Stein's idea and minimax admissible estimation of a multivariate normal mean

We consider estimation of a multivariate normal mean vector under sum of squared error loss.We propose a new class of minimax admissible estimator which are generalized Bayes with respect to a prior distribution which is a mixture of a point prior at the origin and a continuous hierarchical type prior. We also study conditions under which these generalized Bayes minimax estimators improve on the James–Stein estimator and on the positive-part James–Stein estimator.     

Necessary conditions for dominating the James-Stein estimator

This paper develops necessary conditions for an estimator to dominate the James-Stein estimator and hence the James-Stein positive-part estimator. The ultimate goal is to find classes of such dominating estimators which are admissible. While there are a number of results giving classes of estimators dominating the James-Stein estimator, the only admissible estimator known to dominate the James-Stein estimator is the generalized Bayes estimator relative to the fundamental harmonic function in three and higher dimension. The prior was suggested by Stein and the domination result is due to Kubokawa. Shao and Strawderman gave a class of estimators dominating the James-Stein positive-part estimator but were unable to demonstrate admissiblity of any in their class. Maruyama, following a suggestion of Stein, has studied generalized Bayes estimators which are members of a point mass at zero and a prior similar to the harmonic prior. He finds a subclass which is minimax and admissible but is unable to show that any in his class with positive point mass at zero dominate the James-Stein estimator. The results in this paper show that a subclass of Maruyama's procedures including the class that Stein conjectured might contain members dominating the James-Stein estimator cannot dominate the James-Stein estimator. We also show that under reasonable conditions, the “constant” in shrinkage factor must approachp-2 for domination to hold.     

Estimation of the Scale Matrix and its Eigenvalues in the Wishart and the Multivariate F Distributions

In this paper, the problem of estimating the scale matrix and their eigenvalues in a Wishart distribution and in a multivariate F distribution (which arise naturally from a two-sample setting) are considered. A new class of estimators which shrink the eigenvalues towards their arithmetic mean are proposed. It is shown that the new estimator which dominates the usual unbiased estimator under the squared error loss function. A simulation study was carried out to study the performance of these estimators.     

Second-order nonlinear least squares estimation

The ordinary least squares estimation is based on minimization of the squared distance of the response variable to its conditional mean given the predictor variable. We extend this method by including in the criterion function the distance of the squared response variable to its second conditional moment. It is shown that this “second-order” least squares estimator is asymptotically more efficient than the ordinary least squares estimator if the third moment of the random error is nonzero, and both estimators have the same asymptotic covariance matrix if the error distribution is symmetric. Simulation studies show that the variance reduction of the new estimator can be as high as 50% for sample sizes lower than 100. As a by-product, the joint asymptotic covariance matrix of the ordinary least squares estimators for the regression parameter and for the random error variance is also derived, which is only available in the literature for very special cases, e.g. that random error has a normal distribution. The results apply to both linear and nonlinear regression models, where the random error distributions are not necessarily known.     

具有特殊协方差结构的 SURE 模型中参数估计的若干结果

本文讨论具有特殊协方差结构似乎不相关回归方程（SURE）模型中参数的估计问题．除非另有说明,损失函数将取为二次损失和矩阵损失．本文证明了回归系数的线性可估函数的最小二乘估计是极小极大的且在矩阵损失函数下是可容许的;还分别在仿射交换群和平移群下导出了存在回归系数的线性可估函数的一致最小风险同变（UMRE）估计的充要条件,并证明了在仿射交换和二次损失下不存在协方差阵和方差的UMRE估计．     

一类独立随机变量均值中参数的估计

本文给出了独立随机变量均值中参数ａ＝（ａ１，…，ａｐ）′的估计量（ａ１Ｘ１，…，ａｐＸｐ）′在加权平方和损失下为可容许估计的充要条件及在一般损失Ｌｖ下为可容许估计的充分条件（在线性估计类中）。     

均匀分布参数的无偏估计及其分布

讨论了均匀分布未知参数无偏估计量的分布密度,利用无偏估计量构造出一些新的样本函数,并且利用给出的样本函数推导出了未知参数的置信区间.所得到结果改善了现有的估计,易于计算.     

