Bootstrap bandwidth selection in kernel density estimation from a contaminated sample

In this paper we consider kernel estimation of a density when the data are contaminated by random noise. More specifically we deal with the problem of how to choose the bandwidth parameter in practice. A theoretical optimal bandwidth is defined as the minimizer of the mean integrated squared error. We propose a bootstrap procedure to estimate this optimal bandwidth, and show its consistency. These results remain valid for the case of no measurement error, and hence also summarize part of the theory of bootstrap bandwidth selection in ordinary kernel density estimation. The finite sample performance of the proposed bootstrap selection procedure is demonstrated with a simulation study. An application to a real data example illustrates the use of the method. This research was supported by ‘Projet d’Actions de Recherche Concertées’ (No. 98/03-217) from the Belgian government. Financial support from the IAP research network nr P5/24 of the Belgian State (Federal Office for Scientific, Technical and Cultural Affairs) is also gratefully acknowledged.     

正态分布参数函数的最小最大估计

The estimation of the functionθ=exp{αμ bσ2} of parameters (μ,σ2) in normal distribution N(μ,σ2) is discussed. And when the prior distributions ofμandσ2 are independent, under the loss function L(θ,δ)=(θ-1×δ-1)2, the Bayesian estimation and the existence and computing method on minimax estimation are deeply discussed.     

Applications of extreme points to distortion estimates

The main object of this paper is to introduce and investigate a class of analytic functions with negative coefficients defined by the Wright generalized hypergeometric function. By using the extreme points theory we obtain coefficient estimates and distortion theorems in the class of functions.     

Normal Euler classes of knotted surfaces and triple points on projections

We present a new formula relating the normal Euler numbers of embedded surfaces in -space and the number of triple points on their projections into -space. This formula generalizes Banchoff's formula between normal Euler numbers and branch points on the projections.

     

Estimation of the extreme value and the extreme points

Summary Letf be a continuous function defined on some domainA andX ₁,X ₂, ... be iid random variables. We estimate the extreme value off onA by studying the limiting distribution of min {f(X ₁), ...,f(X _n )} or max {f(X ₁), ...,f(X _n )} properly normalized. Sufficient conditions for the existence of the limiting distribution as well as a characterization of the limiting distribution relative to the extreme points off will be provided. A discussion of the multidimensional case is also carried out. Partially supported by CNPq-No. 301508/84.     

Spherical basis functions and uniform distribution of points on spheres

The main purpose of the present paper is to employ spherical basis functions (SBFs) to study uniform distribution of points on spheres. We extend Weyl's criterion for uniform distribution of points on spheres to include a characterization in terms of an SBF. We show that every set of minimal energy points associated with an SBF is uniformly distributed on the spheres. We give an error estimate for numerical integration based on the minimal energy points. We also estimate the separation of the minimal energy points.     

On statistical limit points of double sequences

We investigate the structure of the set of all statistical limit points of a double sequence and prove certain results, mainly showing that this set can be characterized as a F_σ-set.     

The Asymptotic Variance of the Continuous-Time Kernel Estimator with Applications to Bandwidth Selection

We derive simple expressions for the asymptotic variance of the kernel-density estimator of a stationary continuous-time process in one and d dimensions and relate convergence rates to sample path smoothness. Important applications include methods for selecting optimal smoothing parameters and construction of confidence bands for testing hypotheses about the density. In a simulation study the results are applied to bandwidth selection for discrete-time processes that can be modelled as continuous-time processes sampled at a high rate.     

Density of the set of positive proper minimal points in the set of minimal points

One of the important problems of vector optimization concerns the density of the set of positive proper minimal points in the set of minimal points. We use the concepts of dentable point and approximating cones to derive sufficient conditions guaranteeing that the set of minimal points is contained in the closure of the set of positive proper minimal points. The result can be applied to obtain a density result for the unit ball in 1 ^p , 1<p<+, which does not follow from any other well-known density theorem.The author would like to thank Professor W. T. Fu for helpful comments. Moreover, the author is grateful to Professor H. P. Benson and the referees for valuable remarks and suggestions concerning a previous draft of this paper.     

核估计在医疗费用分布研究中的应用

医疗费用的分布是健康保险的精算基础之一.本文研究了某团体一年的住院医疗费用记录,运用非参数模型中的核密度估计方法,对每次住院费用对数的分布密度进行了估计,并用它进一步估计了每次住院费用的均值和方差,取得了较好的效果.在此基础上,本文对拟合结果及其对健康保险的启示作了一些讨论.     

