Kernel density estimation for linear processes: Asymptotic normality and optimal bandwidth derivation

The problem of estimating the marginal density of a linear process by kernel methods is considered. Under general conditions, kernel density estimators are shown to be asymptotically normal. Their limiting covariance matrix is computed. We also find the optimal bandwidth in the sense that it asymptotically minimizes the mean square error of the estimators. The assumptions involved are easily verifiable.Supported in part by NSF grant DMS-9403718.     

有辅助信息时有限总体分布函数的新估计量

本文提出两个新的估计量,利用观察数据中的总体辅助信息来估计有限总体分布函数,并通过两个人工总体的模拟实验,比较新的估计量、传统的估计量及Rao,Kover&Mantel(1990)提出的估计量的相对平均误差与相对标准差。结果表明,从相对标准差的角度分析,两个新的估计量有一个是四个估计量中精度最好的一个,另一个也有很好的表现;而且它们在模型有所偏差时都具备了较好的稳健性。     

2SUR回归模型系数估计的均方误差比较

对于2SUR回归模型的参数估计问题,给出了一些一航均方误差矩阵比较结果,据此提出了一类线性估计和一类基于离差阵广义非限定估计的非线性两步估计,并获得了该两步估计类的一些有限样本性质。     

Probability density estimation with surrogate data and validation sample

The probability density estimation problem with surrogate data and validation sample is considered. A regression calibration kernel density estimator is defined to incorporate the information contained in both surrogate variates and validation sample. Also, we define two weighted estimators which have less asymptotic variances but have bigger biases than the regression calibration kernel density estimator. All the proposed estimators are proved to be asymptotically normal. And the asymptotic representations for the mean squared error and mean integrated square error of the proposed estimators are established, respectively. A simulation study is conducted to compare the finite sample behaviors of the proposed estimators.     

Histogram-kernel error and its application for bin width selection in histograms

Histogram and kernel estimators are usually regarded as the two main classical data-based non- parametric tools to estimate the underlying density functions for some given data sets. In this paper we will integrate them and define a histogram-kernel error based on the integrated square error between histogram and binned kernel density estimator, and then exploit its asymptotic properties. Just as indicated in this paper, the histogram-kernel error only depends on the choice of bin width and the data for the given prior kernel densities. The asymptotic optimal bin width is derived by minimizing the mean histogram-kernel error. By comparing with Scott’s optimal bin width formula for a histogram, a new method is proposed to construct the data-based histogram without knowledge of the underlying density function. Monte Carlo study is used to verify the usefulness of our method for different kinds of density functions and sample sizes.     

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??The Bayes estimators of variance components are derived under weighted square loss function for the balanced one-way classification random effects model with the assumption that variance component has the conjugate prior distribution. The superiorities of the Bayes estimators for variance components to traditional ANOVA estimators are studied in terms of the mean square error (MSE) criterion. Finally, a remark for main results is given.     

A class of estimators with estimated optimum values in sample surveys

Srivastava and Jhajj (1981) proposed a class of estimators for population mean of a character using auxiliary information and optimum values involving unknown parameters. From the practical point of view, their results have very little utility. In view of practical utility, we propose a class of estimators with estimated optimum values. Further, it is shown that the proposed class with estimated optimum values attains the same minimum mean square error of the class of estimators based on optimum values.     

强相关数据回归函数的小波估计

讨论了在强相关数据情形下对回归函数的小波估计,并且给出了估计量的均方误差的一个渐近展开表示式． 对研究估计量的优劣,所推导的近似表示式显得非常重要．对一般的回归函数核估计,如果回归函数不是充分光滑,这个均方误差表示式并不成立A·D2但对小波估计,即使回归函数间断连续,这个均方误差表示式仍然成立．因此,小波估计的收敛速度要比核估计来得快,从而小波估计在某种程度上改进了现有的核估计．     

The superiority of empirical bayes estimation of parameters in partitioned normal linear model

In this article,the empirical Bayes（EB）estimators are constructed for the estimable functions of the parameters in partitioned normal linear model.The superiorities of the EB estimators over ordinary least-squares（LS）estimator are investigated under mean square error matrix（MSEM）criterion.     