IMPROVED ESTIMATES OF THE COVARIANCE MATRIX IN GENERAL LINEAR MIXED MODELS

In this article, the problem of estimating the covariance matrix in general linear mixed models is considered. Two new classes of estimators obtained by shrinking the eigenvalues towards the origin and the arithmetic mean, respectively, are proposed. It is shown that these new estimators dominate the unbiased estimator under the squared error loss function. Finally, some simulation results to compare the performance of the proposed estimators with that of the unbiased estimator are reported. The simulation results indicate that these new shrinkage estimators provide a substantial improvement in risk under most situations.     

A Comparison of Restricted and Unrestricted Estimators in Estimating Linear Functions of Ordered Scale Parameters of Two Gamma Distributions

The problem of estimating linear functions of ordered scale parameters of two Gamma distributions is considered. A necessary and sufficient condition on the ratio of two coefficients is given for the maximum likelihood estimator (MLE) to dominate the crude unbiased estimator (UE) in terms of mean square error. A modified MLE which satisfies the restriction is also suggested, and a necessary and sufficient condition is also given for it to dominate the admissible estimator based solely on one sample. The estimation of linear functions of variances in two sample problem and also of variance components in a one-way random effect model is mentioned.     

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or the variance parameter of the normal distribution with a normal-inverse-gamma prior, we analytically calculate the Bayes posterior estimator with respect to a conjugate normal-inverse-gamma prior distribution under Stein's loss function. This estimator minimizes the Posterior Expected Stein's Loss (PESL). We also analytically calculate the Bayes posterior estimator and the PESL under the squared error loss function. The numerical simulations exemplify our theoretical studies that the PESLs do not depend on the sample, and that the Bayes posterior estimator and the PESL under the squared error loss function are unanimously larger than those under Stein's loss function. Finally, we calculate the Bayes posterior estimators and the PESLs of the monthly simple returns of the SSE Composite Index.     

On Bayes and unbiased estimators of loss

We consider estimation of loss for generalized Bayes or pseudo-Bayes estimators of a multivariate normal mean vector, θ. In 3 and higher dimensions, the MLEX is UMVUE and minimax but is inadmissible. It is dominated by the James-Stein estimator and by many others. Johnstone (1988, On inadmissibility of some unbiased estimates of loss,Statistical Decision Theory and Related Topics, IV (eds. S. S. Gupta and J. O. Berger), Vol. 1, 361–379, Springer, New York) considered the estimation of loss for the usual estimatorX and the James-Stein estimator. He found improvements over the Stein unbiased estimator of risk. In this paper, for a generalized Bayes point estimator of θ, we compare generalized Bayes estimators to unbiased estimators of loss. We find, somewhat surprisingly, that the unbiased estimator often dominates the corresponding generalized Bayes estimator of loss for priors which give minimax estimators in the original point estimation problem. In particular, we give a class of priors for which the generalized Bayes estimator of θ is admissible and minimax but for which the unbiased estimator of loss dominates the generalized Bayes estimator of loss. We also give a general inadmissibility result for a generalized Bayes estimator of loss. Research supported by NSF Grant DMS-97-04524.     

Estimation of the entropy of a multivariate normal distribution

Motivated by problems in molecular biosciences wherein the evaluation of entropy of a molecular system is important for understanding its thermodynamic properties, we consider the efficient estimation of entropy of a multivariate normal distribution having unknown mean vector and covariance matrix. Based on a random sample, we discuss the problem of estimating the entropy under the quadratic loss function. The best affine equivariant estimator is obtained and, interestingly, it also turns out to be an unbiased estimator and a generalized Bayes estimator. It is established that the best affine equivariant estimator is admissible in the class of estimators that depend on the determinant of the sample covariance matrix alone. The risk improvements of the best affine equivariant estimator over the maximum likelihood estimator (an estimator commonly used in molecular sciences) are obtained numerically and are found to be substantial in higher dimensions, which is commonly the case for atomic coordinates in macromolecules such as proteins. We further establish that even the best affine equivariant estimator is inadmissible and obtain Stein-type and Brewster–Zidek-type estimators dominating it. The Brewster–Zidek-type estimator is shown to be generalized Bayes.     

