We consider the estimation of Σ of the p-dimensional normal distribution Np(0,Σ) under the restriction where the eigenvalues of Σ have an upper or lower bound. From a decision-theoretic point of view, we evaluate the performance of the REML (restricted maximum likelihood estimator) with Stein′s loss function and propose another estimator that dominates the REML.
In this article we evaluate P (X < Ω) and P (X > Ω), when X has a matrix variate Kummer-beta or Kummer-gamma distribution and Ω is a non-random positive definite matrix. These
results are obtained in series involving invariant polynomials of matrix arguments.
For a multivariate normally distributed n × p random matrix Y with mean μ and covariance ΣY = V1⊗Σ1 + V2⊗Σ2, necessary and sufficient conditions, under which Y′WY follows a Wishart distribution, are obtained, where W is a symmetric matrix, V1 and V2 are known nonnegative definite matrices, and Σ1 and Σ2 are unknown nonnegative definite parameter matrices. Several examples are given to illustrate our main results.
It was Azzalini (1985) who introduced the univariate skew
normal distribution family with a shape parameter , and
then extended skew normal distribution family by adding an
additional shape parameters . Azzalini and Dalla Valle (1996)
extended the results to the multivariate case. Basic properties for the
univariate and multivariate cases were summarized by Azzalini
(2005). Chen and Gupta (2005) considered the matrix variate skew
normal distribution family and proposed the moment generating
function and demonstrated that the distribution of the quadratic form of the skew normal matrix variate
follows a Wishart distribution. Their results were generalized by Harrar and Gupta (2008). In this paper, we generalize the univariate extended skew normal
distribution family to the matrix variate case. The moment
generating function, the distribution of the quadratic form and the
linear form, and the marginal and conditional distributions of this
family are studied.
BACKGROUND. Mucocele is a benign expansile cyst-like lesion seen in the paranasal sinuses, which has a tendency to expand and erode the surrounding sinus walls. These mucoceles develop as a result of obstruction of the sinus ostium and superadded infection. As such, the frontoethmoidal area is the most common site, followed by the maxillary sinus and the sphenoid sinus. Rarely, mucocele may also develop in abnormally aerated bones, such as middle turbinate (concha bullosa), clinoid process and pterygoid process.
CASE REPORT. We report two infrequent cases of mucocele of concha bullosa, clinically presenting as the cause of nasal obstruction, which were completely removed endoscopically, and patients were symptomatically relieved.
CONCLUSION. Mucocele of the middle turbinate represents a diagnostic challenge to surgeons both in terms of symptoms and risk of complications. Therefore, the condition should be considered as a possible cause of progressive nasal obstruction by otolaryngologists, and careful examination of the nasal cavity is necessary to determine the existence of this rare condition.