits. For large samples,…

状态空间模型中信号提取、影响诊断与交叉验证的快速算法AI 解读

A fast algorithm is developed for computing the conditional mean and variance of the signal given the observations in a signal plus noise model. The resulting recursions can be applied immediately to provide new and efficient formulae for smoothing part or all of the state vector. The ideas of studentized residuals and leverage from regression analysis are generalized to state space models, and the new algorithm is used to compute the various measures. The results are also…

Durbin–Watson 检验的三次样条扩展AI 解读

We propose a new test for model misspecification in linear regression, based on a measure of ‘smoothness’ of the regression residuals. This ‘curvature statistic’ is related to a property of cubic splines and leads to a convenient, powerful test for patterns. The distribution of the curvature statistic may be derived from assumptions of normality. For equally spaced data, the calculations are simplified and a table of significance points of the distribution is given.…

条件泛函估计量的强一致收敛速率AI 解读

Strong uniform consistency rates are established for kernel type estimators of functionals of the conditional distribution function, under general conditions. The present treatment unifies a number of specific problems previously studied separately in the literature. Some of these applications we treat in detail, including regression curve estimation, density estimation, estimation of conditional df's, $L$-smoothing and $M$-smoothing. Various previous results in the…

多元密度的可容许核估计量AI 解读

A kernel density estimator is defined to be admissible if no other kernel estimator has (among all densities and sample sizes) uniformly smaller mean integrated squared error. Admissible kernel density estimators are precisely those using kernels with nonnegative Fourier transforms bounded by 1. Several examples are given.

密度泛函的单侧推断AI 解读

This paper discusses the possibility of truly nonparametric inference about functionals of an unknown density. Examples considered include: discrete functionals, such as the number of modes of a density and the number of terms in the true model; and continuous functionals, such as the optimal bandwidth for kernel density estimates or the widths of confidence intervals for adaptive location estimators. For such functionals it is not generally possible to make two-sided…

非参数回归估计的统一方法AI 解读

There exists a variety of nonparametric regression estimators, with cubic smoothing splines, k-nearest-neighbor (k-NN) estimators, and various types of kernel estimators among the most popular. A class of kernel estimators with local bandwidth depending on the density of the design points is introduced, where the degree of design adaptation may be expressed by a single parameter α in [0, 1]. The adaptation to the design is such that the bandwidth is made larger, to a degree…

平滑样条的贝叶斯置信区间AI 解读

The frequency properties of Wahba's Bayesian confidence intervals for smoothing splines are investigated by a large-sample approximation and by a simulation study. When the coverage probabilities for these pointwise confidence intervals are averaged across the observation points, the average coverage probability (ACP) should be close to the nominal level. From a frequency point of view, this agreement occurs because the average posterior variance for the spline is similar to…

核密度估计的渐近性能下界AI 解读

We consider an arbitrary sequence of kernel density estimates $f_n$ with kernels $K_n$ possibly depending upon $n$. Under a mild restriction on the sequence $K_n$, we obtain inequalities of the type $E\big(\int|f_n - f|\big) \geq (1 + o(1))\Psi(n, f),$ where $f$ is the density being estimated and $\Psi(n, f)$ is a function of $n$ and $f$ only. The function $\psi$ can be considered as an indicator of the difficulty of estimating $f$ with any kernel estimate.

密度与强度估计中平滑参数选择器的等价性AI 解读

Kernel smoothing is an attractive method for the nonparametric estimation of either a probability density function or the intensity function of a nonstationary Poisson process. In each case the amount of smoothing, controlled by the bandwidth, that is, smoothing parameter, is crucial to the performance of the estimator. Bandwidth selection by cross-validation has been widely studied in the context of density estimation. A bandwidth selector in the intensity estimation case…