The Moore-Penrose inverse of a singular Wishart matrix is studied. When the scale matrix equals the identity matrix the mean and dispersion matrices of the Moore-Penrose inverse are known. When the scale matrix has an arbitrary structure no exact results are available. We complement the existing literature by deriving upper and lower bounds for the expectation and an upper bound for the dispersion of the Moore-Penrose inverse. The results show that the bounds become large when the number of rows (columns) of the Wishart  matrix are close to the degrees of freedom of the distribution.

Bilinear models with three types of effects are considered: fixed effects, random effects and latent variable effects. Explicit estimators are proposed.

A bilinear regression model with rank restrictions imposed on the mean-parameter matrix and on the dispersion matrix is studied. Maximum likelihood inspired estimates are derived. The approach generalizes classical reduced rank regression analysis and principal component analysis. It is shown via a simulation study and a real example that even for small dimensions the method works as well as reduced rank regression analysis whereas the approach in this article also can be used when the dimension is large.

The totally ordered conditional independence (TOCI) model N(K) is defined to be the set of all normal distributions on R^I such that for each adjacent pair (K_i, K_i+1)  K, the components of a multivariate normal vector x  R^I, indexed by the set difference { K_i+1 \ K_i } are mutually conditionally independent given the variables indexed by K_i. Here K = {K₁  …  K_q } is a totally ordered set of subsets of a finite index set I. It is shown that TOCI models constitute a proper subset of lattice conditional independence (LCI) models. It follows that like LCI models, for the TOCI models the likelihood function and parameter space can be factored into the products of conditional likelihood functions and disjoint parameter spaces, respectively, where each conditional likelihood function corresponds to an ordinary multivariate normal regression model. 

The joint distribution of standardized traces of $\frac{1}{n}XX'$ and of $\Big(\frac{1}{n}XX'\Big)^2$, where the matrix $X:p\times n$ follows a matrix normal distribution is proved asymptotically to be multivariate normal under condition $\frac{{n}}{p}\overset{n,p\rightarrow\infty}{\rightarrow}c>0$. Proof relies on calculations of asymptotic moments and cumulants obtained using a recursive formula derived in Pielaszkiewicz et al. (2015). The covariance matrix of the underlying vector is explicitely given as a function of $n$ and $p$.

In this article small area estimation with multivariate data that follow a monotonic missing sample pattern is addressed. Random effects growth curve models with covariates are formulated. A likelihood based approach is proposed for estimation of the unknown parameters. Moreover, the prediction of random effects and predicted small area means are also discussed.

In this paper, we give a general recursive formula for , where  denotes a real Wishart matrix. Formulas for fixed n, p  are presented as well as asymptotic versions when i.e. when the so called Kolmogorov condition holds. Finally, we show  application of the asymptotic moment relation when deriving moments for the Marchenko-Pastur distribution (free Poisson law). A numerical  illustration using implementation of the main result is also performed.

The general multivariate analysis of variance model has been extensively studied in the statistical literature and successfully applied in many different fields for analyzing longitudinal data. In this article, we consider the extension of this model having two sets of regressors constituting a growth curve portion and a multivariate analysis of variance portion, respectively. Nowadays, the data collected in empirical studies have relatively complex structures though often demanding a parsimonious modeling. This can be achieved for example through imposing rank constraints on the regression coefficient matrices. The reduced rank regression structure also provides a theoretical interpretation in terms of latent variables. We derive likelihood based estimators for the mean parameters and covariance matrix in this type of models. A numerical example is provided to illustrate the obtained results.

In this article small area estimation with multivariate data that follow a monotonic missing sample pattern is addressed. Random effects growth curve models with covariates are formulated. A likelihood based approach is proposed for estimation of the unknown  parameters. Moreover, the prediction of random effects and predicted small area means are also discussed.

For a normal population likelihood ratio test, Rao’s score test and Wald’s score test for testing covariance structures are compared in the situation when the number of variables and the sample size are growing. Expressions of all three test statistics are derived under the general null-hypothesisΣ=Σ0, using matrix derivative techniques. The special casesΣ=γIpandΣ=Ipare also under consideration. The tests are compared in a simulation experiment with sample sizes varying from 100 to 5000 and dimensionalities from 2 to 50. When the number of variables is growing Rao’s score test behaves most adequately.