On the distribution of sample scale-free scatter matrices

Abstract

This paper addresses certain distributional aspects of a scale-free scatter matrix denoted by R that is stemming from a matrix-variate gamma distribution having a positive definite scale parameter matrix B. Under the assumption that B is a diagonal matrix, a structural representation of the determinant of R is derived; the exact density functions of products and ratios of determinants of matrices possessing such a structure are obtained; a closed form expression is given for the density function of R. Moreover, a novel procedure is utilized to establish that certain functions of the determinant of the sample scatter matrix are asymptotically distributed as chi-square or normal random variables. Then, representations of the density function of R that respectively involve multiple integrals, multiple series and Gauss’ hypergeometric function are provided for the general case of a positive definite scale parameter matrix, and an illustrative numerical example is presented. Cutting-edge mathematical techniques have been employed to derive the results. Naturally, they also apply to the conventional sample correlation matrix which is encountered in various multivariate inference contexts.

Appendices

Appendix

Proof of Theorem 3.4

We will need the asymptotic expansion of the gamma function, namely,

$$\begin{aligned} \Gamma (z+\delta )\!=\!\sqrt{2\pi }z^{z+\delta -\frac{1}{2}}\textrm{e}^{-z-\sum _{k=1}^{\infty }\frac{(-1)^kB_{k+1}(\delta )}{k(k+1)z^{k}}} \mathrm{as\ } |z|\rightarrow \infty \text{ and } ~\delta \text{ is } \text{ bounded, } \end{aligned}$$

(A.1)

where $B_k(\delta )$, $k=1,2,\ldots ,$ are the Bernoulli polynomials. Lists of the first few Bernoulli polynomials and the first few Bernoulli numbers, along with the above expansion and further details, are given in Mathai (1993). In this case, only $B_2(\delta )$ and $B_3(\delta )$ will be needed for deriving the asymptotic normality. For illustrative purposes, we will list $B_k(\delta )$ for $k=2$ and 3:

$$\begin{aligned} B_2(\delta )=\delta ^2-\delta +\frac{1}{6};~~B_3(\delta )=\delta ^3-\frac{3}{2}\delta ^2+\frac{1}{2}\delta . \end{aligned}$$

(A.2)

It was established that the h-th moment of |R| for the case where the scale parameter matrix B is diagonal is

$$\begin{aligned} E[|R|^h]=\frac{\Gamma (\alpha -\frac{1}{2}+h)\cdots \Gamma (\alpha -\frac{p-1}{2}+h)}{\Gamma (\alpha -\frac{1}{2})\cdots \Gamma (\alpha -\frac{p-1}{2})}\left[ \frac{\Gamma (\alpha )}{\Gamma (\alpha +h)}\right] ^{p-1}. \end{aligned}$$

(A.3)

Consider the two-term approximation of a gamma function which is obtained by expanding (A.1) up to $k=1$, namely,

$$\begin{aligned} \Gamma (z+\delta )\approx \sqrt{2\pi }z^{z+\delta -\frac{1}{2}}\textrm{e}^{-z+\frac{1}{2z}(\delta ^2-\delta +\frac{1}{6})}\ \, \mathrm{as\ } |z|\rightarrow \infty \text{ and } ~\delta \text{ is } \text{ bounded }. \end{aligned}$$

(A.4)

We now expand all the gamma functions appearing in (A.3) by making use of (A.4) assuming that $|\alpha |\rightarrow \infty$, by writing $h=it,i=\sqrt{(-1)},\ t$ being a parameter, so that $E[|R|^{it}]$ becomes the characteristic function of $\ln |R|$ or $E[\textrm{e}^{it\ln |R|}]$. Then,

$$\begin{aligned} \left[ \frac{\Gamma (\alpha )}{\Gamma (\alpha +it)}\right] ^{p-1}\approx \textrm{e}^{-\frac{(p-1)}{2\alpha }(-t^2-it)} \end{aligned}$$

(A.5)

and

$$\begin{aligned} \prod _{j=1}^{p-1}\frac{\Gamma (\alpha -\frac{j}{2}+it)}{\Gamma (\alpha -\frac{j}{2})}\approx \textrm{e}^{\frac{1}{2\alpha }[-\frac{(p-1)p}{2}it-(p-1)t^2]}. \end{aligned}$$

(A.6)

On combining (A.5) and (A.6), one has the following two-term approximation:

$$\begin{aligned} E[|R|^h]\approx \textrm{e}^{\frac{1}{2\alpha }\frac{p(p-1)}{2}(it)}. \end{aligned}$$

(A.7)

