Best quadratic unbiased estimation of the variance matrix in normal regression

In 1973 Balestra examined the linear model y=XB+u, where u is a normally distributed disturbance vector, with variance matrix Ω. Ω has spectral decomposition \(\sum\limits_{i = 1}^r {\lambda _i M_i } \) , and the matrices M_i are known. Estimation of ω is thus equivalent with estimation of the λ_i. Balestra presented the best quadratic unbiased estimator of λ_i. In the present paper a derivation will be given which is based on a procedure developed by this writer (1980).     

On invariant quadratic unbiased estimation of variance components

The present study deals with three different invarint quadratic unbiased estimators (IQUE) for variance components namely quadratic least squares estimators (QLSE), weighted quadratic least squares estimators (WQLSE) and Mitra type estimators (MTE). The variance and covariances of these three different estimators are presented for unbalanced one-way random model. The relative performances of these estimators are assessed based on different optimality criteria like, D-optimality, T-optimality and M-optimality together with variances of these estimators. As a result, it has been shown that MTE has optimal properties.     

Bayesian estimation of the dispersion matrix of a multivariate normal distribution

The estimation of the dispersion matrix of a multivariate normal distribution with zero mean on the basis of a random sample is discussed from a Bayesian view. An inverted-Wishart distribu- tion for the dispersion is taken, with its defining matrix of intraclass form. Some consistency properties are described. The posterior distribution is found and its mode investigated as a possible estimate in preference to that of maximum likelihood     

On unbiased and improved loss estimation for the mean of a multivariate normal distribution with unknown variance

There is now a sizeable literature dealing with point estimation using Stein-type estimators. As discussed in Rukhin (In: Gupta, S.S., Berger, J.O. (Eds.), Statistical Decision Theory and Related Topics, Vol. IV, Springer, New York, pp. 409–418), instances arise in practice in which an estimation rule is to be accompanied by an estimate of its loss, which is unobservable. In the context of estimating the mean vector of a multi-normal distribution assuming a known population variance, Johnstone (In: Gupta, S.S., Berger, J.O. (Eds.), Statistical Decision Theory and Related Topics, Vol. IV, Springer, New York, pp. 361–379) derived an estimator that dominates the unbiased estimator of the quadratic loss incurred by the James–Stein estimator. By applying the Stein's lemma, this note generalizes Johnstone's (In: Gupta, S.S., Berger, J.O. (Eds.), Statistical Decision Theory and Related Topics, Vol. IV, Springer, New York, pp. 361–379) analysis to the setting of the unknown population variance. Computational evidence is provided about the risk magnitude of loss estimators associated with the James–Stein point estimator and its positive-part version.     

Parameter estimation for multivariate diffusion processes with the time inhomogeneously positive semidefinite diffusion matrix

Statistical inference for the diffusion coefficients of multivariate diffusion processes has been well established in recent years; however, it is not the case for the drift coefficients. Furthermore, most existing estimation methods for the drift coefficients are proposed under the assumption that the diffusion matrix is positive definite and time homogeneous. In this article, we put forward two estimation approaches for estimating the drift coefficients of the multivariate diffusion models with the time inhomogeneously positive semidefinite diffusion matrix. They are maximum likelihood estimation methods based on both the martingale representation theorem and conditional characteristic functions and the generalized method of moments based on conditional characteristic functions, respectively. Consistency and asymptotic normality of the generalized method of moments estimation are also proved in this article. Simulation results demonstrate that these methods work well.     

On estimation of the scale matrix of the multivariate f distribution

Let F _p×phave a multivariate F distribution with a scale p×p matrix Δ and degrees of freedom k₁ and k₂ such that k_i - p - 1 > 0, i = 1,2. The estimation of Δ under entropy and squared error loss functions are considered. In both cases a new class of orthogonally invariant estimators are obtained which dominate the best unbiased estimator.     

Robust estimation in the multivariate normal model with variance components

Fisher consistent and Fréchet differentiable statistical functionals have been already used by Bednarski and Zontek [Robust estimation of parameters in a mixed unbalanced model. Ann Statist. 1996;24(4):1493–1510] to get a robust estimator of parameters in a two-way crossed classification mixed model. This way of robust estimation appears also in the variance components model with a commutative covariance matrix [Zmy?lony, Zontek. Robust M-estimator of parameters in variance components model. Discuss Math Probab Stat. 2002;22:61–71]. In this paper it is shown that a modification of this method does not involve any assumptions about commutation of covariance matrix. The theoretical results have been completed with computer simulation studies. Robustness of considered estimator and possibility of approximation of the estimator's distribution with some multivariate normal distribution for both model and contaminated data have been confirmed there.     

