The Exact Analytical Expression for the Moments of the Test-Statistic

In order to evaluate $\L _{th}$ for a given $P_{FA}$ it is necessary to know the PDF (or the CDF) of the test-statistic $\L$. While the PDF of the random variable $\L$ is not known in analytical closed form, the exact analytical expression of the statistical moments of any order for the test-statistic monotonic function $V = \L ^{1/N}$ can be found by using the well-known complex Wishart distribution for the matrix ${\bf A}$ given in [3]

$$\begin{array}{c} W({\it A},\S ,N)=\frac{\vert A\vert^{N-p}}{I(\S )}e^{-Sp(\S ^{-1}A)}, \end{array}$$ (7)

where $I(\S )=\frac{1}{K(\S )} =\pi^{\frac{1}{2}p(p-1)}\Gamma(n)\ldots\Gamma(N-p+1)\vert\S \vert^N$

If the null hypothesis $H_0$ is true, the matrix ${\bf A}$ is $W({\it A},\S _0,N)$ distributed.

The $h$-th order moment of the random variable

$V=\L ^{\frac{1}{N}}=\frac{\vert A\vert}{(\frac{SpA}{p})^p}$

can be straightforward determined from multidimensional integral

$$\begin{array}{l} M[V^h]=\int...\int\vert A\vert^h\frac{1}{\biggl(... ... _0,N)\times \ \times \vert A\vert^{N-p}exp{[-Sp(\S _0^{-1}A)]dA} \end{array}$$ (8)

The integration procedure is carry out here over area $dA=da_{11}da_{12}...da_{p-1,p}da_{pp}$, where A are all non-negative defined Hermitian $[p \times p]$ matrices. The integral (8) can be transformed into

$$\begin{array}{l} M[V^h]=\frac {p^{hp}K(\S _0,N)} {K(\S _0,N+h)}\i... ...xp[-Sp(\S _0^{-1}A)]\} \frac{1}{(\sum\limits_{i=1}^pa_{ii})^{ph}}dA \end{array}$$ (9)

The element of integration (9) in figured brackets is also Wishart distribution (7) with $N+h$ degrees of freedom $W(A,\S _0,N+h)$. After integration of (9) over all nondiagonal $W(A,\S _0,N+h)$ distribution elements $\it {a_{ij}}$ $(i\ne j)$, the partial joint distribution of the diagonal elements $a'_{11}...a'_{pp}$ of the matrix ${\bf A}$ is obtained. For hypothesis $H_0$ this distribution is transformed into product of one-dimensional distributions $W({\it a}_{ii},{\bf\sigma}^2,N+h)$. The one-dimensional complex Wishart distribution $W({\it a}_{ii},{\bf\sigma}^2,N+h)$ is the distribution of the $i$-th diagonal element of the matrix ${\bf A}$. Note, that $W({\it a}_{ii},{\bf\sigma}^2,N+h)$ is $\chi^2$ distribution with $2(N+h)$ degrees of freedom:

$$W(a_{ii},{\bf\sigma}^2,N+h)= (\frac{a_{ii}^{N+h-1}}{\Gamma(N+h){\bf\sigma}^{2(N+h)}})exp{(\frac{-a_{ii}}{{\bf\sigma}^2})}$$ (10)

Thus, the expression for the $h$-th order moment of the test-statistic $V$ can be reduced to the integral

$M[V^h]=\underbrace{p^{hp}\frac{K(\S _0,N)}{K(\S _0,N+h)}}_{g}\int...\int \frac{... ...}{{\bf\sigma}^2}}}{{\bf\sigma}^{2(N+h)}\Gamma(N+h)} da_{11}da_{22}\dots da_{pp}$

and can be represented as

$M[V^h]=g<\frac{1} {(\sum\limits_{i=1}^pa_{ii}')^{ph}}>$,

where $<.>$ denotes expectation value.

Diagonal elements $a_{ii}'$ of the matrix ${\bf A}$ are independent and identically $\chi^2$ distributed random values with $2(N+h)$ degrees of freedom and variances ${\bf\sigma}^2/2$. Therefore, the random variable $B=\sum a_{ii}'$ is also $\chi^2$ distributed with $2(N+h)p$ degrees of freedom. Thereby, $h$-th order moment of the random variable $V$ is proportional to the $r$-th order moment $(r=-ph)$ of the random variable $B$ which can be expressed through known one-dimensional integral

$$\begin{array}{l} M[B^r]=\int\limits_0^{\infty}B^r\frac{B^{(N+h)p-... ...c{{\bf\sigma}^{2Np}\Gamma(Np)}{{\bf\sigma}^{2(N+h)p}\Gamma((N+h)p)} \end{array}$$ (11)

For practical calculations of the high order moments of the test-statistic $V$ it is conveniently to transform the last expression into product using properties of gamma-function:

$M[V^h]=g\frac{{\bf\sigma}^{2pN}\Gamma(pN)}{{\bf\sigma}^{2p(N+h)}\Gamma(p(N+h))}... ...\times\ \times\frac {\Gamma(N+h)...\Gamma(N+h-p+1)}{\Gamma(N)...\Gamma(N-p+1)}$

So, the exact analytical expression for the $h$-th order moment of the test-statistic $V$ is

$$M[V^h]= p^{hp}\frac{\prod\limits_{i=1}^{p}\prod\limits_{j=1}^{h}(j+N-i)}{\prod\limits_{i=1}^{ph}(i+pN-1)}$$ (12)

It is easy to see that the distribution of the random variable $V$ is concentrated in interval [0,1]. So, there are moments of any order (12) for $V$, and they completely determine its distribution function [5]. It is worth to notice that the characteristic function of the random variable $V$ is expressed through moments as a converging Taylor series.