2 Probability Theory 2.3 Generating and Sampling Two Signed Bernoulli Random Variables with Given Correlation 3 Random Matrix Theory

2.4 Distributions of Functions of Random Variables and Matrices

This is a consolidated post containing two former posts on distributions of functions of variables.

2.4.1 Distribution of A Simple Function of Gamma Variables

In this post, we look at a nifty result presented in [11] where the probability density function (pdf) of the function $\dfrac{r_{1}}{c+r_{2}}$ two independent Gamma distributed random variables $r_{1}\sim\mathcal{G}(k_{1},\theta_{1})$ and $r_{2}\sim\mathcal{G}(k_{2},\theta_{2})$ is derived.

The derivation is an exercise (for the scalar case) in computing the pdf of functions of variables by constructing the joint pdf and marginalizing based on the Jacobian. A similar approach can also be used for matrix-variate distributions (which will probably be a good topic for another post.)

Theorem 6.

Let $c>0,$ and let $r_{1}\sim\mathcal{G}(k_{1},\theta_{1})$ and $r_{2}\sim\mathcal{G}(k_{2},\theta_{2})$ be independent random variables, then, the pdf of $r=\dfrac{r_{1}}{c+r_{2}}$ , denoted by $p_{r}(r;k_{1},\theta_{1},k_{2},\theta_{2},c),$ is given by

K_{\mathrm{r}}r^{k_{1}-1}\exp\left(-\frac{rc}{\theta_{1}}\right)U\left(k_{2},k% _{1}+k_{2}+1;c\left(\frac{r}{\theta_{1}}+\frac{1}{\theta_{2}}\right)\right),

(2.42)

where

U(a,b;z)=\frac{1}{\Gamma(a)}\int_{0}^{+\infty}\exp(-zx)x^{a-1}(1+x)^{b-a-1}% \mathrm{d}x

is the hypergeometric U-function [15, Chapter 13, Kummer function], and $K_{\mathrm{r}}$ is a constant ensuring that the integral over the pdf equals one.

Proof.

As $r_{1}$ and $r_{2}$ are independent, their joint pdf, denoted by $p_{r_{1},r_{2}}(r_{1},r_{2};k_{1},\theta_{1},k_{2},\theta_{2}),$ is given by

\frac{1}{\Gamma(k_{1})\theta_{1}^{k_{1}}\Gamma(k_{2})\theta_{2}^{k_{2}}}r_{1}^% {k_{1}-1}r_{2}^{k_{2}-1}\exp\left(-\frac{r_{1}}{\theta_{1}}-\frac{r_{2}}{% \theta_{2}}\right).

(2.43)

Applying transformation $r=\frac{r_{1}}{c+r_{2}}$ with Jacobian $\frac{\mathrm{d}r_{1}}{\mathrm{d}r}=c+r_{2},$ we obtain the transformed pdf, $p_{r,r_{2}}(r,r_{2};k_{1},\theta_{1},k_{2},\theta_{2},c)$ , as

\displaystyle K_{\mathrm{r}}^{\prime}r^{k_{1}-1}(c+r_{2})^{k_{1}}r_{2}^{k_{2}-% 1}\exp\left(-\frac{rc}{\theta_{1}}-\left(\frac{r}{\theta_{1}}+\frac{1}{\theta_% {2}}\right)r_{2}\right),

(2.44)

where $K_{\mathrm{r}}^{\prime}$ is a constant ensuring that the integral over the pdf equals one. Next, $p_{r}(r;k_{1},\theta_{1},k_{2},\theta_{2},c)$ is obtained by marginalization as

p_{r}(r;k_{1},\theta_{1},k_{2},\theta_{2},c)=\int_{0}^{+\infty}p_{r,r_{2}}(r,r% _{2};k_{1},\theta_{1},k_{2},\theta_{2},c)\mathrm{d}r_{2},

(2.45)

where the integration is conducted using the integral representation of the hypergeometric U-function from [15, Chapter 13, Kummer function] to obtain the expression in the theorem statement. ∎

Version History

1.

First published: 19th Oct. 2021 on aravindhk-math.blogspot.com
2.

Modified: 17th Dec. 2023 – Style updates for LaTeX

2.4.2 Distribution of the Determinant of a Complex-Valued Sample Correlation Matrix

In this post, we look at the distribution of the determinant of the sample correlation matrix of the realizations of a complex-valued Gaussian random vector. The distribution for real-valued Gaussian random vector was developed in [18], and we largely follow the developed framework. Thanks to Prashant Karunakaran for bringing this problem and the many applications to my attention in late 2017/early 2018.

