Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

About the matrix variate problem involved in the distribution of $\mathbf{E}^{-1}\mathbf{H}$

José A. Díaz-García, Francisco J. Caro-Lopera

Abstract

This work studies the distribution of the nonsymmetric matrix $\mathbf{E}^{-1}\mathbf{H}$. This random product is of fundamental interest under the general multivariate linear hypothesis setting. Specifically when $\mathbf{H}$ and $\mathbf{E}$ are seen as the sums of squares and the sums of products due to the hypothesis and due to the error, respectively.

About the matrix variate problem involved in the distribution of $\mathbf{E}^{-1}\mathbf{H}$

Abstract

This work studies the distribution of the nonsymmetric matrix . This random product is of fundamental interest under the general multivariate linear hypothesis setting. Specifically when and are seen as the sums of squares and the sums of products due to the hypothesis and due to the error, respectively.

Paper Structure

This paper contains 6 sections, 4 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries results
  3. Notation
  4. Jacobians
  5. Matrix variate beta type II distribution
  6. Main result

Key Result

Theorem 2.1

Let $\mathbf{Y} \in GL(m)$ and $\mathbf{C}, \mathbf{X} \in \mathfrak{P}_{m}$. Define $\mathbf{Y} = \mathbf{XC}$, then where $\lambda_{i}= \mathop{\rm ch}_{i}\nolimits(\mathbf{X})$, $i = 1, \dots,m$, $\mathbf{G} \in \mathcal{O}(m)$ and $(\mathbf{G}'d\mathbf{G})$ is the Haar measure on $\mathcal{O}(m)$, see mh:05.

Theorems & Definitions (12)

  • Theorem 2.1
  • proof
  • Remark 2.1
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.1
  • Definition 3.1: Jordan decomposition
  • Proposition 3.1
  • ...and 2 more