# The exact distribution of Bartlett's criterion for testing equality of covariance matrices

A. M. Mathai

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## To cite this version:

A. M. Mathai. The exact distribution of Bartlett's criterion for testing equality of covariance matrices. Annales de l'ISUP, 1970, XIX (2), pp.1-15. hal-04083354

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Submitted on 27 Apr 2023

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Distributed under a Creative Commons Attribution 4.0 International License
# THE EXACT DISTRIBUTION OF BARTLETT'S CRITERION FOR TESTING EQUALITY OF COVARIANCE MATRICES

## A.M. MATHAI*

### RESUMÉ

Ce qui suit donne la distribution exacte du critère de Bartlette pour constater l'hypothèse que les matrices de covariance soient égales dans un ensemble de q distributions, indépendantes, normales, p-variantes. La distribution exacte du critère, sous l'hypothèse nulle est donnée par une fonction G de Meijer aussi bien que par des fonctions simples et computables, si les tailles de l'échantillon sont égales. La technique des transformées inverses de Mellin et le Calcul des résidus sont appliqués en dérivant la distribution. La distribution cumulative est donnée aussi par une forme computable, qui calcule des points de pourcentage.

### SUMMARY

This article gives the exact distribution of Bartlett's criterion for testing the hypothesis that the covariance matrices are equal in a set of q independent p-variate normal distributions. The exact distribution of the criterion, under the null hypothesis, is obtained in terms of a Meijer's G-function as well as in simple computable functions, when the sample sizes are equal. The technique of inverse Mellin transform and the Calculus of Residues are used to obtain the distribution. The cumulative distribution is also given in a computable form so that percentage points can be computed.

## 1. INTRODUCTION

Consider a set of q independent p-variate normal populations with covariance matrices $\Sigma_1, \dots, \Sigma_q$ and consider the hypothesis,

$$ H_1: \Sigma_1 = \dots = \Sigma_q. \qquad (1.1) $$

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<footnote>* Department of Mathematics, McGill University, Montreal, Canada.</footnote>
Bartlett (1937) has suggested a criterion $V_1$ as a modification to the likelihood ratio criterion for testing $H_1$, where,

$$V_1 = \prod_{q=1}^{n_q} \frac{|A_q|^{n_q/2}}{|A|^{n/2}} \qquad (1.2)$$

where, $n_q = N_q - 1$, $n = \sum n_q = N - q$ and $N_q$, $g = 1, \dots, q$ are the sample sizes. The numerator of $V_1$ is proportional to a power of a weighted geometric mean of the sample generalized variances and the denominator is proportional to a power of the determinant of a weighted arithmetic mean of the sample covariance matrices. The h-th moment, about the origin, of $V_1$ is given in Anderson (1958, p. 253) as,

$$E(V_1^h) = \prod_{i=1}^{p} \left\{ \prod_{q=1}^{q} \Gamma \left[ \frac{(n_q + hn_q + 1 - i)/2}{\Gamma[(n_q + 1 - i)/2]} \right] \right\} \frac{\Gamma[(n+1-i)/2]}{\Gamma[(n+hn+1-i)/2]} \quad (1.3)$$

Since the variation of $V_1$ is finite, the moment sequence in (1.3) uniquely determine the distribution of $V_1$, Rao (1965, p. 86). Hence the density of $V_1$ can be obtained by taking the inverse Mellin transform of (1.3). For $p = 1, q = 2$ it can be shown that the density of $V_1$ is a function of an F-statistic. The exact distribution of $V_q$ for $p = 2, q = 2, n_1 = n_2$, is evaluated in Anderson (1958, p. 257) which was also discussed by Pearson and Wilks (1933). Brown (1949) and Scheffé (1942) studied the unbiasedness of the test based on $V_1$. Box (1949) has given an approximation which is also discussed in Anderson (1958, p. 255).

This article gives the exact distribution of $V_1$ for the most general values of $p$ and $q$, when the sample sizes are equal. The same technique can be used to obtain the exact distribution when the sample sizes are not restricted to be equal. But in the general case, computation of the percentage points is difficult due to a large number of parameters. But the distribution obtained in this article enables one to compute percentage points. The distribution is worked out for the case $q \ge p$, $n_1=n_2=\dots=n_q=n'$. It is pointed out that when the sample sizes are equal the distributions for the case $p > q$ can be easily obtained in a similar fashion. For simplicity the distribution of $V_1^{2/n'}$ is obtained. In section 2 a few definitions and
preliminary results are given. These are used in the development of the
results.

