I nternat. J. Math. Math. Sci.
(1981) 147-154
Vol. 4 No.
147
ON THE NONCENTRAL DISTRIBUTION OF
THE RATIO OF THE EXTREME ROOTS OF
THE WISHART MATRIX
V.B. WAIKAR
Department of Mathematics & Statistics
Miami University
45056
Oxford, Ohio
USA
(Received February 6, 1979)
ABSTRACT.
The distribution of the ratio of the extreme latent roots of the Wis-
hart matrix is useful in testing the sphericity hypothesis for a multivariate
normal population.
Let X be a p x n matrix whose columns are distributed inde-
pendently as multivariate normal with zero mean vector and covariance matrix
Further, let S
XX’
and let
11
>
> i
P
> 0 be the characteristic roots of S.
Thus S has a noncentral Wishart distribution.
tion of
fp
polynomials.
ip/l I
1
is derived.
The density of
fp
is given in terms of zonal
These results have applications in nuclear physics also.
KEY WORDS AND PHRASES.
Extreme roots, Wih distribution, Zonal polynom
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i
In this paper, the exact distribu-
Primary 62H10; Secondary 62E15.
INTRODUCTION.
The distribution of the ratio of the extreme latent roots of the Wishart
V.B. WAIKAR
148
matrix is useful in testing the sphericity hypothesis for a multivariate normal
population.
In the central (null) case, Sugiyama (1970) derived the density of
the ratio of the smallest to the largest root of the Wishart matrix when the
associated covariance matrix is the identity matrix.
(1973) derived an alternate expression which
Waikar and Schuurmann
is much superior to that given by
Sugiyama (1970) from the point of view of computing and in fact we computed some
In
tables of the percentage points which are also included in the above paper.
this paper, the author has derived an exact expression for the ratio of the
smallest to the largest root of the noncentral Wishart matrix.
has applications in nuclear physics
see Wigner
This research
(1967) ].
Constantine [i, p. 1277] defines the non-central Wishart distribution.
Anderson
2, p. 409] relates the Wishart and non-central Wishart distributions
to each other and to neighboring areas of multivariate analysis.
James
3, p.475]
gives a brief exposition of this area of multivariate analysis.
2.
PRELIMINARIES
If A is a square, nonsingular matrix its inverse and determinant are denoted
-I
respectively by A
and
The transpose, trace and exponential of the trace of
a matrix B are denoted respectively by
B’,
tr B and etr B.
Also I
note respectively a p x p identity matrix and a p x p null matrix.
we define as in James (1964).
ap; b l,...,bq;S)
pFq(al’
k=O
K
(al)K’’" (ap)K C K(s)
k.’
(b I) ...(bq)
K
K
and
P
F
,ap bl,...,bq; S,T)
(al)K" (ap) K C K(s) C K(T)
q(a I,
k:0 EK (bl)K
(bq)K CK(I p)
k.’
P
and 0
P
de-
In addition,
NONCENTRAL DISTRIBUTION OF EXTREME ROOTS OF THE WISHART MATRIX
where S and T are p x p symmetric matrices and
the integer k satisfying (i) k
(ii)
k
I
+
+
k.
k
P
I
> k
l,...,kp)
is a partition of
> 0 and
> k
>
2
(k
K
149
p
Further
P
(a
(a)
(a)k
l)/2)k.
(i-
i=l
i
a(a + l)...(a + i
(a)
k)
and finally C (S) is the zonal polynomial as defined in James (1964) and satis-
fies (tr S)
k
C
(S).
A special case of the above is
K
SI -a
IIP
IF0(a;S)
Note that if one of the a. ’s above is a negative integer say a,
l
-n then
for k _> pn + 1 all the coefficients vanish so that the function p F q reduces to
a
(finite) polynomial of degree pn (see Constantine (1963) p. 1276).
throughout the paper, whenever a partition say
(kl,..., kp)
K
Further,
of a nonnegative
integer k is defined, it will be implied that
(i)
k
> k
>
I
> 0
and
p--
(ii)
I +
+ kp
k
k
The following three lemmas are needed in the sequel.
LEMMA 2.1.
Let k and d be two nonnegative integers and let
(dl,...,dp)
and
diag (gl
gp )"
denote partitions respectively of k and d.
(bl,...,bp)
LEMMA 2.2.
Let
K
Further let G
Then
C
where 8
(kl,...,k)
P
K
8
(G).C6(G)
gK
(2 i)
C (G)
8
is a partition of the integer k
+
b.
Let G be as defined in Lemma 2.1 and further let G
(kl,...,kp+l)
..
be a partition of a nonnegative integer k.