Estimation of parameters in a two-parameter exponential distribution using ranked set sample

In situations where the experimental or sampling units in a study can be easily ranked than quantified, McIntyre (1952,Aust. J. Agric. Res.,3, 385–390) proposed that the mean ofn units based on aranked set sample (RSS) be used to estimate the population mean, and observed that it provides an unbiased estimator with a smaller variance compared to a simple random sample (SRS) of the same sizen. McIntyre's concept ofRSS is essentially nonparametric in nature in that the underlying population distribution is assumed to be completely unknown. In this paper we further explore the concept ofRSS when the population is partially known and the parameter of interest is not necessarily the mean. To be specific, we address the problem of estimation of the parameters of a two-parameter exponential distribution. It turns out that the use ofRSS and its suitable modifications results in much improved estimators compared to the use of aSRS.     

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.     

协方差的二次型容许估计

本文研究方差的二次型估计的容许性,在平方损失下,我们给出了一个二次型估计在二次型估计类中是协方差的容许估计的充要条件。     

协方差的非齐次估计的可容许性

本文研究协方差的非齐次二次估计的可容许性,在平方损失下,我们给出了一个非齐次二次估计在非齐次二次估计类中是协方差的容许估计的充要条件.     

Wavelet estimation of conditional density with truncated, censored and dependent data

In this paper we define a new nonlinear wavelet-based estimator of conditional density function for a random left truncation and right censoring model. We provide an asymptotic expression for the mean integrated squared error (MISE) of the estimator. It is assumed that the lifetime observations form a stationary α-mixing sequence. Unlike for kernel estimators, the MISE expression of the wavelet-based estimators is not affected by the presence of discontinuities in the curves. Also, asymptotic normality of the estimator is established.     

A Markov Chain Monte Carlo comparison of variance estimators for the sampling of particulate mixtures

During the sampling of particulate mixtures, samples taken are analyzed for their mass concentration, which generally has non‐zero sample‐to‐sample variance. Bias, variance, and mean squared error (MSE) of a number of variance estimators, derived by Geelhoed, were studied in this article. The Monte Carlo simulation was applied using an observable first‐order Markov Chain with transition probabilities that served as a model for the sample drawing process. Because the bias and variance of a variance estimator could depend on the specific circumstances under which it is applied, Monte Carlo simulation was performed for a wide range of practically relevant scenarios. Using the ‘smallest mean squared error’ as a criterion, an adaptation of an estimator based on a first‐order Taylor linearization of the sample concentration is the best. An estimator based on the Horvitz–Thompson estimator is not practically applicable because of the potentially high MSE for the cases studied. The results indicate that the Poisson estimator leads to a biased estimator for the variance of fundamental sampling error (up to 428% absolute value of relative bias) in case of low levels of grouping and segregation. The uncertainty of the results obtained by the simulations was also addressed and it was found that the results were not significantly affected. The potentials of a recently described other approach are discussed for extending the first‐order Markov Chain described here to account also for higher levels of grouping and segregation. Copyright © 2013 John Wiley & Sons, Ltd.     

Some closed form robust moment‐based estimators for the MEM(1,1)

In this paper, we extend the closed form moment estimator (ordinary MCFE) for the autoregressive conditional duration model given by Lu et al (2016) and propose some closed form robust moment‐based estimators for the multiplicative error model to deal with the additive and innovational outliers. The robustification of the closed form estimator is done by replacing the sample mean and sample autocorrelation with some robust estimators. These estimators are more robust than the quasi‐maximum likelihood estimator (QMLE) often used to estimate this model, and they are easy to implement and do not require the use of any numerical optimization procedure and the choice of initial value. The performance of our proposal in estimating the parameters and forecasting conditional mean μ_t of the MEM(1,1) process is compared with the proposals existing in the literature via Monte Carlo experiments, and the results of these experiments show that our proposal outperforms the ordinary MCFE, QMLE, and least absolute deviation estimator in the presence of outliers in general. Finally, we fit the price durations of IBM stock with the robust closed form estimators and the benchmarks and analyze their performances in estimating model parameters and forecasting the irregularly spaced intraday Value at Risk.     

Maximum integrated likelihood estimator of the interest parameter when the nuisance parameter is location or scale

The problem of estimation of an interest parameter in the presence of a nuisance parameter, which is either location or scale, is studied. Two estimators are considered: the usual maximum likelihood estimator and the estimator based on maximization of the integrated likelihood function. The estimators are compared, asymptotically, with respect to the bias and with respect to the mean squared error. The examples are given.