## 2. SOME DEFINITIONS

2.1. Meijer's G-function. This is one of the most general Special Functions and is defined as,

$$ G_{p,q}^{m,n}(x|a_1, \dots, a_p) = (1/2\pi i) \int_L \frac{\prod_{j=1}^{m} \Gamma(b_j+s) \prod_{j=1}^{n} \Gamma(1-a_j-s) x^{-s} ds}{\prod_{j=m+1}^{q} (1-b_j-s) \prod_{j=n+1}^{p} \Gamma(a_j+s)} \quad (2.1) $$

where an empty product is interpreted as unity, $i = (-1)^{1/2}$, $0 \le m \le q$, $0 \le n \le p$ and no poles of $\Gamma(b_j+s)$, $j = 1, \dots, m$ coincides with any pole of $\Gamma(1-a_k-s)$, $k=1, \dots, n$. The existence of the function and the contour L is discussed in Erdélyi (1953, p. 207). In our problem in this article, the parameters satisfy the conditions of existence of the G-function and hence this aspect won't be discussed here. A series expansion, into a computable form, of the G-function is given in Erdélyi (1953, p. 208) and other expansions are available in Meijer (1946). But it can be easily seen that the parameters in our problem violate the conditions under which these expansions are given. Hence, these expansions are not of much help in representing the exact distribution of $V_1$ into a computable form.

2.2. The $\psi$-function. The $\psi$-function is defined as,

$$ \psi(z) = \frac{d}{dz} \log \Gamma(z) = -\gamma + (z-1) \sum_{n=0}^{\infty} [(n+1)(z+n)]^{-1} \quad (2.2) $$

where $\gamma$ is the Euler's constant: $\gamma = 0.5772156649 \dots$

2.3. The generalized Zeta function :

$$ \zeta(s,v) = \sum_{n=0}^{\infty} (v+n)^{-s}, \quad R(s) > 1, \ v \neq 0, -1, \dots \quad (2.3) $$

where $R(\cdot)$ means the real part of $(\cdot)$. Some properties of $\psi$-function are available in Erdélyi (1953) and the properties of Zeta function are available in Erdélyi (1953) and in Titchmarsh (1951).
2.4. The Gauss-Legendre multiplication formula :

$$ \Gamma(mz) = (2\pi)^{(1-m)/2} m^{mz-1/2} \prod_{r=0}^{m-1} \Gamma(z+r/m), m = 2,3,\dots \quad (2.4) $$

### 3. THE EXACT DISTRIBUTION

$$ E(V_1^h) = \left\{ \prod_{i=1}^{p} \prod_{q=1}^{q} \frac{\Gamma[(n+1-i)/2]}{\Gamma[(n_q+1-i)/2]} \right\} \left\{ \prod_{i=1}^{p} \prod_{q=1}^{q} \frac{\Gamma[(n_q+hn+i-1)/2]}{\Gamma[(n+hn+1-i)/2]} \right\} \quad (3.1) $$
where $n = \sum_{q=1}^{q} n_q$, $q \ge 2$, $p \ge 1$.

Now consider the case,

$$ n_1 = \dots n_q = n', \quad n = \sum n_q = qn' $$

Then,

$$ E(V_1^h) = C \prod_{i=1}^{p} \frac{\Gamma^q[(n'+hn'+1-i)/2]}{\Gamma[(qn'+qhn'+1-i)/2]} \quad (3.2) $$

where,

$$ C = \prod_{i=1}^{p} \left\{ \Gamma\left[\left(\frac{qn'+1-i}{2}\right)\right] / \Gamma\left[\left(\frac{n'+1-i}{2}\right)\right] \right\}. \quad (3.3) $$

By using (2.4) one can write,

$$ \Gamma\left[\frac{(qn'+qhn'+1-i)}{2}\right] = \left(2\pi\right)^{\frac{(1-q)}{2}} \frac{q^{\frac{(qn'+qhn'-i)}{2}}}{\prod_{r=0}^{q-1} \Gamma\left\{\frac{hn'+n'+(1-i)/q}{2} + r/q\right\}} \quad (3.4) $$

Hence,

$$ E\left[\left(\frac{V_1^{2/n'}}{n'}\right)^h\right] = C' \prod_{i=1}^{p} \left\{ \frac{\Gamma^q\left[\frac{h+(n'+1-i)}{2}\right] (q^q)^{-h}}{\prod_{r=0}^{q-q} \Gamma\left[\frac{h+n'+(1-i)}{2q} + r/q\right]} \right\} \quad (3.5) $$
$$ = C' \frac{\Gamma^q(\alpha) \Gamma^q(\alpha - 1/2) \Gamma^q(\alpha - 1) \dots \Gamma^q[\alpha - (p-1)/2] (q^q)^{-h}}{\prod_{t=0}^{p-1} \prod_{r=0}^{q-1} \Gamma(\alpha - t/2q + r/q)} \quad (3.6) $$

where,

$$ \alpha = h+n'/2, \quad C' = C/(2\pi)^{(1-q)/2} q^{(qn'-1)/2}. \quad (3.7) $$

Now the density of $V_1^{2/n'}$ is obtained from (3.5) by taking the inverse Mellin transform of (3.6). Let $f(v)$ denote the density of $V_1^{2/n'}$. Then,

