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核密度估计与边缘化一致性AI 解读

Kernel density estimates, as commonly applied, generally have no exact model-based interpretation since they violate conditions that define coherent joint distributions. The issue of marginalization consistency is considered here. It is shown that most commonly used kernel functions violate this condition. It is also shown that marginalization consistency holds only for classes of kernel estimates based on Laplacian, or double-exponential kernels whose window width…

核密度估计中最优的数据驱动带宽选择AI 解读

A bandwidth selection method is proposed for kernel density estimation. This is based on the straightforward idea of plugging estimates into the usual asymptotic representation for the optimal bandwidth, but with two important modifications. The result is a bandwidth selector with the, by nonparametric standards, extremely fast asymptotic rate of convergence of n−½ where n ↑ ∞ denotes sample size. Comparison is given to other bandwidth selection methods, and small sample…

异方差与自相关一致的协方差矩阵估计AI 解读

This paper is concerned with the estimation of covariance matrices in the presence of heteroskedasticity and autocorrelation of unknown forms. Currently available estimators that are designed for this context depend upon the choice of a lag truncation parameter and a weighting scheme. No results are available regarding the choice of lag truncation parameter for a fixed sample size, regarding data-dependent automatic lag truncation parameters, or regarding the choice of…

学生化 U 统计量的 Edgeworth 展开与 Bootstrap 近似AI 解读

The asymptotic accuracy of the estimated one-term Edgeworth expansion and the bootstrap approximation for a Studentized $U$-statistic is investigated. It is shown that both the Edgeworth expansion estimate and the bootstrap approximation are asymptotically closer to the exact distribution of a Studentized $U$-statistic than the normal approximation. The conditions needed to obtain these results are weak moment assumptions on the kernel $h$ of the $U$-statistic and a…

投影寻踪型回归模型的估计AI 解读

Since the pioneering work of Friedman and Stuetzle in 1981, projection-pursuit algorithms have attracted increasing attention. This is mainly due to their potential for overcoming or reducing difficulties arising in nonparametric regression models associated with the so-called curse of dimensionality, that is, the amount of data required to avoid an unacceptably large variance increasing rapidly with dimensionality. Subsequent work has, however, uncovered a dependence on…

多元自适应回归样条AI 解读

A new method is presented for flexible regression modeling of high dimensional data. The model takes the form of an expansion in product spline basis functions, where the number of basis functions as well as the parameters associated with each one (product degree and knot locations) are automatically determined by the data. This procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties. Unlike recursive partitioning,…

1851—1985年北半球爆炸性火山活动的探索性分析AI 解读

Studies of regional climate following volcanic eruptions suggest that explosive volcanism exerts a short-term cooling effect. To understand the effects of explosive volcanism on historic temperatures, it is necessary to identify changes through time in the frequency of explosive eruptions. This article presents an exploratory analysis of a record of explosive eruptions in the Northern Hemisphere over the period 1851–1985. This record is modeled as a nonstationary Poisson…

综述：非参数密度估计的近期发展AI 解读

Advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research, and especially the development of nonparametric data analysis procedures. In particular, theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics, such as nonparametric regression, nonparametric discrimination, and nonparametric pattern recognition. This…

非参数密度估计的近期发展AI 解读

Advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research, and especially the development of nonparametric data analysis procedures. In particular, theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics, such as nonparametric regression, nonparametric discrimination, and nonparametric pattern recognition. This…

1851—1985年北半球爆炸性火山活动的探索性分析AI 解读

Studies of regional climate following volcanic eruptions suggest that explosive volcanism exerts a short-term cooling effect. To understand the effects of explosive volcanism on historic temperatures, it is necessary to identify changes through time in the frequency of explosive eruptions. This article presents an exploratory analysis of a record of explosive eruptions in the Northern Hemisphere over the period 1851–1985. This record is modeled as a nonstationary Poisson…

非参数回归中的最优局部带宽选择AI 解读

SUMMARY Kernel estimators of a regression function are investigated. The bandwidths are locally chosen by a data-driven method based on the minimization of a local cross-validation criterion. This method is shown to be asymptotically optimal with respect to local quadratic measures of errors. Monte Carlo experiments are presented, and finally the method is applied to some data of medical interest.

