Moments of the multivariate Beta distribution

Moments of probability distribution are an important topic in statistics. Given the moment sequence, the probability distribution is unique under some mind conditions. To prove the convergence of random variables, we can prove the convergence of its moment sequences instead. To accomplish such goal, the analytical form of moments is a prerequisite. The techniques to compute moments for different distributions differ. In this article, we focus on the Beta distribution of 2 by 2 matrix.

David introduces an extension of multivariate extension for Beta distribution, denoted as $\mathbf{B}(\alpha,\beta;I_{p})$ (see [1]). It is a random $p\times p$ symmetric matrix $W$ whose density function is given by

$B_{p}(\alpha,\beta)$ is called the multivariate Beta function (see [5]); $\lvert W\rvert$ is the determinant of matrix $W$ and $S_{p,p}^{++}$ is the collection of positive definite matrix. When $p=1$, the distribution reduces to normal Beta distribution for $0<x<1$.

This extension may have useful applications in multivariate statistical problems but little is known about the analytical property of such extension. For example, it is unknown whether the moments $\mathbb{E}[f(W)]$ can be written in concise form, where $f$ is a monomial about the positive-definite matrix $W$.

Konno has derived the formula of the moment up to second order (see [4]). In this paper, we focus on the case $p=2$ and deduce the analytical form of moments for $\mathbf{B}(\alpha,\beta;I_{2})$. This formula includes the expectation and variance, which are the first and second order moment respectively. Our moments formula, as far as we know, is novel and can be used directly in the computation related with multivariate Beta models instead of approximating numerical integration.

In this article, the following notation convention is adopted: $W=\begin{pmatrix}X&Z\\ Z&Y\end{pmatrix}$ is the symmetric random matrix to be considered. Its density function is given by Equation (1), which can also be treated as the joint density function of $X,Y,Z$. $\lvert W\rvert=XY-Z^{2}$. Let $\mathbb{E}_{\alpha,\beta}[f(X,Y,Z)]=\int f(x,y,z)p(w)dw$ denotes the expectation with $\mathbf{B}(\alpha,\beta;I_{2})$ where $f(\cdot,\cdot,\cdot)$ is an arbitrary function with three variables. We will compute $\mathbb{E}_{\alpha,\beta}[f(X,Y,Z)]$ when $f(X,Y,Z)$ takes the monomial form: $f(X,Y,Z)=X^{m}Y^{r}Z^{2t}$.

In this section we will compute $\mathbb{E}_{\alpha,\beta}[f(X,Y,Z)]$ for $f(X,Y,Z)=X^{m}$ and show that $X$ is one dimensional Beta distribution. To accomplish our goals, we need the following lemma:

Using the above Lemma, we give the main conclusion of this section:

In this section, we further compute $\mathbb{E}_{\alpha,\beta}[X^{m}Y^{r}Z^{t}]$. By symmetric property $\mathbb{E}_{\alpha,\beta}[X^{m}Y^{r}Z^{2t+1}]=0$. Therefore we only need to consider the case when the power of $Z$ is even. Firstly We consider the case when $r=0$:

From Theorem 2, we can get the general formula for the mixed moment when $m\geq r$:

Since $\mathbb{E}_{\alpha,\beta}[X^{m}Y^{r}Z^{2t}]=\mathbb{E}_{\alpha,\beta}[X^{r}Y^{m}Z^{2t}]$, when $m<r$ we can exchange $m$ with $r$ and then use Corollary 1.

In this section, we will give a natural example which illustrates how our result can be used. We will consider the random matrix $S=QQ^{T}$ where $Q$ is $n\times k$ random orthogonal matrix. We are interested in how $\mathbb{E}[S_{11}^{m}S_{12}^{2t}]$ changes as $k\to\infty$ when $r=\frac{k}{n}$ is fixed.

From Proposition 7.2 of [2], $\begin{pmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\end{pmatrix}$ is exactly 2 by 2 random matrix of Beta distribution with parameter $B(\frac{k}{2},\frac{n-k}{2},I_{2})$. Using Theorem 2, we could write $\mathbb{E}[S_{11}^{m}S_{12}^{2t}]\sim\frac{(2t-1)!!}{2^{t}}\frac{r^{t}(1-t)^{t+m}}{n^{t}}=O(n^{-t})$. That is, $\mathbb{E}[S_{11}^{m}S_{12}^{2t}]$ decreases in the order of $n^{-t}$.

We have derived the formula of moments for multivariate Beta distribution of 2 by 2 matrix. This result is helpful for analyzing other statistical properties of multivariate Beta distribution.