$$ f(v) = C' v^{-1} (1/2 \pi i) \int_L \frac{\prod_{j=0}^{p-1} \Gamma^q(\alpha - j/2) (v q^j)^{-h} \, dh}{\prod_{t=0}^{p-1} \prod_{r=0}^{q-1} \Gamma(\alpha - t/2q + r/q)} \quad (3.8) $$

where $i = (-1)^{1/2}$ and $L$ is a suitable contour. The existence of the function (3.8) is discussed in section 2.1. Now $f(v)$ can be written as a Meijer's G-function. That is,

$$ f(v) = C' v^{-1} G_{pq,pq}^{pq,0} \left[ \begin{array}{l} n'/2 + (2r-t)/2q, \quad r=0,1,\dots,q-1; \\ n'/2 - j/2, \quad j=0,1,\dots,p-1 \end{array} \right. \\ \qquad \left. \begin{array}{l} t=0,1,\dots,p-1 \\ \text{repeated } q \text{ times} \end{array} \right] , \quad 0 < v < 1. \quad (3.9) $$

But $f(v)$ of (3.9) can not be put into a computable form by the results available in the literature. Hence we will evaluate (3.8) with the help of the Calculus of Residues since the contour exists which encloses the poles of the integrand of (3.8). Hence $f(v)$ is available as the sum of the residues of the integrand in (3.8) at the poles of (3.6). The poles of (3.6) are available from the equations,

$$ \alpha - j/2 = -v, \quad v = 0,1,\dots; \quad j = 0,1,\dots,p-1. \quad (3.10) $$

But some of the numerator Gammas in (3.6) may get cancelled with the denominator Gammas. Here we will examine the nature of the poles of $\Gamma[\alpha + (2r-t)/2q]$, $t = 0,1,\dots,p-1$; $r = 0,1,\dots,q-1$, and we will examine
6

to see whether,

$$ (2r-t)/2q = \pm v/2, \quad v = 0,1,...; \quad r = 0,1,...,q-1; \quad t = 0,...,p-1. $$

$$ (2r-t)/2q \neq 1,3/2,2,... \quad \text{since } r = 0,1,...,q-1. \qquad (3.11) $$

$$ \begin{cases} \left(-\frac{1}{2}\right) \Rightarrow t & 2r+q = q,q+2,...,3q-2 \\ -1 \Rightarrow t & 2r+2q = 2q,2q+2,...,4q-2 \\ \dots & \end{cases} \qquad (3.12) $$

$$ (2r-t)/2q = 0 \Rightarrow r = t/2 = 0,1,...,(p-1)/2 \text{ or } (p-2)/2. \qquad (3.13) $$

$$ (2r-t)/2q = 1/2 \Rightarrow r = (t+q)/2 = q/2,q/2 + 1/2,...,q/2+(p-1)/2. \qquad (3.14) $$

If $q \ge p$ then only (3.13) and (3.14) can come. Further, (3.13) gives, $r = 0,1,...,(p-1)/2$ (that is, $(p+1)/2$ in number) if $p$ is odd, $= 0,1,...,(p-2)/2$ ($p/2$ in number) if $p$ is even;

and (3.14) gives,

$$ \begin{aligned} r &= q/2,q/2+1,...,q/2+(p-1)/2 \quad ((p+1)/2 \text{ in number}) \text{ if } q\text{-even},p\text{-odd}; \\ &= q/2,q/2+1,...,q/2+(p-2)/2 \quad (p/2 \text{ in number}) \text{ if } q\text{-even}, p\text{-even}; \\ &= q/2+1/2,q/2+3/2,...,q/2+(p-1)/2 \quad (p/2 \text{ in number}) \text{ if } q\text{-odd}, p\text{-even}; \\ &= q/2+1/2,q/2+3/2,...,q/2+(p-2)/2 \quad ((p-1)/2 \text{ in number}) \text{ if } q\text{-odd}, p\text{-odd}. \end{aligned} $$

When $q \ge p$, we need consider only three cases, namely,
case I. $p$-even; case II: $p$-odd,$q$-even; case III: $p$-odd,$q$-odd.

The Gamma products in (3.6), excluding the constant $C'$, reduces to,

$$ \text{Case I: } \left\{ \Gamma^{q-p/2} (\alpha-1/2) \Gamma^q (\alpha-3/2) \dots \Gamma^q [\alpha-(p-1)/2] \right\} \quad (3.15) \\ \times \left\{ \Gamma^{q-p/2} (\alpha) \Gamma^q (\alpha-1) \dots \Gamma^q [\alpha-(p-2)/2] \right\} / \left\{ (\alpha-1/2)^{p/2} \beta(\alpha) \right\}; $$

