Comparison of estimators of a dispersion matrix under generalized pitman nearness criterion

Some equivariant estimators of the dispersion matrix of a multivariate normal population are compared using the generalized Pitman nearness criterion based on the entropy loss function. It is shown that, under the group of lower triangular transformations, a best equivariant estimator does not exist. Existence of best estimators in certain subclasses are discussed and the performances of two commonly used estimators are compared. Some properties of central chi-square distributions are obtained and used to derive the main results.     

Direct methods for generating extreme characteristic roots of certain random matrices

Computer generation of extreme characteristic roots of random matrices is considered. The usual approach in Monte-Carlo applications is to randomly generate the matrix and then compute desired characteristic roots. There are, however, theoretical results about the distribution of individual characteristic roots which might be used as a basis for computing algorithms. This alternative approach is considered for the Wishart and Beta matrices.     

Characterization of the Wishart Distribution on Homogeneous Cones in the Bobecka and Wesolowski Way

The aim of this article is to extend the characterization of the ordinary Wishart on symmetric matrices as given by Bobecka and Weso?owski (2002 Bobecka , K. , Weso?owski , J. ( 2002 ). The Lukacs–Olkin–Rubin theorem without invariance of the “quotient” . Studia Math. 152 : 147 – 160 . [Google Scholar]) to the Wishart distribution on homogeneous cones.     

Subjective Bayesian Analysis of the Elliptical Model

For the multivariate elliptical model subjective Bayesian estimators of the location vector and some functions of the characteristic matrix with the normal-inverse Wishart and the normal-Wishart as prior, respectively, are derived. Fang and Li (1999 Fang, K.T., Li, R.Z. (1999). Bayesian statistical inference on elliptical matrix distributions. J. Multivariate Anal. 70: 66–85.[Crossref], [Web of Science ®] , [Google Scholar]) considered the elliptical model for Bayesian analysis for an objective prior structure. In addition, the newly developed results are applied to the multivariate normal- and t-distribution. A performance study is done to evaluate the normal-gamma and normal-inverse gamma distributions as suitable priors. A practical application for the posterior distributions of the multivariate t-distribution is included by means of Gibbs sampling and a Metropolis-Hastings algorithm.     

Improved Tolerance Factors for Multivariate Normal Distributions

The Q_os and Qm are two leading estimators of the probability of misclassification which are based on the asymptotic expansion of the the expected value of the Error Rate, P_i. The estimators are, however, not suitable for estimating the Error rates for certain ranges of the parameters p , n₁, n₂ and ß.We investigate the regions in which they produce unacceptable estimates , and show that the Q_os is, in general, better than the Q_m in producing acceptable estimates     

Scheffés mixed model for multivariate repeated measures:a relative efficiency evaluation

Scheffé’s mixed model, generalized for application to multivariate repeated measures, is known as the multivariate mixed model (MMM). The primary advantages the MMM are (1) the minimum sample size required to conduct an analysis is smaller than for competing procedures and (2) for certain covariance structures, the MMM analysis is more powerful than its competitors. The primary disadvantage is that the MMM makes a very restrictive covariance assumption; namely multivariate sphericity. This paper shows, first, that even minor departures from multivariate sphericity inflate the size of MMM based tests. Accordingly, MMM analyses, as computed in release 4.0 of SPSS MANOVA (SPSS Inc., 1990), can not be recommended unless it is known that multivariate sphericity is satisfied. Second, it is shown that a new Box-type (Box, 1954) Δ-corrected MMM test adequately controls test size unless departure from multivariate sphericity is severe or the covariance matrix departs substantially from a multiplicative-Kronecker structure. Third, power functions of adjusted MMM tests for selected covariance and noncentrality structures are compared to those of doubly multivariate methods that do not require multivariate sphericity. Based on relative efficiency evaluations, the adjusted MMM analyses described in this paper can be recommended only when sample sizes are very small or there is reason to believe that multivariate sphericity is nearly satisfied. Neither the e-adjusted analysis suggested in the SPSS MANOVA output (release 4.0) nor the adjusted analysis suggested by Boik (1988) can be recommended at all.     

On approximating a linear combination of central wishart matrices with positive coefficients

Consider linear combinations of central Wishart matrices with positive coefficients. By equating the expected values and the generalized variance of variances and covariances, we propose approximating the distributions of those linear combinations by central Wishart distributions. Some Monte Carlo results are given to demonstrate the closeness of this approximation.     

Linear. empirical bayes estimation in the case of the wishart distribution

We consider independent pairs (X₁,∑₁), (X₂,∑₂),…,(X_n∑_n), where each Si is distributed according to some unknown density function g(∑) and, given ∑_i = ∑, X has a conditional density function g(x|∑) of the Wishart type. In each pair, the first component is observable but the second is not. After the (n + l)-th observation X_n+i is obtained, the objective is to estimate ∑ _n+i corresponding to X_n+i. This estimator is called an empirical Bayes (EB) estimator of ∑. We construct a linear EB estimator of ∑ and examine its precision.     

Euler characteristic heuristic for approximating the distribution of the largest eigenvalue of an orthogonally invariant random matrix

The Euler characteristic heuristic has been proposed as a method for approximating the upper tail probability of the maximum of a random field with smooth sample path. When the random field is Gaussian, this method is proved to be valid in the sense that the relative approximation error is exponentially smaller. However, very little is known about the validity of the method when the random field is non-Gaussian. In this paper, as a milestone to developing the general theory about the validity of the Euler characteristic heuristic, we examine the Euler characteristic heuristic for approximating the distribution of the largest eigenvalue of an orthogonally invariant non-Gaussian random matrix. In this particular example, if the probability density function of the random matrix converges to zero sufficiently fast at the boundary of its support, the approximation error of the Euler characteristic heuristic is proved to be small and the approximation is valid. Moreover, for several standard orthogonally invariant random matrices, the approximation formula for the distribution of the largest eigenvalue and its asymptotic error are obtained explicitly. Our formulas are practical enough for the purpose of numerical calculations.     

Hyper Inverse Wishart Distribution for Non-decomposable Graphs and its Application to Bayesian Inference for Gaussian Graphical Models

While conjugate Bayesian inference in decomposable Gaussian graphical models is largely solved, the non-decomposable case still poses difficulties concerned with the specification of suitable priors and the evaluation of normalizing constants. In this paper we derive the DY-conjugate prior ( Diaconis & Ylvisaker, 1979 ) for non-decomposable models and show that it can be regarded as a generalization to an arbitrary graph G of the hyper inverse Wishart distribution ( Dawid & Lauritzen, 1993 ). In particular, if G is an incomplete prime graph it constitutes a non-trivial generalization of the inverse Wishart distribution. Inference based on marginal likelihood requires the evaluation of a normalizing constant and we propose an importance sampling algorithm for its computation. Examples of structural learning involving non-decomposable models are given. In order to deal efficiently with the set of all positive definite matrices with non-decomposable zero-pattern we introduce the operation of triangular completion of an incomplete triangular matrix. Such a device turns out to be extremely useful both in the proof of theoretical results and in the implementation of the Monte Carlo procedure.