消费者需求中的偏好变化检验：间接可分的半参数模型AI 解读

This article considers the hypothesis that preference changes may partly explain observed consumption patterns. A model is developed based on indirect weak separability. The kernel estimator of the regression function models the unknown shape of demad functions, and a parametric structure models seasonality, dynamics, and preference change. This semiparametric specification is applied to U.S. meat demand. The results support the notion that changes in consumer preferences,…

高效的偏差校正非参数谱估计AI 解读

We consider estimation of a spectral density at a particular frequency. A linear model is developed based on the local second order behaviour of kernel spectral estimates and a second stage estimate is derived from this model. The estimate is shown to be asymptotically more efficient than the kernel estimate with the optimal, but unknown, bandwidth whenever the smoothing parameter is chosen so that it oversmooths the kernel estimate. A small-sample simulation study shows…

迈向可实施的贝叶斯非参数密度估计AI 解读

SUMMARY Nonparametric density estimators smooth the empirical distribution function and are sensitive to the choice of smoothing parameters. This paper develops an hierarchical Bayes formulation for the smoothing problem. The prior distribution for the density function is the logistic normal process, which is a logistic transform of a Gaussian process. The covariance of the Gaussian process is a smoothing kernel and has parameters that control the degree of smoothness. The…

长度偏倚数据的核密度估计AI 解读

A new kernel density estimator for length biased data which derives from smoothing the nonparametric maximum likelihood estimator is proposed and investigated. It has various advantages over an alternative method suggested by Bhattacharyya, Franklin & Richardson (1988): it is necessarily a probability density, it is particularly better behaved near zero, it has better asymptotic mean integrated squared error properties and it is more readily extendable to related problems…

端点附近的平滑最优核估计量AI 解读

Kernel estimators for smooth curves like density, spectral density or regression functions require modifications when estimating near endpoints of the support, both for practical and asymptotic reasons. The construction of such boundary kernels as solutions of a variational problem is addressed and representations in orthogonal polynomials are given, including explicit solutions for the most important cases. Based on explicit formulae for certain functional of the kernels,…

核密度估计与边缘化一致性AI 解读

Kernel density estimates, as commonly applied, generally have no exact model-based interpretation since they violate conditions that define coherent joint distributions. The issue of marginalization consistency is considered here. It is shown that most commonly used kernel functions violate this condition. It is also shown that marginalization consistency holds only for classes of kernel estimates based on Laplacian, or double-exponential kernels whose window width…

核密度估计中最优的数据驱动带宽选择AI 解读

A bandwidth selection method is proposed for kernel density estimation. This is based on the straightforward idea of plugging estimates into the usual asymptotic representation for the optimal bandwidth, but with two important modifications. The result is a bandwidth selector with the, by nonparametric standards, extremely fast asymptotic rate of convergence of n−½ where n ↑ ∞ denotes sample size. Comparison is given to other bandwidth selection methods, and small sample…

离散化误差对核密度估计带宽选择的影响AI 解读

It is well known that the cross-validation score function of discretized data often tends to -∞ as the bandwidth tends to zero. This disturbing property causes some difficulty in applying cross-validation to discretized data. Since all data are rounded to some degree, the phenomenon limits the application of cross-validation in practice. Based on characteristic functions, the cause of the difficulty is explained, and a simple modification is suggested. Under some conditions,…

某些非参数极大似然估计量的大偏差概率AI 解读

Let $(X, \mathscr{A})$ be a measurable space and $\{P_{\vartheta,\tau}\mid\mathscr{A}: \vartheta \in \Theta, \tau \in T\}$ a family of probability measures. Given an appropriate estimator sequence for $\vartheta$, we define a sequence of asymptotic maximum likelihood estimators for $\tau$ and give bounds for its large deviation probabilities under conditions which are natural for the application to the estimation of mixing distributions. This paper generalizes earlier…

非参数回归中的自适应 M 估计AI 解读

A method for robust nonparametric regression is discussed. We consider kernel $M$-estimates of the regression function using Huber's $\psi$-function and extend results of Hardle and Gasser to the case of random designs. A practical adaptive procedure is proposed consisting of simultaneously minimising a cross-validatory criterion with respect to both the smoothing parameter and a robustness parameter occurring in the $\psi$-function. This method is shown to possess a…

带宽估计量的渐近分布AI 解读

The problem of automatic bandwidth selection for a kernel regression estimate is studied. Since the bandwidth considerably affects the features of the estimated curve, it is important to understand the behavior of bandwidth selection procedures. The bandwidth estimate considered here is the minimizer of Mallows' criterion. Though it was established that the bandwidth estimate is asymptotically normal, it is well recognized that the rate of convergence is extremely slow. In…

概率密度点值估计风险的下界AI 解读

This article considers estimation of f(0) of a density function satisfying a Lipshitz condition in a neighborhood of 0. A nonstandard use of the Cramer-Rao inequality yields numerical lower bounds on the minimax squared error risk of any estimator. These bounds are then compared with the minimax risk of the asymptotically optimal kernel-type estimator. The asymptotic bounds obtained (as the sample size n → ∞) are not quite as good as those in Donoho and Liu (in press a, b),…

核密度估计的自助法带宽选择AI 解读

A bootstrap-based choice of bandwidth for kernel density estimation is introduced. The method works by estimating the integrated mean squared error (IMSE) for any given bandwidth and then minimizing over all bandwidths. A straightforward application of the bootstrap method to estimate the IMSE fails because it does not capture the bias component. A smoothed bootstrap method based on an initial density estimate is described that solves this problem. It is possible to…