$$ \text{Case II: } \left\{ \Gamma^{q-(p+1)/2} (\alpha-1/2) \Gamma^q (\alpha-3/2) \dots \Gamma^q [\alpha-(p-2)/2] \right\} \quad (3.16) \\ \times \left\{ \Gamma^{q-(p+1)/2} (\alpha) \Gamma^q (\alpha-1) \dots \Gamma^q [\alpha-(p-1)/2] \right\} / \left\{ (\alpha-1/2)^{(p+1)/2} \beta(\alpha) \right\}; $$
$$ \text{Case III: } \{ \Gamma^{q-(p-1)/2} (\alpha-1/2) \Gamma^q (\alpha-3/2) \dots \Gamma^q [\alpha-(p-2)/2] \} \quad (3.17) \\ \times \{ \Gamma^{q-(p+1)/2} (\alpha) \Gamma^q (\alpha-1) \dots \Gamma^q [\alpha-(p-1)/2] \} / \{(\alpha-1/2)^{(p-1)/2} \beta(\alpha)\}, $$

where,

$$ \beta(\alpha) = \prod_{(t,r) \in \Gamma_{\alpha}} \left[ \alpha + \frac{(2r-t)}{2q} \right] \quad \text{and} \quad a = \{(t,r) \mid r=0,1,\dots,q-1\}; \quad (3.18) \\ t=0,1,\dots,p-1; \quad r \neq t/2; \quad r \neq (t+q)/2. $$

Now the poles of the Gamma products in (3.15), (3.16) and (3.17) can be obtained by equating to zero, the various factors of the following, where the exponents denote the orders of the poles.

$$ \text{Case I: } \prod_{j=0}^{\infty} \left[ \alpha - \frac{(p-1)}{2+j} \right]^j \quad \text{and} \quad \prod_{j=0}^{\infty} \left[ \alpha - \frac{p-2}{2+j} \right]^j \quad j; \quad (3.19) $$

$$ \text{Case II: } \prod_{j=0}^{\infty} \left[ \alpha - \frac{p-2}{2+j} \right]^j \quad \text{and} \quad \prod_{j=0}^{\infty} \left[ \alpha - \frac{p-1}{2+j} \right]^j \quad j; \quad (3.20) $$

$$ \text{Case III: } \prod_{j=0}^{\infty} \left[ \alpha - \frac{p-2}{2+j} \right]^j \quad \text{and} \quad \prod_{j=0}^{\infty} \left[ \alpha - \frac{p-1}{2+j} \right]^j \quad j, \quad (3.21) $$

where,

$$ a_j = \begin{cases} q(j+1), & j = 0,1,\dots,p/2-2 \\ q(p/2), & j = p/2-1 \\ (p/2) (q-1), & j \ge p/2; \end{cases} \qquad b_j = \begin{cases} q(j+1), & j=0,\dots,p/2-2 \\ (p/2) (q-1), & j \ge p/2-1. \end{cases} \quad (3.22) $$

$$ c_j = \begin{cases} q(j+1), & j=0,\dots,(p-1)/2-2 \\ q(p-1)/2, & j=(p-1)/2-1 \\ q(p-1)/2 - (p+1)/2, & j \ge (p-1)/2; \end{cases} \qquad d_j = \begin{cases} q(j+1), & j=0,\dots,(p+1)/2-2 \\ (q-1)(p+1)/2, & j \ge (p+1)/2-1. \end{cases} \quad (3.23) $$

$$ e_j = \begin{cases} q(j+1), & j=0,\dots,(p-1)/2-2 \\ q(p-1)/2, & j=(p-1)/2-1 \\ (q-1)(p-1)/2, & j \ge (p-1)/2. \end{cases} \qquad f_j = d_j. \quad (3.24) $$

Here $a_j$ to $f_j$ are all > 0. If any exponent is zero then the corresponding
Gamma product is absent in (3.6). From the structure of the Gammas and their poles in cases I, II, III, it is apparent that the exact distributions in all the cases can be obtained in the same fashion as in any one of the cases. Hence we will obtain the exact density in one of the cases, for example, case II. By comparison, the densities in the other cases can be easily written down.

### 4. THE EXACT DISTRIBUTION (p-ODD, q-EVEN, $q \ge p$).

The aim now is to evaluate (3.8) for case II. Since the contour L exists, $f(v)$ can be written as,

$$f(v) = C' v^{-1} \sum_{j=0}^{\infty} (R_j + R'_j), \quad 0 < v < 1 \qquad (4.1)$$

where $R_j$ is the residue of (3.8) at the pole $\alpha = (p-2)/2 - j$ of order $c_j$ and $R'_j$ is the residue at the pole $\alpha = (p-1)/2 - j$ of order $d_j$. From the Calculus of Residues,

$$R_j = \frac{1}{(c_j - 1)!} \frac{\partial^{c_j - 1}}{\partial h^{c_j - 1}} \{ [\alpha - (p-2)/2 + j]^{c_j} B(v q^q)^{-h} \}, \quad \text{at } \alpha = (p-2)/2 - j \quad (4.2)$$

where $B$ is the Gamma product (3.16). Now (4.2) can be simplified as,

$$R_j = \frac{(v q^q)^{-h}}{(c_j - 1)!} \sum_{j_1=0}^{c_j-1} \left( \frac{c_j-1}{j_1} \right) (-\log v q^q)^{c_j-1-j_1} \quad (4.3)$$

$$\times \frac{\partial^{j_1}}{\partial h^{j_1}} [\alpha - (p-2)/2 + j]^{c_j} B, \quad \text{at } \alpha = (p-2)/2 - j.$$

