多元正态分布熵的Stein型和Brester-Zidek型估计

基于一些随机样本,在Linex损失下估计期望及方差阵都未知的多元正态分布的熵。在仅依赖于|S|的估计类中,熵的最优仿射同变估计δ_c*是可容许估计,但在一些范围更大的估计类中,δ_c*是不可容许估计。文章首先用Stein型估计δ_?ST去改进δ_c*,但Stein型估计不是光滑的,然后用具有光滑性的Brester-Zidek型估计去改进δ_c*,进一步研究知Brester-Zidek估计是可容许估计,也是Bayes估计。     

Empirical bayes shrinkage estimation of reliability in the exponential distribution

In this paper we propose two empirical Bayes shrinkage estimators for the reliability of the exponential distribution and study their properties. Under the uniform prior distribution and the inverted gamma prior distribution these estimators are developed and compared with a preliminary test estimator and with a shrinkage testimator in terms of mean squared error. The proposed empirical Bayes shrinkage estimator under the inverted gamma prior distribution is shown to be preferable to the preliminary test estimator and the shrinkage testimator when the prior value of mean life is clsoe to the true mean life.     

The admissible minimax estimator in Gauss–Markov model under a balanced loss function

In this paper, we investigate the admissible minimax estimator (AME) of regression coefficient in Gauss–Markov model under a balanced loss function. In the class of homogeneous linear estimators, we obtain the AME under two occasions, respectively. We also prove that the AME is a shrinkage estimator of the best linear unbiased estimator (BLUE). Furthermore, we prove that the AME dominates the BLUE under certain conditions.     

On the efficiency of a testimator for the weibull shape parameter

In estimating the shape parameter of a two-parameter Weibull distribution from a failure-censored sample, a recently popular procedure is to employ a testimator which is a shrinkage estimator based on a preliminary hypothesis test for a guessed value of the parameter. Such an adaptive testimator is a linear compound of the guessed value and a statistic. A new compounding coefficient is numerically shown to yield higher efficiency in many situations compared to some of the existing ones.     

A formal bayes multiple shrinkage estimator

A new minimax multiple shrinkage estimator is constructed. This estimator which can adaptively shrink towards many subspace targets, is formal Bayes with respect to a mixture of harmonic priors. Unbiased estimates of risk and simulation results suggest that the risk properties of this estimator are very similar to those of the multiple shrinkage Stein estimator proposed by George (1986a). A special case is seen to be admissible.     

Some improved estimators for a measures of dispersion of an inverse gaussian distribution

In this paper some improved estimators for the measure of dispersion of an inverse Gaussian distribution have been obtained. If some guessed value of λ is available in the form of a point esitmate λ₀ the shrikage technique has been applied and an estimator has been proposed which has smaller mean squared error than the usual estimator. Since the shrinkage estimator has better performance if the guessed value is in the vicinity of the true value, a shrinkage testimator has also been proposed and compared with the usual estimator.     

A note on finite population prediction under asymetric loss functions

The purpose of the present note is to derive optimal population total predictors relative to the Linex (Zellner, 1986) loss function under some well known superpopulation models. The risk function and Bayes risk are derived and compared with those of usual predictors. Minimax and and admissibility properties of some of the derived predictors are also investigated.     

Bayesian shrinkage estimation based on Rayleigh type-II censored data

In this paper, we construct a Bayes shrinkage estimator for the Rayleigh scale parameter based on censored data under the squared log error loss function. Risk-unbiased estimator is derived and its risk is computed. A Bayes shrinkage estimator is obtained when a prior point guess value is available for the scale parameter. Risk-bias of the Bayes shrinkage estimator is considered. A comparison between the proposed Bayes shrinkage estimator and the risk-unbiased estimator is provided using calculation of the relative efficiency. A numerical example is presented for illustrative and comparative purposes.     

Whither jackknifing in stein-rule estimation

In multi-parameter ( multivariate ) estimation, the Stein rule provides minimax and admissible estimators , compromising generally on their unbiasedness. On the other hand, the primary aim of jack-knifing is to reduce the bias of an estimator ( without necessarily compromising on its efficacy ), and, at the same time, jackknifing provides an estimator of the sampling variance of the estimator as well. In shrinkage estimation ( where minimization of a suitably defined risk function is the basic goal ), one may wonder how far the bias-reduction objective of jackknifing incorporates the dual objective of minimaxity ( or admissibility ) and estimating the risk of the estimator ? A critical appraisal of this basic role of jackknifing in shrinkage estimation is made here. Restricted, semi-restricted and the usual versions of jackknifed shrinkage estimates are considered and their performance characteristics are studied . It is shown that for Pitman-type ( local ) alternatives, usually, jackkntfing fails to provide a consistent estimator of the ( asymptotic ) risk of the shrinkage estimator, and a degenerate asymptotic situation arises for the usual fixed alternative case.     

Sufficient conditions for the admissibility under the LINEX loss function in non-regular case

Consider an estimation problem under the LINEX loss function in one-parameter non-regular distributions where the endpoint of the support depends on an unknown parameter. The purpose of this paper is to give sufficient conditions for a generalized Bayes estimator of a parametric function to be admissible. Also, it is shown that the main result in this paper is an extension of the quadratic loss case. Some examples are given.     

Shrinkage Testimators for the Shape Parameter of Pareto Distribution Using LINEX Loss Function

In this article, shrinkage testimators for the shape parameter of a Pareto distribution are considered, when its prior guess value is available. The choices of shrinkage factor are also suggested. The proposed testimators are compared with the minimum risk estimator among the class of unbiased estimators with the LINEX loss function.     

