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In this paper, a vector parameter method for ridge regression is proposed. We choose the negative gradient of mean square error as vector direction and decide vector norm with the expectation constrains both of mean square error and of residual error. We come to conclusions that the mean square error is a decreasing function of vector norm while the residual error a increasing one. It is the monotonicity of the errors that leads to our expectation constrains. Since two conflict constrains are under consideration, our vector parameter ridge regression is expected to bear both satisfactory mean square error and acceptable residual error. Finally, a multi-collinearity model is given as an example.     

带随机量测噪声的线性系统的初始状态马尔柯夫估计的均方收敛性

一、引言设线性离散系统X(k+1)=GX(k),Y(k)=CX(k)+V(k) (1.1)式中 G,C 分别为 n×n,m×n 常阵,且 G 为非奇异阵,V(k)为 m×1量测噪声矢量,{V(k)}为平稳白噪声序列,即     

正态-逆Wishart先验下多元线性模型中经验Bayes估计的优良性

在正态-逆Wishart先验下研究了多元线性模型中参数的经验Bayes估计及其优良性问题.当先验分布中含有未知参数时,构造了回归系数矩阵和误差方差矩阵的经验Bayes估计,并在Bayes均方误差(简称BMSE)准则和Bayes均方误差阵(简称BMSEM)准则下,证明了经验Bayes估计优于最小二乘估计.最后,进行了Monte Carlo模拟研究,进一步验证了理论结果.     

Bessel变换的一个下界

Bessel逆问题在物理、化学和工程学等诸多领域有重要应用.解决线性逆问题的传统方法不适合处理具有奇异性曲线边缘的二元函数.鉴于切波对这一类函数的最优表示能力,相关文献采用切波方法研究Bessel逆问题,构造了目标函数的切波域值估计器,得到了它在函数空间V中积分均方差收敛阶的上界.在此基础上利用统计理论给出其最小最大风险的一个下界,证明了在估计Bessel逆问题时此估计器是最优的.     

Error bounds via a new extremum variational principle,mean square residual and weighted mean square residual

Error bounds for a wide class of nonlinear one-dimensional boundary value problems are derived from a new extremum variational principle. A new least-squares approximate technique, based on a weighted mean square residual, is established. Also, the value of the weighted mean square residual and value of the classical mean square residual are used for error estimate. The results are illustrated by four examples.     

线性模型中回归系数和误差方差的同时Bayes估计的优良性

在线性模型中回归系数与误差方差具有正态-逆Gamma先验时,导出了回归系数与误差方差的同时Bayes估计.在均方误差矩阵准则和Bayes Pitman closeness准则下,研究了回归系数的Bayes估计相对于最小二乘(LS)估计的优良性,还讨论了误差方差的Bayes估计在均方误差准则下相对于LS估计的优良性.     

The Superiorities of Simultaneous Empirical Bayes Estimation for the Regression Coefficients and Error-Variance in Linear Model

When the hyperparameters of prior distribution are partly known in linear model, the simultaneous parametric empirical Bayes estimators (PEBE) of the regression coefficients and error variance are constructed. The superiority of PEBE over the least squares estimator (LSE) of regression coefficients is investigated in terms of the the mean square error matrix (MSEM) criterion, and the superiority of PEBE over LSE of the error variance is discussed under the the mean square error (MSE) criterion. Finally, when all hyperparameters are unknown, the PEBE of regression coefficients and error variance are reconstructed and the superiority of them over LSE under the MSE criterion are studied by simulation methods.     

Multi-stage nonparametric estimation of density function using orthonormal systems

In this note, the authors propose a new nonparametric method of estimation of density using orthonormal systems iteratively. The asymptotic mean integrated square error of the estimate at each stage is less than or equal to that of the preceding stage. The new estimate is better, in some cases, than the traditional estimate based upon orthonormal functions from the point of view of the mean integrated square error in the limit.     

Gauss-Markov估计关于误差分布的稳健性

对于一般线性模型y=Xβ+ε,本文讨论了在广义均方误差准则及均方误差矩阵准则下,未知参数β的可估函数Xβ的Gauss-Markov估计关于误差分布的稳健性,分别给出了误差项ε的最大分布类,使得误差项ε的分布在此范围内变动时,Gauss-Markov估计在相应准则下是最优估计.     

i.i.d情况下非参数回归模型的误差密度估计的收敛性质

本文研究了i．i．d情况下非参数回归的误差密度估计的一致收敛和均方收敛，给出了一定条件下误差密度的估计量f^n(x)的一致收敛速度和均方收敛速度。     

The area within a curve

The area of a simple closed convex curve can be estimated in terms of the number of points of a square lattice that lie within the curve. We obtain the usual error bound without integration using a form of the Hardy—Littlewood—Ramanujan circle method, and also present simple estimates for the mean square error.     