Let,

$$\Delta = [\alpha - (p-2)/2 + j]^{c_j} B \qquad (4.4)$$

Then,

$$D = \frac{\partial^{j_1}}{\partial h^{j_1}} \Delta = \frac{\partial^{j_1-1}}{\partial h^{j_1-1}} A \quad \Delta = \left\{ \sum_{j_2=0}^{j_1-1} \left( \frac{j_1-1}{j_2} \right) A^{(j_1-1-j_2)} \right\} \quad (4.5)$$

$$\sum_{j_3=0}^{j_2-1} \left( \frac{j_2-1}{j_3} \right) A^{(j_2-1-j_3)} \dots \Delta,$$
where,

$$A = \frac{\partial}{\partial h} \log \Delta$$

and $A^{(r)}$ denotes the r-th partial derivative of $A$ with respect to $h$. Let $A_o^{(r)}, A_o, \Delta_o$ and $D_o$ denote the values of $A^{(r)}, A, \Delta$, and $D$ at $\alpha = (p-2)/(2+j)$ respectively. Then $R_j$ can be written as,

$$R_j = \frac{(v q^q)^{n'/2} - (p-2)/2 + j}{(c_j - 1)!} \sum_{j_1=0}^{c_j-1} \frac{(c_j^{-1})}{j_1} (-\log v q^q)^{c_j-1-j_1} D_o \quad (4.7)$$

and

$$R'_j = \frac{(v q^q)^{n'/2} - (p-1)/2 + j}{(d_j - 1)!} \sum_{j_1=0}^{d_j-1} \frac{(d_j^{-1})}{j_1} (-\log v q^q)^{d_j-1-j_1} D'_o \quad (4.8)$$

where $D'_o$ is obtained from $D_o$ by replacing $\Delta$ of $R_j$ by

$[\alpha-(p-1)/2 + j]^{d_j}$ B. Here $R_j$ and $R'_j$ are in computable forms and the various quantities $A_o, A_o^{(r)}, \Delta_o$ in $R_j$ and the corresponding quantities $A'_o, A'_o^{(r)}, \Delta'_o$ in $R'_j$ will be evaluated with the help of the following lemmas.

### LEMMA 4.1.

$$\frac{\partial^{r+1}}{\partial h^{r+1}} \{ \log \prod_{j=1}^{k} \Gamma(\alpha_j + h) \} = \sum_{j=1}^{k} \psi(\alpha_j + h) \quad \text{for } r=0, \quad (4.9)$$

$$= (-1)^{r-1} r! \sum_{j=1}^{k} \zeta(r+1, \alpha_j+h) \quad \text{for } r \ge 1.$$

This result follows from the definitions (2.2) and (2.3).

### LEMMA 4.2.

$$[\alpha-(p-2)/2+j]^{c_j} \frac{\Gamma^{q-(p+1)/2}(\alpha-1/2) \Gamma^q(\alpha-3/2) \dots \Gamma^{q-\alpha-(p-2)/2} \zeta(\alpha-1/2) (p+1)/2}{\alpha-1/2} \quad \dots$$
10

...

$$ = \Gamma^{q-(p+1)/2} (\alpha-1/2) \Gamma^q (\alpha-3/2) \dots \Gamma^q [\alpha-(p-2)/2]^{j+2}] \quad (4.10) $$

$$ \times \Gamma^{q(j+2)} [\alpha-(p-2)/2+j+1] / \{(\alpha-1/2)^{(p+1)/2} [\alpha-(p-2)/2+j-1]^q\} \\ \times [\alpha-(p-2)/2+j-2]^{q(j-1)} \dots [\alpha-(p-2)/2]^q, $$

for $j = 0, 1, \dots, (p-1)/2 - 2$;

$$ = \Gamma^{q(p-1)/2} - (p+1)/2 [\alpha-(p-2)/2+j+1] \quad (4.11) $$

$$ / \{[\alpha-(p-2)/2+j-1]^{q(p-1)/2} (p+1)/2 \dots (\alpha+1/2)^{q(p-1)/2} (p+1)/2 \\ (\alpha-1/2)^{q(p-1)/2} (\alpha-3/2)^q [(p-1)/2 - 1] \dots [\alpha-(p-2)/2]^q \} $$

for $j \ge (p-1)/2 - 1$.