Second-order asymptotic comparison of estimators under φ-divergence loss

Our main concern is on the second-order asymptotic optimality problem of estimators. The φ-divergence loss is used as a criterion for evaluating the performance of estimators. In the comparison problem of any two estimators, the condition that one estimator dominates another estimator under the φ-divergence risk is given by evaluating the second-order term in the difference between the risks. As a result, it is proved that the condition is characterized by a peculiar value of the φ-divergence loss, which is called the divergence-loss coefficient. Furthermore, it is shown that the comparison based on the φ-divergence loss does not correspond with that based on any standard loss functions including the mean squared error, the absolute loss and the 0-1 loss. In addition, a necessary and sufficient condition for an estimator to be second-order admissible is derived.     

A Note on the Estimation of Binomial Probabilities

I am concerned with the admissibility under quadratic loss of certain estimators of binomial probabilities. The minimum variance unbiased estimator is shown to be admissible for Pr(X = 0) and Pr(X = n), but it is inadmissible for Pr(X = k), where 0 < k < n. An example is given of an admissible maximum likelihood estimator (MLE). It is conjectured that the MLE is always admissible.     

Shrinkage estimator of regression model under asymmetric loss

This article investigates the performance of the shrinkage estimator (SE) of the parameters of a simple linear regression model under the LINEX loss criterion. The risk function of the estimator under the asymmetric LINEX loss is derived and analyzed. The moment-generating functions and the first two moments of the estimators are also obtained. The risks of the SE have been compared numerically with that of pre-test and least-square estimators (LSEs) under the LINEX loss criterion. The numerical comparison reveals that under certain conditions the LSE is inadmissible, and the SE is the best among the three estimators.     

Estimation of the MSE matrix of the stein estimator

Uniformly minimum-variance unbiased (UMVU) estimators of the total risk and the mean-squared-error (MSE) matrix of the Stein estimator for the multivariate normal mean with unknown covariance matrix are proposed. The estimated MSE matrix is helpful in identifying the components which contribute most to the total risk. It also contains information about the performance of the shrinkage estimator with respect to other quadratic loss functions.     

Wavelet threshold based on Stein's unbiased risk estimators of restricted location parameter in multivariate normal

In this paper, the problem of estimating the mean vector under non-negative constraints on location vector of the multivariate normal distribution is investigated. The value of the wavelet threshold based on Stein''s unbiased risk estimators is calculated for the shrinkage estimator in restricted parameter space. We suppose that covariance matrix is unknown and we find the dominant class of shrinkage estimators under Balance loss function. The performance evaluation of the proposed class of estimators is checked through a simulation study by using risk and average mean square error values.     

The Bayes rule of the parameter in (0,1) under Zhang’s loss function with an application to the beta-binomial model

Abstract

For the restricted parameter space (0,1), we propose Zhang’s loss function which satisfies all the 7 properties for a good loss function on (0,1). We then calculate the Bayes rule (estimator), the posterior expectation, the integrated risk, and the Bayes risk of the parameter in (0,1) under Zhang’s loss function. We also calculate the usual Bayes estimator under the squared error loss function, and the Bayes estimator has been proved to underestimate the Bayes estimator under Zhang’s loss function. Finally, the numerical simulations and a real data example of some monthly magazine exposure data exemplify our theoretical studies of two size relationships about the Bayes estimators and the Posterior Expected Zhang’s Losses (PEZLs).     

Risk performance of a pre-test estimator for normal variance with the Stein-variance estimator under the LINEX loss function

In this paper, we derive the exact formula of the risk function of a pre-test estimator for normal variance with the Stein-variance (PTSV) estimator when the asymmetric LINEX loss function is used. Fixing the critical value of the pre-test to unity which is a suggested critical value in some sense, we examine numerically the risk performance of the PTSV estimator based on the risk function derived. Our numerical results show that although the PTSV estimator does not dominate the usual variance estimator when under-estimation is more severe than over-estimation, the PTSV estimator dominates the usual variance estimator when over-estimation is more severe. It is also shown that the dominance of the PTSV estimator over the original Stein-variance estimator is robust to the extension from the quadratic loss function to the LINEX loss function.     

Bayesian Estimation of Bivariate Exponential Distributions Based on Linex and Quadratic Loss Functions: A Survival Approach with Censored Samples

In this article, four bivariate exponential (BVE) distributions with subject to right censoring samples are presented. Bayesian estimates of the parameters of BVE are obtained through Linex and quadratic loss functions. Gamma prior distribution has been suggested to reforming the posterior function. The estimations and standard errors of parameters have also been obtained through simulation method. Markov chain Monte Carlo (MCMC) method is employed for the case of Block-Buse bivariate distribution because there was no closed form for estimator criteria. Simulation studies have been conducted to show that the computation parts can be implemented easily and comparing the estimated values due to two methods and with the true values as well.     

Characterization of Admissible Linear Estimators in Multivariate Linear Model with Respect to Inequality Constraints under Matrix Loss Function

In this article, we study the characterization of admissible linear estimators in a multivariate linear model with inequality constraint, under a matrix loss function. In the homogeneous class, we present several equivalent, necessary and sufficient conditions for a linear estimator of estimable functions to be admissible. In the inhomogeneous class, we find that the necessary and sufficient conditions depend on the rank of the matrix in the constraint. When the rank is greater than one, the necessary and sufficient conditions are obtained. When the rank is equal to one, we have necessary conditions and sufficient conditions separately. We also obtain the necessary and sufficient conditions for a linear estimator of inestimable function to be admissible in both classes.