C (G
K
I)
t=0
T
b
K,
C (G)
I
diag(l,G).
Then
(2.2)
V.B. WAIKAR
150
(tl,...,t)
P
where
is a partition of t.
The above two lemmas are stated in Khatri and Pillai (1968).
cients in (2.1) and the b-coefficients in
The g-coeffi-
(2.2) were tabulated by Khatri and
Pillai (1968) for various values of the arguments and can be obtained from them.
Throughout this paper, the following notations will be used:
P
p(p-l)/4
F (a)
P
F(a- (i- 1)/2)
i=l
p
p-i
p
(a
i,j=t
aj)
i
(a
H
i=t j=i+l
aj),
i
0 < t < p.
The following lemma can be proved by making trivial modification in the proof of
the Lemma given in Sugiyama (1967).
LEMMA 2.3.
R
diag
I
Let R
(rl,...,rp_l,l).
positive integer k.
[a
DP(t)
diag(r I,
O<rl
<"
-<r p_l
(pt + k)
Further let K
where 0 < r
(kI,
I
kp)
<...<
rp_ I
< i and let
be a partition of the
Then
<I
[RIt-(p+l)/2[p_
(rp(p/2)/
where
p
2)CK(Ip)
p(p-l)/4
F (a,<)
P
LEMMA 2.4.
rp_I)
p-i
1
R[ CK()
(r
i
(Fp(t,m)Fp((p + )/2))Irp (t
P
i=l
r(a + k
i
Let A be any p x p matrix and let
of a nonnegatlve integer k.
H
i>j=l
(i
rj)dR
(2.3)
+ (p + i)/2, )
1)/2).
(kl,...,k)
P
be a partition
Then
k
C (I
where
(gl
gp)
P
+ A)
L L a ,v C
(A)
g=0
C(Ip)/Cy(l p
(2.4)
is a partition of g.
The above lemma is stated in Constantine (1963) and some tabulations of acoefficients are also given in the same paper.
151
NONCENTRAL DISTRIBUTION OF EXTREME ROOTS OF THE WISHART MATRIX
3.
DENSITY OF THE RATIO ’OF THE SMALLEST TO THE LARGEST ROOT OF THE WISHART MATRIX,
Let X be a p x n matrix whose columns are distributed independently as multi-
variate normal with zero mean vector and covariance matrix ). and let p
XX’
ther, let S
and let
>
i
2
>’’’>
_<
n.
Fur-
> 0 be the characteristic roots of S.
p
Thus S has a noncentral Wishart distribution and the joint density of its roots
i
P
as derived by James (1964) is
(3.1)
-I
I
0F0(I
L) etr(-L)
ILl(n
i)/2
p
P
x
i<j:l
>
(Ai- Aj)’
and k(p,n)
HP
making the transformation
AI AI’ fi
i
Al,f
as
where L
diag
(Al’’’’’p)
the joint density of
h2(1,f 2
-n/2
p(n-2)/2
e
i
P
H
i>j=2
(n-p-l)/2
F
k:0
/2/(2Pn/2pp(n/2)
Ai/li
Pp(p/2)).
diag
p in (3.1), we obtain
2
CK(I
K
.
(fi- fj)
I-i
)
k
i
(f2’’’"
0F0(1 F)
h(l,f2,...,f P ):
I
C [diag(l
<
p-I
-F)]
k.’ C (I)
p
f
p
and
K
i
(kl,...,kp)
On using Lemmas 2.2 and 2.4 to expand C
as
Now, on
-PI etr(1F) IFI
0 <
etr(1F)
> 0
P
(3.2)
L
where F
A
fp)
k(p,n)I’.I
llp_ I
2
AI >’">
<
0 <
=’
f2
<’" "< f
< I
P
is a partition of the integer k.
[ diag(l
I
F)]
and further, writing
and then expanding it we can rewrite the above density as
V.B. WAIKAR
15 2
II -n/2 [(n-2)/2 e-P1 IFI II FI (n-p-l)/2
k(p,n)
(f
. ..
i>j=2
k
t
b
t=O
T
where
K,
(a
f
i
kffi0
CT(1)
C (-F)
a
C (I)
y
ap_ I)
I
J
__z;y T
g=O y
=
k’
C (I)
0 <
i
<
a
0 <
f2
(gl,...,gp
I
L
C=(F) Cy(-F)
g,yn
(-I) g
(nl,...,np_ I)
where
C
Z L
Also then
Cn(F) C(F) L
are given by Lemma 2.1.
h4(1,f2
Now note that
+ g
and the coefficients
is a partition of the integer d.
gn
C (F)
is a partition of
(a + g) + d and the coefficients
Thus the density in (3.3) becomes
,fp)
k(p,n)
t
g=O y
a
g,y
C (F)
d’.