LEMMA 4.3.

$$ [\alpha-(p-1)/2+j]^{d_j} \Gamma^{q-(p+1)/2}(\alpha) \Gamma^q(\alpha-1) \dots \Gamma^q[\alpha-(p-1)/2] \quad (4.12) $$

$$ = \Gamma^{q-(p+1)/2}(\alpha) \Gamma^q(\alpha-1) \dots \Gamma^q[\alpha-(p-1)/2+j+2] $$

$$ \Gamma^{q(j+2)} [\alpha-(p-1)/2+j+1] / \{[\alpha-(p-1)/2+j-1]^q\} $$

$$ [\alpha-(p-1)/2+j-2]^{q(j-1)} \dots [\alpha-(p-1)/2]^q, $$

for $j = 0, 1, \dots, (p+1)/2 - 2$;

$$ = \Gamma^{(q-1)(p+1)/2} [\alpha-(p-1)/2+j+1] / \{[\alpha-(p-1)/2+j-1]^{(q-1)(p+1)/2} \quad (4.13) $$

$$ \dots (\alpha)^{(q-1)(p+1)/2} (\alpha-1)^q [(p+1)/2 - 1] (\alpha-2)^q [(p+1)/2 - 2] $$

... [α-(p-1)/2]q, for j ≥ (p+1)/2 - 1.

Now $\Delta_o, A_o, A_o^{(r)}$ are available from the Gamma product in (3.16) and lemmas 4.1 and 4.2. The corresponding quantities $\Delta_o', A_o', A_o^{(r)}$ are available from (3.16) and lemmas 4.1 and 4.3. That is,
$$ \Delta_o = \{ \Gamma^{q-(p+1)/2} [(p-2)/2 - j-1/2] \Gamma^q [p-2)/2 - j - 3/2] \} \quad (4.14) $$

$$ \dots \Gamma^q(2) \Gamma^{q(j+2)(1)} \} \{ \Gamma^{q-(p+1)/2} [(p-2)/2 - j] \Gamma^q(p-2)/2 - j - 1] $$

$$ \dots \Gamma^q [(p-2)/2 - j - (p-1)/2] \} / \{ [(p-2)/2 - j - 1/2]^{(p+1)/2} $$

$$ (-1)^{q(j)} (-2)^{q(j-1)} \dots (-j)^q \beta[(p-2)/2 - j] \} $$

$$ \text{for } j = 0, 1, \dots, (p-1)/2 - 2; $$

$$ = \{ \Gamma^{q(p-1)/2} - (p+1)/2 (1) \} \{ \Gamma^{q-(p+1)/2} [(p-2)/2 - j] \} \quad (4.15) $$

$$ \Gamma^q [(p-2)/2 - j - 1] \dots \Gamma^q [(p-2)/2 - j - (p-1)/2] \} / \{ ((-1)^2) $$

$$ \dots [(p-2)/2 - j + 1/2]^q [(p-1)/2 - (p+1)/2] [(p-2)/2 - j - 1/2]^{q(p-1)/2} $$

$$ [(p-2)/2 - j - 3/2]^q [(p-1)/2 - 1] \dots (-j)^q \beta[(p-2)/2 - j] \} $$

$$ \text{for } j \ge (p-1)/2 - 1. $$

$$ A_o = [q - (p+1)/2] \psi [(p-2)/2 - j - 1/2] + q \psi [(p-2)/2 - j - 3/2] \quad (4.16) $$

$$ + \dots + q \psi (2) + q(j+2) \psi (1) + [q - (p+1)/2] \psi [(p-2)/2 - j] $$

$$ + q \psi[(p-2)/2 - j - 1] + \dots + q \psi[(p-2)/2 - j - (p-1)/2] $$

$$ - [(p+1)/2] \{ 1 / [(p-2)/2 - j - 1/2] \} - q [j/(-1) + (j-1)/(-2) $$

$$ + \dots + 1/(-j) \} , \sum_{(t,r) \in \alpha} \psi [(p-2)/2 - j + (2r-t)/2q] , $$

$$ \text{for } j = 0, 1, \dots, (p-1)/2 - 2; $$

$$ = [q(p-1)/2 - (p+1)/2] \psi (1) + [q - (p+1)/2] \psi [(p-2)/2 - j] \quad (4.17) $$

$$ + q \psi[(p-2)/2 - j] + q \psi[(p-2)/2 - j - 1] + \dots + q \psi[(p-2)/2 - j - (p-1)/2] $$

$$ - [q(p-1)/2 - (p+1)/2] \{ 1/(-1) + 1/(-2) + \dots + 1/[(p-2)/2 - j + 1/2] \} $$

$$ - q \{ [(p-1)/2]/[(p-2)/2 - j - 1/2] + [(p-1)/2 - 1] / [(p-2)/2 - j - 3/2] $$

$$ + \dots + 1/(-j) \} - \sum_{(t,r) \in \alpha} \psi [(p-2)/2 - j + (2r-t)/2q] , \quad \text{for } j \ge (p-1)/2 - 1. $$
$A_o^{(r)}$ for $r \ge 1$ are obtained from $A_o$ by multiplying $A_o$ by $(-1)^{r-1} r!$
replacing $\psi(\cdot)$ by $\zeta(r+1, \cdot)$ and raising the denominators of the terms not
containing $\psi(\cdot)$ to the power $r+1$. For example, corresponding to the
terms $(-q)[j/(-1)+(j-1)/(-2)+\dots+1/(-j)]$ in $A_o$, we have,
$(-1)^{r-1}r! (+q)[j/(-1)^{r+1}+(j-1)/(-2)^{r+1}+\dots+1/(-j)^{r+1}]$ in $A_o^{(r)}$. In a
similar fashion $A_o^{\prime(r)}$ are obtained from $A_o'$ and hence we will give only
$A_o'$ and $\Delta_o'$ in the following discussion.