非参数模型均值函数结构变点的Bootstrap检测

本文检测非参数回归模型均值函数结构变点,针对均值函数跃度的长期均值为零时,基于残量的CUSUM统计量对均值函数结构变点检验无效的问题,本文提出了一种基于均值函数的核估计的检验统计量,得到统计量在原假设和备择假设下的极限分布,并构造Bootstrap方法对非参数回归模型均值函数结构变点进行检验,证明了检验和估计的一致性;模拟结果表明本文方法明显优于已有方法。     

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This paper studies the estimation of change point in mean and variance function of a non-parametric regression model based on kernel estimation and wavelet method. First, kernel estimation of mean function is developed and it is used to estimate the position and jump size of mean change. Second, wavelet methods are applied to derive the variance estimator which is used to estimate the location and jump size of the change point in variance. The asymptotic properties of these estimators are proved. Finally, the results from a numerical simulations and comparison study show that validate the effectiveness of our method.     

Statistical inference in massive data sets

Analysis of massive data sets is challenging owing to limitations of computer primary memory. In this paper, we propose an approach to estimate population parameters from a massive data set. The proposed approach significantly reduces the required amount of primary memory, and the resulting estimate will be as efficient if the entire data set was analyzed simultaneously. Asymptotic properties of the resulting estimate are studied, and the asymptotic normality of the resulting estimator is established. The standard error formula for the resulting estimate is proposed and empirically tested; thus, statistical inference for parameters of interest can be performed. The effectiveness of the proposed approach is illustrated using simulation studies and an Internet traffic data example. Copyright © 2012 John Wiley & Sons, Ltd.     

正态分布在学生学习成绩评估中的应用

本文将正态分布应用于学生学习成绩评估中，将原始分数转换为标准分数后，各科的分数方可比较或相加，从而改变了将原始分数直接相加比较的那种不够科学也不甚合理的传统做法。     

时间序列中方差的结构变点的小波识别(英文)

本文给出了时间序列中方差的小波系数的两种估计:连续估计和离散估计.这两种估计可以用来检测时间序列中方差的结构变点.利用这两种估计我们给出了方差变点的位置和跳跃幅度的估计,并且显示出这些估计可达到最佳收敛速度.同时,我们还给出了这些估计的收敛速度以及检验统计量的渐进分布!     

非指数分布数族参数的经验Bayes估计的收敛速度

在“平方损失”下,研究了非指数分布族参数θ的经验Bayes估计,首先利用概率密度函数的核估计,构造了位置参数的经验Bayes（EB）估计量,在适当的条件下获得了它的收敛速度．     

Bootstrap choice of tuning parameters

Consider the problem of estimating θ=θ(P) based on datax _n from an unknown distributionP. Given a family of estimatorsT _{n, β} of θ(P), the goal is to choose β among β∈I so that the resulting estimator is as good as possible. Typically, β can be regarded as a tuning or smoothing parameter, and proper choice of β is essential for good performance ofT _{n, β} . In this paper, we discuss the theory of β being chosen by the bootstrap. Specifically, the bootstrap estimate of β, , is chosen to minimize an empirical bootstrap estimate of risk. A general theory is presented to establish the consistency and weak convergence properties of these estimators. Confidence intervals for θ(P) based on , are also asymptotically valid. Several applications of the theory are presented, including optimal choice of trimming proportion, bandwidth selection in density estimation and optimal combinations of estimates.     

负二项分布两种参数估计及其比较

负二项分布在昆虫空间分布中有重要应用,其参数常用矩法或零频率法来估计。本文讨论两种估计的性质,得到大样本情形下,零频率较大时,零频率估计较矩估计有更高精度的结论;通过计算机模拟,验证结论。     

Efficiency and Robustness of a Resampling M-Estimator in the Linear Model

In the literature, there are basically two kinds of resampling methods for least squares estimation in linear models; the E-type (the efficient ones like the classical bootstrap), which is more efficient when error variables are homogeneous, and the R-type (the robust ones like the jackknife), which is more robust for heterogeneous errors. However, for M-estimation of a linear model, we find a counterexample showing that a usually E-type method is less efficient than an R-type method when error variables are homogeneous. In this paper, we give sufficient conditions under which the classification of the two types of the resampling methods is still true.