Let $\boldsymbol{x}$ be a Gaussian random vector of length $p$ with mean $\boldsymbol{\mu}\in\mathbb{C}^{p}$ and covariance $\boldsymbol{\Sigma}\in\mathbb{C}^{p\times p}.$ Let $\boldsymbol{x}^{(1)},\boldsymbol{x}^{(2)},\dots,\boldsymbol{x}^{(n)}$ denote $n$ realizations, $n\geq p,$ of $\boldsymbol{x}.$ In the terminology of [18], the adjusted sample covariance matrix is given by

\boldsymbol{S}=\frac{1}{n}\sum_{i=1}^{n}(\boldsymbol{x}^{(i)}-\bar{\boldsymbol% {x}})(\boldsymbol{x}^{(i)}-\bar{\boldsymbol{x}})^{\mathrm{H}},

where $\bar{\boldsymbol{x}}$ is the sample mean given by

\bar{\boldsymbol{x}}=\frac{1}{n}\sum_{i=1}^{n}\boldsymbol{x}^{(i)}.

Note that the adjusted sample covariance matrix is positive semi-definite.

The correlation matrix $\boldsymbol{R}$ is defined as:

\boldsymbol{R}=\boldsymbol{D}^{-\frac{1}{2}}\boldsymbol{S}\boldsymbol{D}^{-% \frac{1}{2}},

where $\boldsymbol{D}=\mathrm{Diag}(\boldsymbol{S})$ is a diagonal matrix with the diagonal elements of $\boldsymbol{S}$ on the main diagonal. Hence, $\boldsymbol{R}$ has unit diagonal elements and is independent of the variance of the elements of $\boldsymbol{x}.$

Now, for real-valued $\boldsymbol{x},$ the determinant of $\boldsymbol{R},$ denoted by $|\boldsymbol{R}|,$ is shown in [18, Theorem 2] to be a product of $p-1$ Beta-distributed scalar variables $\mathrm{Beta}(\frac{n-i}{2},\frac{i-1}{2}),$ $i=1,\dots,p-1.$ The density of the product can be given in terms of the $\mathrm{MeijerG}$ function as follows [18, Theorem 2]:

g_{\mathbb{R}}(x;n,p)=\frac{\left[\Gamma(\frac{n-1}{2})\right]^{(p-1)}}{\Gamma% (\frac{n-2}{2})\dots\Gamma(\frac{n-p}{2})}\mathrm{MeijerG}^{\begin{bmatrix}p-1% &0\\ p-1&p-1\end{bmatrix}}\left(x\middle|\begin{matrix}\frac{n-3}{2},\dots,\frac{n-% 3}{2}\\ \frac{n-4}{2},\dots,\frac{n-(p+2)}{2}\end{matrix}\right).

Analogously, for complex-valued $\boldsymbol{x},$ $|\boldsymbol{R}|$ is a product of $p-1$ Beta-distributed scalar variables $\mathrm{Beta}(n-i,i),$ $i=1,\dots,p-1.$ The density of the product can now be given in terms of the $\mathrm{MeijerG}$ function, in a straightforward manner, as follows.

g_{\mathbb{C}}(x;n,p)=\frac{\left[\Gamma(n-1)\right]^{(p-1)}}{\Gamma(n-1)\dots% \Gamma(n-p+1)}\mathrm{MeijerG}^{\begin{bmatrix}p-1&0\\ p-1&p-1\end{bmatrix}}\left(x\middle|\begin{matrix}n-1,\dots,n-1\\ n-2,\dots,n-p\end{matrix}\right).

In the following, a Mathematica program for numerical simulation and the corresponding output are provided.

⬇

gC[x_, n_,p_] := (Gamma[n])^(p - 1) /

Product[Gamma[n - i], {i, 1, p - 1}] MeijerG[{{},

Table[n - 1, {i, 1, p - 1}]}, {Table[n - i, {i, 2, p}], {}}, x]

r[x_] := Module[{d}, d = DiagonalMatrix[Diagonal[x]];

MatrixPower[d, -1/2] . x . MatrixPower[d, -1/2]]

\[ScriptCapitalD] =

MatrixPropertyDistribution[Det[r[(xr + I xi) .

ConjugateTranspose[xr + I xi]]], {xr \[Distributed]

MatrixNormalDistribution[IdentityMatrix[p], IdentityMatrix[n]],

xi \[Distributed]

MatrixNormalDistribution[IdentityMatrix[p],

IdentityMatrix[n]]}] ;

data = Re[RandomVariate[\[ScriptCapitalD], 100000]] ;

\[ScriptCapitalD]1 = SmoothKernelDistribution[data] ;

Plot[{PDF[\[ScriptCapitalD]1, x], gC[x, n, p]}, {x, 0, 1},

PlotLabels -> {"Numerical", "Analytical"},

AxesLabel -> {"u", "p(u)"}]

The following figure shows that the numerical and analytical results match perfectly for the example case $n=4,k=6.$

Version History

1.

First published: 12th Dec. 2021 on aravindhk-math.blogspot.com
2.

Modified: 17th Dec. 2023 – Style updates for LaTeX