$$
\Delta_o' = \left\{ \Gamma^{q-(p+1)/2} \left[ (p-1)/2 - \frac{j-1}{2} \right] \Gamma^{q} \left[ \frac{(p-1)}{2j-3} \right] \right\} \quad (4.18)
$$

$$
\dots \Gamma^q \left[ \frac{(p-1)}{2j-\frac{p-2}{2}} \right] \left\{ \Gamma^{\frac{q-(p+1)}{2}} \left[ \frac{(p-1)}{2j} \right] \right\}
$$

$$
\Gamma^q \left[ \frac{(p-1)}{2j-1} \right] \dots \Gamma^q \left( \frac{2}{2j} \Gamma^{q(j+2)} \left( \frac{1}{2} \right) \right) \quad / \quad \left[ \frac{(p-1)}{2j-1/2} \right]^{(p+1)/2}
$$

$$
(-1)^{q(j)} \quad (-2)^{q(j-1)} \quad \dots \quad (-j)^q \quad \beta \left[ \frac{(p-1)}{2-j} \right] ,
$$

$$
\text{for } j = 0, 1, \dots, \frac{p+1}{2} - 2;
$$

$$
= \left\{ \Gamma^{\frac{q-(p+1)}{2}} \left[ \frac{(p-1)}{2j-1/2} \right] \Gamma^{\frac{q}{2}} \left[ \frac{(p-1)}{2j-3/2} \right] \right\} \quad (4.19)
$$

$$
\dots \Gamma^q \left[ \frac{(p-1)}{2j-\frac{p-2}{2}} \right] \Gamma^{\frac{q-(p+1)}{2}} \left( \frac{1}{2} \right) \left\{ \left[ \frac{(p-1)}{2j-\frac{1}{2}} \right]^{\frac{p+1}{2}} \right\}
$$

$$
\left[ \frac{(-1)}{2} \dots \frac{(p-1)}{2j} \right]^{\left( \frac{q-1}{2} \right)} \left( \frac{p+1}{2} \right)
$$

$$
\left[ \frac{(p-1)}{2j-1} \right]^q \left[ \frac{p+1}{2} \right]^{\frac{1}{2}} \dots \left[ \frac{(-j)^q}{2} \right] \beta \left[ \frac{(p-1)}{2} \right]
$$

$$
\text{for } j \ge \frac{(p+1)}{2} - 1 .
$$

$$
A_o' = \left[ \frac{q-(p+1)}{2} \right] \psi \left[ \frac{(p-1)}{2j-1} \right] + q \psi \left[ \frac{(p-1)}{2j-3} \right] \quad (4.20)
$$

$$
+ \dots + q \psi \left[ \frac{(p-1)}{2j-\frac{p-2}{2}} \right] + \left[ \frac{q-(p+1)}{2} \right] \psi \left[ \frac{(p-1)}{2j} \right]
$$

$$
+ q \psi \left[ \frac{(p-1)}{2j-1} \right] + \dots + q \psi \left( \frac{2}{2j} \right) + q \left( \frac{j+2}{2j} \right) \psi \left( \frac{1}{2j} \right)
$$

$$
- \left[ \frac{(p+1)}{2} \right] \left\{ \frac{1}{\left[ \frac{(p-1)}{2j-1} \right]} \right\} - q \left[ \frac{j}{(-1)} + \frac{j-1}{(-2)} \right]
$$

$$
+ \dots + \frac{1}{(-j)} \left\{ - \sum_{(t,r)} \epsilon_{\alpha} \psi \left[ \frac{(p-1)}{2j} + \frac{2r-t}{2q} \right] \right\},
$$