(d I,
bp_ I)
is a
(-(n- p-
where
(b I,
(tl,...,tp_ I)
i)/2, F)
p
d=0
B
< i
Further
IF0(_(n
dp_ I)
P
n(F)
is a partition of a
are given by Lemma 2.1.
ii Fl(n-p-1)/2
<’’’< f
is a partition of g and the b
are given by Lemmas 2.2 and 2.4 respectively.
and a
a’
a=0
’
Ca(F)
is a partition of the integer a, T
partition of the integer t, y
where
I-I k
C (I
p
(33)
I:1 -/2
C (I) (-1)
Cy(1)
g
(-(n
n
’Y
p
d.’
d=O
i)/2)
g,
NONCENTRAL DISTRIBUTION OF EXTREME ROOTS OF THE WISHART MATRIX
(n-2)/2 + k
g8
8
+
a
e
-P1 FI
0 <
n
i>j=2
’
Now, on making the transformation
and then integrating out
P
(fi- f J Cs(F)
0 <
f2 <’’’< f P <
i i’ ri fl/fp
<
i
r2,...,rp_ 1
h5( 1
P
a-_O
[.
k=0
b
t=0
1
]-1
K
Cx (I).
(-i)
g
y
1)/2) a
L
d
l-p(p+l)-2+a+g+d
.2
0 < f
’
P
/1
out
i
(0 <
i
<
P
D
+ 2 )’ (3.5)
2
< i
)’ we get the marginal density of
as
h6(f p)
k(p,n)
k
i L i i
a=0
t f0 "r
I I
lgn=y
=0
n
b
a.’
’ll -n/2
"
g=0 y
(-(n- p
d’.
CK (I
k’
k=0
a
P
k; C (I)
K
(-(n- p
<
rp_ 1
f
< i
((p + 2)/2)is given by Lemma 2.3.
Now, on integrating
P
i
2
<...<
P
as
P
. IK
a
(n-2)/2+k+a-P1
f
and f
g=0 y
n
Lgot,y
LL
d=0
n
where D
i
t
"r
0 <
2,...,p- i, f
i
C (I
k(p,’n)II -n/2
k
i.
over the surface 0 < r
(using Lemma 2.3) we get the joint density of
f )=
153
C
...1(I) -1)
C (I) (-i)
g
cy ()
1)/2) 6
18
gn,
B
9-i (p. +2 2)
V.B. WAIKAR
154
-( + k + a)
p
i
+
r (@
"z
k
+ )
(3.6)
(p+l )-2+a+g+d
p
REMARK 2.
An important observation is that in the special case when (n- p
1)/2
is
which
an integer, the summation over d becomes finite (see Remark i) in (3.6)
means the noncentral density of f
P
i
P
/I
involves only two infinite sums.
The author wishes to acknowledge the support from Miami
ACKNOWLEDGMENT.
University in the form of a Summer Research Fellowship.
REFERENCES
Some noncentral distribuiton problems in multivariate
analysis, Ann. Math. Statist., 34 (1963) 1270-1285.
i.
Constantine, A. G.
2.
Anderson, T. W.
3.
James, A. T.
4.
Khatri, C. G. and Pillai, K. C. S.
Ann. Math. Statist., 17 (1946) 409-431.
Distributions of matrix variates and latent roots derived from
normal samples, Ann. Math. Statist., 35 (1964) 475-501.
On the noncentral distributions of two
test criteria in multivariate analysis of variance, Ann. Math. Statist.,
39 (1968) 215-226.
5.
Krishnaiah, P. R. and Waikar, V. B. Simultaneous tests for equality of latent
roots against certain alternatives
II, Aeros2ace Research Laboratories
Technical Report ARL, (1969) 69-0178.
6.
Sugiyama, T. On the distribution of the largest latent root of the covariance
matrix, Ann. Math. Statist., 38 (1967) 1148-1151.
7.
Suglyama, T. Joint distribution of the extreme roots of a covariance matrix,
Ann. Math. Statist., 41 (1970) 655-657.
8.
Waikar, V. B. and Schuurmann, F. J. Exact joint density of the largest and
smallest roots of the Wishart and MANOVA matrices, Utilltas Mathematlca,
4 (1973) 253-260.
9.
Wigner, E. P.
Random matrices in physics, SIAM Review, 9 (1967) 1-23.