On the estimation of the mean vector of a multivariate normal distribution under symmetry

The problem of estimation of the mean vector of a multivariate normal distribution with unknown covariance matrix, under uncertain prior information (UPI) that the component mean vectors are equal, is considered. The shrinkage preliminary test maximum likelihood estimator (SPTMLE) for the parameter vector is proposed. The risk and covariance matrix of the proposed estimato are derived and parameter range in which SPTMLE dominates the usual preliminary test maximum likelihood estimator (PTMLE) is investigated. It is shown that the proposed estimator provides a wider range than the usual premilinary test estimator in which it dominates the classical estimator. Further, the SPTMLE has more appropriate size for the preliminary test than the PTMLE.     

A family of admissible minimax estimators of the mean of a multivariate,normal distribution

Let X has a p-dimensional normal distribution with mean vector θ and identity covariance matrix I. In a compound decision problem consisting of squared-error estimation of θ, Strawderman (1971) placed a Beta (α, 1) prior distribution on a normal class of priors to produce a family of Bayes minimax estimators. We propose an incomplete Gamma(α, β) prior distribution on the same normal class of priors to produce a larger family of Bayes minimax estimators. We present the results of a Monte Carlo study to demonstrate the reduced risk of our estimators in comparison with the Strawderman estimators when θ is away from the zero vector.     

The exact mean squared error risks of preliminary test type estimators for the multivariate normal mean

The exact mean squared error risks of the preliminary test estimtor and the Sclove modified Stein rule estimator (Sclove, Morris and Radhakrishnan, 1972) for the multivariate normal mean are computed and their risks are compared with the risks of Stein estimators.     

Empirical bayes estimation of the mean in a multivariate normal distribution

We consider the problem of estimating the mean vector of a multivariate normal distribution under a variety of assumed structures among the parameters of the sampling and prior distributions. We adopt a pragmatic approach. We adopt distributional familites, assess hyperparmeters, and adopt patterned mean and coveariance structures when it is relatively simple to do so; alternatively, we use the sample data to estimate hyperparameters of prior distributions when assessment is a formidable task; such as the task of assessing parameters of multidimensional problems. James-Stein-like estimators are found to result. In some cases, we've been abl to show that the estimators proposed uniformly dominate the MLE's when measured with respect to quadratic loss functions.     

Truncated linear estimation of a bounded multivariate normal mean

We consider the problem of estimating the mean θ$\theta$ of an _Np(θ,_Ip)${\mathrm{N}}_{\mathrm{p}}(\theta ,I_p)$ distribution with squared error loss ∥δ−θ∥²$\parallel \delta -\theta {\parallel }^2$ and under the constraint ∥θ∥≤m$\parallel \theta \parallel \le m$, for some constant m>0$m>0$. Using Stein's identity to obtain unbiased estimates of risk, Karlin's sign change arguments, and conditional risk analysis, we compare the risk performance of truncated linear estimators with that of the maximum likelihood estimator _δmle${\delta }_{\mathrm{mle}}$. We obtain for fixed (m,p)$(m,p)$ sufficient conditions for dominance. An asymptotic framework is developed, where we demonstrate that the truncated linear minimax estimator dominates _δmle${\delta }_{\mathrm{mle}}$, and where we obtain simple and accurate measures of relative improvement in risk. Numerical evaluations illustrate the effectiveness of the asymptotic framework for approximating the risks for moderate or large values of p.     

Improved minimax estimation of a normal precision matrix

Let S_{p × p} have a Wishart distribution with parameter matrix Σ and n degrees of freedom. We consider here the problem of estimating the precision matrix Σ^?1 under the loss functions L₁(σ) tr (σ) - log |σ| and L₂(σ) = tr (σ). James-Stein-type estimators have been derived for an arbitrary p. We also obtain an orthogonal invariant and a diagonal invariant minimax estimator under both loss functions. A Monte-Carlo simulation study indicates that the risk improvement of the orthogonal invariant estimators over the James-Stein type estimators, the Haff (1979) estimator, and the “testimator” given by Sinha and Ghosh (1987) is substantial.     

On estimating a function of the mean of a multivariate normal distribution

The unique minimum variance of unbiased estimator is obtained for analysis functions of the mean of a multivariate normal distribution with either unknown covariance matrix or with covariance matrix of the form σ²v where σ² is unknown.     