$$
\text{for } j = 0, 1, \dots, \frac{p+1}{2} - 2 ;
$$
$$ 
\begin{aligned}
&= \left[ \frac{q-(p+1)}{2} \right] \psi \left[ \frac{(p-1)}{2-j-1/2} \right] + q \psi \left[ \frac{(p-1)}{2-j-3/2} \right] + \\
& \quad + q \psi \left[ \frac{(p-1)}{2-j-(p-2)/2} \right] + \left[ \frac{(q-1)}{p+1} \right] \psi \left( \frac{1}{1} \right) \\
& \quad - \left[ \frac{(p+1)}{2} \right] \left\{ \frac{1}{\left[ \frac{(p-1)}{2-j-1/2} \right]} - \frac{1}{(q-1) \left[ \frac{(p+1)}{2} \right]} \right\} \\
& \quad + \frac{1}{(-2)} + \dots + \frac{1}{\left[ \frac{(p-1)}{2-j} \right]} - q \left\{ \frac{1}{\left[ \frac{(p+1)}{2-1} \right]} - \frac{1}{\left[ \frac{(p-1)}{2-j-1} \right]} \right\} \\
& \quad + \frac{1}{\left[ \frac{(p+1)}{2-2} \right]} - \frac{1}{\left[ \frac{(p-1)}{2-j-2} \right]} + \dots + \frac{1}{(-j)} \\
& \quad - \sum_{(t,r) \in \alpha} \psi \left[ \frac{(p-1)}{2-j} + \frac{(2r-t)}{2q} \right], \quad \text{for } j \ge \frac{(p+1)}{2} - 1.
\end{aligned}
\quad (4.21) 
$$

Here (4.14), (4.15), (4.18) and (4.19) can be simplified into factorials and the Gammas with positive arguments. $A_o$ and $A_o^{(r)}$ can be simplified by using the properties of $\psi$-functions. $A_o^{(r)}$ and $A_o^{(r)}$ can be simplified by using the properties of Zeta functions. But these simplifications are not done because for computational elegance the above representations seem to be better rather than their simplified forms. If necessary, simplifications can be achieved by using the results Erdélyi (1953, p.3, (5)-(7), p.16, (8)-(12)).

Lemmas corresponding to lemmas 4.2 and 4.3 can be written down for case I and case III by simple comparison. Then by using these lemmas the density function for case I and III can be written down by comparing the density function in case II. Hence the discussion is omitted. For the cases $q < p$, the technique developed in this article can be used, after obtaining the nature of the cancellation of the Gammas with the help of (3.12). It is evident that the cancellation depends upon the values of $p$ and $q$. Hence this case will no be discussed here.

Thus the density is case II is obtained from (4.1) and the various quantities appearing in (4.1) are obtained from (4.7), (4.8), (4.14)-(4.21). In section 5 we will indicate the method of obtaining the cumulative distribution function.

## 5. THE CUMULATIVE DISTRIBUTION.

The cumulative distribution,

$$ F(x) = \int_{0}^{x} f(v) \, dv, \quad 0 < x < 1 \qquad (5.1) $$
is obtained from (4.1) with the help of the following lemma. [Lemma 5.1].
For $b > 0$, $k = 0, 1, \dots$, $0 < x < 1$,

$$
\int_0^u x^b (-\log x)^k \, dx = \left[ \frac{u^{b+1}}{(k+1)} \right] \sum_{r=0}^k \left[ \frac{(k+1)k \dots (k-r+1)}{(b+1)^{r+1}} \right] \quad (5.2) \\
\times \frac{(-\log u)^{k-r}}{(b+1)^r} .
$$

This lemma is easily proved by successive integration by parts. The application of lemma 5.1 in (4.1) is apparent and hence the details of the discussion is omitted. Now the density function in (4.1) as well as the cumulative distribution function in (5.1) are in computable forms. There are only three parameters, namely, p,q,n'. Hence the percentage points can be computed without any difficulty.
RÉFÉRENCES

[1] ANDERSON (T.W.) (1958). *An Introduction to Multivariate Statistical Analysis*. Wiley, New York.

[2] BARTLETT (M.S.) (1937). *Properties of sufficiency and statistical tests*. Proc. Roy. Soc. A, 160, 268-282.

[3] BOX (G.E.P.) (1949). *A general distribution theory for a class of like- lihood criteria*. Biometrika. 36, 317-346.

[4] BROWN (G)) (1939). On the power of the L₁-test for equality of se- veral variances. Ann. Math. Statist. 10, 119-128.

[5] ERDELYI (A. et al.) (1953). *Higher Transcendental Functions*. Vol. I. McGraw-Hill, New York.

[6] MEIJER (C.S.) (1946). On the G-function. I-VIII, Proc. Kon. Neder. Acad. Van. Wet. 49, 227-237, 344-356, 457-469, 632-641, 756-772, 936-943, 1063-1072, 1165-1175).

[7] PEARSON (E.S.) & WILKS (S.S.) (1933). Methods of statistical analysis appropriate for k samples of two variables. Biometrika. 25, 353-378.

[8] RAO (C.R.) (1965). *Linear Statistical Inference and Its Applications*. Wiley, New York.

[9] SCHEFFE (H.) (1942). On the ratio of the variances of two normal pop- ulations. Ann. Math. Statist. 13, 371-388.

[10] TITCHMARSH (E.C.) (1951). *The Theory of Riemann Zeta Function*. Oxford University Press.

Department of Mathematics
McGill University

MONTREAL 110. CANADA
Reçu en Février 1970.