无重复因析试验中一种散度效应的迭代估计方法

在无重复因析试验下,基于李济洪,王钰等(2010)提出的散度效应的AMH估计,给出了一种散度效应的迭代估计方法(称为IAMH估计),通过模拟试验验证了此方法比常用的AMH,MH估计具有更小的均方误差.     

对数正态分布基于恒加试验的参数估计

讨论对数正态分布场合有非常数尺度参数恒加试验的参数估计,由最小均方误差准则导出基于完全样本恒加试验的点估计和近似区间估计.     

Comparison of two orthogonal series methods of estimating a density and its derivatives on an interval

We compare the merits of two orthogonal series methods of estimating a density and its derivatives on a compact interval—those based on Legendre polynomials, and on trigonometric functions. By examining the rates of convergence of their mean square errors we show that the Legendre polynomial estimators are superior in many respects. However, Legendre polynomial series can be more difficult to construct than trigonometric series, and to overcome this difficulty we show how to modify trigonometric series estimators to make them more competitive.     

Optimal kernel estimation of densities

Precise asymptotic behavior for mean integrated squared error (MISE) is determined for sequences of kernel estimators of a density in a broad class, including discontinuous and possibly unbounded densities. The paper shows that the sequence using the kernel optimal at each fixed sample size is asymptotically more efficient than a sequence generated by changing the bandwidth of a fixed kernel shape, regardless of the kernel shape. The class of densities considered are those whose characteristic functions behave at large arguments like the product of a Fourier series and a regularly varying function. This condition may be related to the smoothness of an m-th derivative of the density.Partially supported by National Science Foundation Grant DMS-8711924.     

半参数回归模型的几乎无偏岭估计

提出了半参数回归模型的几乎无偏岭估计,并与岭估计进行了比较,在均方误差意义下,几乎无偏岭估计优于岭估计. 然后讨论了有偏参数的选取问题. 最后,用模拟算例和实际应用说明了几乎无偏岭估计的有效性和可行性.     

Integrated squared error of kernel-type estimator of distribution function

Let X ₁ ,...,X _n be a random sample drawn from distribution function F(x) with density function f(x) and suppose we want to estimate X(x). It is already shown that kernel estimator of F(x) is better than usual empirical distribution function in the sense of mean integrated squared error. In this paper we derive integrated squared error of kernel estimator and compare the error with that of the empirical distribution function. It is shown that the superiority of kernel estimators is not necessarily true in the sense of integrated squared error.     

A note on superkernel density estimators

It is well known that so-called superkernel density estimators have better asymptotic properties than conventional kernel estimators (and generally finite-order estimators) in the case when the density to be estimated is very smooth. In this note, we study asymptotic behavior of the mean integrated square error of superkernel density estimators in the case when the density to be estimated is not very smooth. It turns out that in this case, superkernel estimators still have better asymptotics than finite-order estimators.     

Nonparametric orthogonal series estimators of regression: A class attaining the optimal convergence rate in L2

In this note a class of nonparametric orthogonal series type estimators for regression function fitting is considered. Sufficient conditions are given for the estimators to attain the optimal convergence rate in the mean integrated square error sense. Using results from the theory of numerical integration, examples of estimators are given, for which the above mentioned conditions hold.     

A KERNEL ESTIMATOR OF A DENSITY FUNCTION IN MULTIVARIATE CASE FROM RANDOMLY CENSORED DATA

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对Stein的SLS估计的改进研究

提出一类新的估计——c-(K,S)型估计,证明了在均方误差意义下运用泛岭回归技术可以改进S te in的SLS估计,同时给出了参数的最优值满足的条件.     

Optimizing the smoothed bootstrap

In this paper we develop the technique of a generalized rescaling in the smoothed bootstrap, extending Silverman and Young's idea of shrinking. Unlike most existing methods of smoothing, with a proper choice of the rescaling parameter the rescaled smoothed bootstrap method produces estimators that have the asymptotic minimum mean (integrated) squared error, asymptotically improving existing bootstrap methods, both smoothed and unsmoothed. In fact, the new method includes existing smoothed bootstrap methods as special cases. This unified approach is investigated in the problems of estimation of global and local functionals and kernel density estimation. The emphasis of this investigation is on theoretical improvements which in some cases offer practical potential.