Best linear unbiased estimation of location and scale parameters of the log-logistic distribution

If an assumption, such as homoscedasticity, or some other aspect of an inference problem, such as the number of cases, is altered, our conclusions may change and different parts of the conclusions can be affected in different ways. Most diagnostic procedures measure the influence on one particular aspect of the conclusion - such as model fit or change in parameter estimates. The effect on all aspects of the conclusions can be described by the difference in two log likelihood functions and when the log likelihood functions come from an exponential family or are quasi-likelihoods, this difference can be factored into three terms: one depending only on the alteration, another depending only on the aspects of the conclusions to be considered, and a third term depending on both. The third term is interesting because it shows which aspects of the conclusions are relatively insensitive even to large alterations.     

Distribution theory of quadratic forms for matrix multivariate elliptical distribution

This paper proposes the density and characteristic functions of a general matrix quadratic form X^(?)AX${\mathbf{X}}^{(?)}\mathbf{AX}$, when A=A^(?)$\mathbf{A}={\mathbf{A}}^{(?)}$ is a positive semidefinite matrix, X$\mathbf{X}$ has a matrix multivariate elliptical distribution and X^(?)${\mathbf{X}}^{(?)}$ denotes the usual conjugate transpose of X$\mathbf{X}$. These results are obtained for real normed division algebras. With particular cases we obtained the density and characteristic functions of matrix quadratic forms for matrix multivariate normal, Pearson type VII, t and Cauchy distributions.     

THE non-null distribution of the likelihood ratio criterion for testing the hypothesis that the covariance matrix is diagonal

This article deals with the exact non-null distribution of the likelihood ratio criterion for testing the hypothesis that the covariance matrix in a multinormal distribution is diagonal. The exact non-null moments as well as the exact non-null distribution are derived. The distribution is also expressed in computable form with the help of inverse Mellin transform and calculus of residues. The results obtained in this article are useful in studying the power of testing several correlation coefficients simultaneously.     

On the distribution of quadratic forms in normal random variables

This paper provides necessary and sufficient conditions for a quadratic form in singular normal random variables to be distributed as a given linear combination of independent noncentral chi-square variables. Using this result, an extension of Cochran's theorem to quadratic forms of noncentral chi-square variables is derived.     

Inadmissibility of the uniformly minimum variance unbiased estimator of the inverse gaussian variance

The uniformly minimum variance unbiased estimator (UMVUE) of the variance of the inverse Gaussian distribution is shown to be inadmissible in terms of the mean squared error, and a dominating estimator is given. A dominating estimator to the maximum likelihood estimator (MLE) of the variance and estimators dominating the MLE's and the UMVUE's of other parameters are also given.     

Minimum variance unbiased estimation in a modified power series distribution and some of its applications

In this paper we study the minimum variance unbiased estimation in the modified power series distribution introduced by the author (1974a). Necessary and sufficient conditions for the existence of minimum variance unbiased estimate (MVUE) of the parameter based on sufficient statistics are obtained. These results are, then, applied to obtain MVUE of θ^r (r ≥ 1) for the generalized negative binomial and the decapitated generalized negative binomial distributions (Jain and Consul, 1971). Similar estimates are obtained for the generalized Poisson (Consul and Jain, 1973a) and the generalized logarithmic series distributions (Jain and Gupta, 1973). Several of the well-known results follow trivially from the results obtained here.     

On geometry of the set of admissible invariant quadratic estimators in balanced two variance components model

Two variance components model for which each invariant quadratic admissible estimator of a linear function of variance components (under quadratic loss function) is a linear combination of two quadratic forms,Z ₁,Z ₂, say, is considered. A setD={(d ₁,d ₂)′:d ₁ Z ₁+d ₂ Z ₂ is admissible} is described by giving formulae on the boundary ofD. Different forms of the setD are presented on figures.     

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.     

Moments and quadratic forms of matrix variate skew normal distributions

ABSTRACT

In 2007, Domínguez-Molina et al. obtained the moment generating function (mgf) of the matrix variate closed skew normal distribution. In this paper, we use their mgf to obtain the first two moments and some additional properties of quadratic forms for the matrix variate skew normal distributions. The quadratic forms are particularly interesting because they are essentially correlation tests that introduce a new type of orthogonality condition.