帕累托索赔分布中风险参数的经验贝叶斯估计

本文建立了贝叶斯模型,讨论了帕累托索赔额分布中参数的估计问题,得到了风险参数的极大似然估计、贝叶斯估计和信度估计,并证明了这些估计的强相合性.在均方误差的意义下比较了这些估计的好坏,并通过数值模拟对均方误差进行了验证,结果表明,贝叶斯估计比其他估计具有较小的均方误差.最后,给出了结构参数的估计并证明了经验贝叶斯估计和经验贝叶斯信度估计的渐近最优性.     

Measuring the efficiency of trigonometric series estimates of a density

Some types of density estimators, particularly those based on trigonometric series, converge reasonably quickly to their limit except in the neighbourhood of one or two singularities. In this situation the mean integrated square error, the traditional measure of the efficiency of a density estimator, is an unsatisfactory measure. The notion of partial mean integrated square error is introduced and used to compare the performance of trigonometric series estimators. The results lead to consideration of some new estimators which have excellent properties from the points of view of both efficiency and ease of computation.     

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A kernel-type nonparametric estimator of the intensity function for inhomogeneous spatial point patterns with replicated data is proposed. Asymptotic expansion of the mean square error is derived and the rateof convergence of the integrated square error is also investigated. Two methods, least-square and composite likelihood cross-validation, for selecting the bandwidth are described. The performance of the two procedures are illustrated using simulation data.     

Second moment of error term of mean value for binary Egyptian fractions

We study the mean square of the error term of the mean value for binary Egyptian fractions.We get an asymptotic formula under the Riemann Hypothesis.     

On the role of the shrinkage parameter in local linear smoothing

Summary. It has been shown that local linear smoothing possesses a variety of very attractive properties, not least being its mean square performance. However, such results typically refer only to asymptotic mean squared error, meaning the mean squared error of the asymptotic distribution, and in fact, the actual mean squared error is often infinite. See Seifert and Gasser (1996). This difficulty may be overcome by shrinking the local linear estimator towards another estimator with bounded mean square. However, that approach requires information about the size of the shrinkage parameter. From at least a theoretical viewpoint, very little is known about the effects of shrinkage. In particular, it is not clear how small the shrinkage parameter may be chosen without affecting first-order properties, or whether infinitely supported kernels such as the Gaussian require shrinkage in order to achieve first-order optimal performance. In the present paper we provide concise and definitive answers to such questions, in the context of general ridged and shrunken local linear estimators. We produce necessary and sufficient conditions on the size of the shrinkage parameter that ensure the traditional mean squared error formula. We show that a wide variety of infinitely-supported kernels, with tails even lighter than those of the Gaussian kernel, do not require any shrinkage at all in order to achieve traditional first-order optimal mean square performance. Received: 22 May 1995 / In revised form: 23 January 1997     

一种改进的基于指数平滑神经网络模型的时间序列预测方法

基于指数平滑模型与误差反传神经网络法提出了一个改进的时间序列预测方法.将神经网络模型移植入指数加权滑动平均模型中,充分考虑了时间序列的部分线性性和非线性性对预测结果的影响,是传统的混合模型的一个更合理的改进.最后通过对上证指数时间序列的实证分析,以预测均方误差为检验标准,对五种常用的时间序列预测模型进行了预测精度的比较,而且经验证所提出的改进的时间序列预测模型相对来说具有更小的预测均方误差.     

增加试验数据对岭估计的均方误差的影响

在线性模型中,对于回归系数的岭估计和广义岭估计,本文给出了选择岭参数的条件,在此条件下,岭估计和广义岭估计不仅能改善LS估计,而且增加试验数据时,它们的均方误差都会减少。同时,本文将增加试验数据换成增加附加信息,从而讨论了附加信息对混合岭估计和混合广义岭估计的影响问题。     

导弹最大射程Bayes估计的分析和改进

本文用验前数据的质量因子及估计的相对均方误差分析了导弹最大射程的一类Ｂａｙｅｓ估计的性能，对不同的质量因子，给出了最佳验前数据量的一种近似公式。针对这类Ｂａｙｅｓ估计的冒进问题，本文对它们进行了改进并得到了一类新的估计。最后，通过ＭｏｎｔｅＣａｒｌｏ法比较了这些估计的相对均方误差，验证了新估计的优良性。     

一种有偏估计与最小二乘估计的两种新的相对效率

考察了线性回归模型的回归系数的一类有偏估计,在均方误差矩阵准则下将其与最小二乘估计(LSE)进行比较,导出了这类有偏估计相对于LSE的两种新的相对效率的上、下界.