Now, consider a three-term approximation of all the gamma functions when $\alpha$ is large. The third factor in the asymptotic series of the gamma function is available from (A.1), letting $k=2$, and (A.2). We will consider the approximation of the third factor for $k=2$. By taking $\delta =0$ for the numerator gamma functions and $\delta =it$ for the denominator gamma functions, the third factor can be approximated as follows:

$$\begin{aligned} \left[ \frac{\Gamma (\alpha )}{\Gamma (\alpha +it)}\right] ^{p-1}&\rightarrow \prod _{j=1}^{p-1}\textrm{e}^{-\frac{1}{6\alpha ^2}(0)}/\textrm{e}^{-\frac{1}{6\alpha ^2}[(it)^3-\frac{3}{2}(it)^2+\frac{1}{2}(it)]}\nonumber \\&=\textrm{e}^{\frac{(p-1)}{6\alpha ^2}[(it)^3-\frac{3}{2}(it)^2+\frac{1}{2}(it)]}. \end{aligned}$$

(A.8)

By taking $\delta =-\frac{j}{2}+it$ for the numerator gamma functions and $\delta =-\frac{j}{2}$ for the denominator gamma functions, the third factor in $\prod _{j=1}^{p-1}\frac{\Gamma (\alpha -\frac{j}{2}+it)}{\Gamma (\alpha -\frac{j}{2})}$ tends to

$$\begin{aligned}&\prod _{j=1}^{p-1}\textrm{e}^{-\frac{1}{6\alpha ^2}[(-\frac{j}{2}+it)^3-\frac{3}{2}(-\frac{j}{2}+it)^2+\frac{1}{2}(-\frac{j}{2}+it)]} /\textrm{e}^{-\frac{1}{6\alpha ^2}[(-\frac{j}{2})^3-\frac{3}{2}(-\frac{j}{2})^2+\frac{1}{2}(-\frac{j}{2})]}\nonumber \\&\ =\textrm{e}^{-\frac{\beta }{6\alpha ^2}} \end{aligned}$$

(A.9)

where $\beta =\tfrac{1}{8}(p-1)p(2p+5)(it)+\tfrac{3}{4}p(p-1)t^2+(p-1)(it)^3 -\tfrac{3}{2}(p-1)(it)^2+\tfrac{(p-1)}{2}(it)$.

Then, the three-term approximation of $E[|R|^h]$ is the following for $h=it$:

$$\begin{aligned} E[|R|^{it}]&=E[\textrm{e}^{it\ln |R|}]\\&\approx \textrm{e}^{\frac{p(p-1)}{4\alpha }(it)-\frac{1}{6\alpha ^2}[\frac{1}{8}(p-1)p(2p+5)(it)+\frac{3}{4}p(p-1)t^2]}, \end{aligned}$$

so that

$$\begin{aligned} \!\!\!\!\!\!\!\!\!\! E[\textrm{e}^{it(\alpha \ln |R|)}]&\approx \textrm{e}^{\frac{p(p-1)}{4}(it)-\frac{1}{6\alpha }[\frac{1}{8}(p-1)p(2p+5)(it)]-\frac{1}{2}\frac{p(p-1)}{4}t^2}. \end{aligned}$$

(A.10)

One can readily verify that, for $n=4,5,\ldots ,$ additional factors involving $B_n(\delta )$ will converge to 1 when $|\alpha |\rightarrow \infty$. Accordingly, when $|\alpha |\rightarrow \infty ,$

$$\begin{aligned} E[\textrm{e}^{it(\alpha \ln |R|-\frac{p(p-1)}{4})}]&\rightarrow \textrm{e}^{-\frac{1}{2}\frac{p(p-1)}{4}t^2}, \end{aligned}$$

which implies that

$$\begin{aligned} E[\textrm{e}^{it\frac{{2}}{\sqrt{p(p-1)}}(\alpha \ln |R|-\frac{p(p-1)}{4})}]&\rightarrow \textrm{e}^{-\frac{t^2}{2}}, \end{aligned}$$

so that

$$\begin{aligned} \tfrac{{2}}{\sqrt{p(p-1)}}(\alpha \ln |R|-\tfrac{p(p-1)}{4})&\rightarrow \mathcal{N}_1(0,1).\end{aligned}$$

(A.11)

Theorem 3.4 follows from (A.7) to (A.11).