MSE performance and minimax regret significance points for a HPT estimator under a multivariate t error distribution

In this article, assuming that the error terms follow a multivariate t distribution,we derive the exact formulae forthe moments of the heterogeneous preliminary test (HPT) estimator proposed by Xu (2012b Xu, H. (2012b). MSE performance and minimax regret significance points for a HPT estimator when each individual regression coefficient is estimated. Commun. Stat. Theory Methods 42:2152–2164.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). We also execute the numerical evaluation to investigate the mean squared error (MSE) performance of the HPT estimator and compare it with those of the feasible ridge regression (FRR) estimator and the usual ordinary least squared (OLS) estimator. Further, we derive the optimal critical values of the preliminary F test for the HPT estimator, using the minimax regret function proposed by Sawa and Hiromatsu (1973 Sawa, T., Hiromatsu, T. (1973). Minimax regret significance points for a preliminary test in regression analysis. Econometrica 41:1093–1101.[Crossref], [Web of Science ®] , [Google Scholar]). Our results show that (1) the optimal significance level (α*) increases as the degrees of freedom of multivariate t distribution (ν₀) increases; (2) when ν₀ ? 10, the value of α* is close to that in the normal error case.     

On the minimum mean squared error estimators in a regression model

In this paper we analyze the properties of two estimators oroposed by Farebrother (1975) for linear regression models.     

On the efficiency of affine minimax rules in estimating a bounded multivariate normal mean

Problems involving bounded parameter spaces, for example T-minimax and minimax esyimation of bounded parameters, have received much attention in recent years. The existing literature is rich. In this paper we consider T-minimax estimation of a multivariate bounded normal mean by affine rules, and discuss the loss of efficiency due to the use of such rules instead of optimal, unrestricted rules. We also investigate the behavior of 'probability restricted' affine rules, i.e., rules that have a guaranteed large probability of being in the bounded parameter space of the problem.     

Unbiased estimation of the common mean of a multivariate normal distribution

The problem of unbiased estimation of the common mean of a multivariate normal population is considered. An unbiased estimator is proposed which has a smaller variance than the usual estimator over a large part of the parameter space.     

Equivariant estimation of a normal mean using a normal concomitant variable for covariance adjustment

Equivariant point estimators of one component of a bivariate normal mean vector are considered when the second component is known. Equivariant point estimators are characterized and compared in terms of their risk functions with respect to a normalized squared-error loss function. Specific point estimators that dominate the usual estimator when the squared correlation coefficient is sufficiently large are provided.     

Parametric estimation under a class of multivariate distributions

In this paper the parameters of some members of a class of multivariate distributions, which was constructed by AL-Hussaini and Ateya (2003), are estimated by using the maximum likelihood and Bayes methods.     

Improved pretest nonparametric estimation in a multivariate regression model

Shrinkage pretest nonparametric estimation of the location parameter vector in a multivariate regression model is considered when nonsample information (NSI) about the regression parameters is available. By using the quadratic risk criterion, the dominance of the pretest estimators over the usual estimators has been investigated. We demonstrate analytically and computationally that the proposed improved pretest estimator establishes a wider dominance range for the parameter under consideration than that of the usual pretest estimator in which it is superior over the unrestricted estimator.     

Minimax estimation of normal precisions via expansion estimators

In this paper, the simultaneous estimation of the precision parameters of k normal distributions is considered under the squared loss function in a decision-theoretic framework. Several classes of minimax estimators are derived by using the chi-square identity, and the generalized Bayes minimax estimators are developed out of the classes. It is also shown that the improvement on the unbiased estimators is characterized by the superharmonic function. This corresponds to Stein's [1981. Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9, 1135–1151] result in simultaneous estimation of normal means.     

Estimation of eigenvalues of the scale matrix of the multivariate f distribution

Let F have the multivariate F distribution with a scale matrix Δ. In this paper, the problem of estimating the eigenvalues of the scale matrix Δ is considered. New class of estimators are obtained which dominate the best linear estimator of the form cF. Simulation study is also carried out to compare the performance of these estimators.     

Further remarks on estimating the parameter of a noncentral chi-square distribution

Perlman and Rasmussen (1975) have found estimators of the non-centrality parameter of a noncentral chi-square distribution which have lower mean square error than the maximum likelihood estimator. This paper studies some extensions of their estimators and some related problems.