Proof of Theorem 4.1

On integrating out $s_{jj},\ j=1,\ldots ,k$, from S, one has

$$\begin{aligned} f_{R}(R)\textrm{d}R&=\frac{|B|^{\alpha }}{\Gamma _p(\alpha )}|R|^{\alpha -\frac{p+1}{2}}\int _0^{\infty }\cdots \int _0^{\infty }(s_{11}\cdots s_{pp})^{\alpha -1}\nonumber \\&\qquad \times \textrm{e}^{-\sum _{j=1}^pb_{jj}s_{jj}-2\sum _{i<j}b_{ij}s_{ij}}\textrm{d}s_{11}\wedge \ldots \wedge \textrm{d}s_{pp}\wedge \textrm{d}R, \end{aligned}$$

(A.12)

where the exponent $\sum _{i<j}b_{ij}s_{ij}$ can be re-expressed as follows:

$$\begin{aligned} \sum _{i<j}b_{ij}s_{ij}&=\sum _{i<j}b_{ij}r_{ij}\sqrt{s_{ii}s_{jj}}\nonumber \\&=b_{12}r_{12}\sqrt{s_{11}s_{22}}+\cdots +b_{1,p-1}r_{1,p-1}\sqrt{s_{11} s_{p-1,p-1 }} +b_{1p}r_{1p}\sqrt{s_{11}s_{pp}} \nonumber \\&\ \ \ +b_{23}r_{23}\sqrt{s_{22}s_{33}}+\cdots +b_{2p}r_{2p}\sqrt{s_{22}s_{pp}}\nonumber \\&\ \ \ +\ \cdots \nonumber \\&\ \ \ +b_{p-1,p}r_{p-1,p}\sqrt{s_{p-1,p-1}s_{pp}}\,. \end{aligned}$$

(A.13)

Now, on expanding the exponential terms, one has

$$\begin{aligned} \textrm{e}^{-2\sum _{i<j}b_{ij}s_{ij}}&=\prod _{i<j}\textrm{e}^{-2b_{ij}r_{ij}\sqrt{s_{ii}s_{jj}}}\nonumber \\&=\!\!\sum _{k_{12}=0}^{\infty }\!\!\frac{(-2b_{12}\,r_{12}\sqrt{s_{11}s_{22}})^{k_{12}}}{k_{12}!} \cdots \nonumber \\&\sum _{k_{p-1,p}=0}^{\infty }\!\!\!\!\frac{(-2b_{p-1,p}\,r_{p-1,p} \sqrt{s_{p-1,p-1}s_{pp}})^{k_{p-1,p}}}{k_{p-1,p}!}\nonumber \\&=\sum _{k=0}^{\infty }\ \sum _K\Big [ \prod _{i<j}\frac{(-2b_{ij}r_{ij})^{k_{ij}}}{k_{ij}!}\Big ]\, s_{11}^{\rho _1}\cdots s_{pp}^{\rho _p} \end{aligned}$$

(A.14)

where $\sum _K$ denotes the sum over all $k_{ij},\ i<j$, such that $k=k_{12}+\cdots +k_{1p}+k_{23}+\cdots +k_{2p}+\cdots +k_{p-1,p}$ and $\rho _r$ $=\frac{1}{2}(k_{1r}+k_{2r}+\cdots +k_{r-1,r}+k_{r,r+1}+\cdots +k_{rp})$ with $k_{ij}=0$ whenever $i\ge j$. Note that all the possible combinations of nonnegative integer values that the $k_{ij}$’s can take on in the next-to-last equality are accounted for in the last one. Then, for example, the integrals over $s_{11}$ and $s_{rr}$ can be evaluated as follows whenever $\Re (\alpha )>0$:

$$\begin{aligned} \int _{s_{11}=0}^{\infty }s_{11}^{\alpha -1+\rho _1}\textrm{e}^{-b_{11}s_{11}}\textrm{d}s_{11}&=b_{11}^{-[\alpha +\frac{1}{2}(k_{12}+\cdots +k_{1p})]}\Gamma (\alpha +\tfrac{1}{2}(k_{12}+\cdots +k_{1p})),\nonumber \\ \int _{s_{rr}=0}^{\infty }s_{rr}^{\alpha -1+\rho _r}\textrm{e}^{-b_{rr}s_{rr}}\textrm{d}s_{rr}&=b_{rr}^{-[\alpha +\frac{1}{2}(k_{1r}+\cdots +k_{r-1,r}+k_{r,r+1}+\cdots +k_{rp})]}\nonumber \\&\ \ \ \times \Gamma (\alpha +\tfrac{1}{2}(k_{1r}+\cdots +k_{r-1,r}+k_{r,r+1}+\cdots +k_{rp})), \end{aligned}$$

(A.15)

for $r=2,\ldots ,p$. Therefore, the product of the integrals over $s_{rr},\ r=1,\ldots ,p,$ can be written as

$$\begin{aligned}{} & {} \prod _{r=1}^pb_{rr}^{-(\alpha +\rho _r)}\Gamma (\alpha +\rho _r),\ \Re (\alpha )>0, \nonumber \\{} & {} \quad \rho _r=\tfrac{1}{2}(k_{1r}+\cdots +k_{r-1,r}+k_{r,r+1}+\cdots +k_{rp}). \end{aligned}$$

(A.16)