NOTE ON SOME FORMULAS FOR
~JEIGHTED SUMS OF ZONAL POLYNOMIALS 1
by
NARIAKI SUGIURA
2
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mineo Series No. 610
JANUARY 1969
1 The research was partially sponsored by the National Science Foundation Grant
No. GU-2059, U. S. Air Force Grant AFOSR 68-1415, and the Sakko-kai Foundation.
2 University of North Carolina and Hiroshima University.
Note On Some Formulas for Weighted Sums of Zonal Polynomials t
by
Nariaki Sugiura
1.
Introduction.
The first purpose of this paper is to prove a stronger
result than the previous lemma for the sum of zonal polynomials given in Sugiura
and Fujikoshi [5], which played an important role in deriving the asymptotic
expansions of the non-null distributions of the likelihood ratio criteria in
multivariate analysis by these authors, Sugiura [6] and of the generalized
variance by Fujikoshi [1].
THEOREM 1.
Let
K
be the zonal polynomial of degree
K = {k , k , ... , k }
2
l
p
to a partition
p x p
C (Z)
positive definite matrix
Z.
of
k
corresponding
for a
k
Put
P
=
l: k (k - a)
a
a
a=l
P
l: k (4k - 6ak + 3( 2).
a
a
a
a=l
(1.1)
Then the following equalities hold:
(1. 2)
l: al(K)CK(Z)
(K)
(1. 3)
l: a (K)2C (Z)
l
K
(K)
k(k-l) trZ 2 (trZ) k-2
k(k-l) [{trZ 2 + (trZ)2} (trz)k-2
+ 4(k-2) trZ 3 (trZ)k-3 + (k-2)(k-3)(trZ 2 )2 (trZ)k-4]
l:
(1. 4)
(K)
a (K)C (Z) = k[(trZ)k + 3(k-l) {trZ 2 + (trZ)2} (trZ)k-2
2
K
+ 4(k-l) (k-2) (k-3) trZ 3 (trz)k-3],
where the symbol
l:(K)
means the sum of all possible partition
K
of
k.
t The research was partially sponsored by the National Science Foundation Grant
No. GU-2059, United States Air Force Grant AFOSR-68-l415, and the Sakko-kai
Foundation.
2
Dividing by
k!
on both sides of each of the equations (1.2), (1.3) and
(1.4) and summing with respect to
k
from zero to infinity, we obtain the
lemma given by Sugiura and FUjikoshi [5].
The second purpose of this paper is to give an alternative proof of the
following theorem due to Fujikoshi [2], by using a differential equation for
zonal polynomials obtained recently by James [4].
THEOREM 2. (Fujikoshi).
(1. 5)
(b)
K
=
~
With the same notation as in Theorem 1, put
(b - a;l) (b + 1 - a;l) ... (b + k
a=l
a
_ 1 _ a;l).
Then the following equalities hold:
00
(1. 6)
~
~
k=O
(K)
00
(1. 7)
~
~
k=O
(K)
(b) a (K)C (2)
1
K
k!
=
K
(b) a (K)2c (z)
l
K
k!
%II -
= ~ II -
K
zl-b {(2b + 1) trW 2 + (trW)2}
zl-b {(2b + 1)(2b 2 + b + 2) (trW 2 )2
+ 2(2b 2 + b + 2)trW2(trW)2 + 2(8b 2 + lOb + 5)trW 4 + 8(2b + 1)trW3 trW
+ b(trW)4 + 8(2b 2 + 3b + 2) trW 3 + l2(2b + 1)trW 2 trW + 4(trW)3
+ 2(2b + 1) (trW)2 + 2(2b + 3) trW2} ,
where the positive definite matrix
less than one, and
2.
W = 2(1 - Z)
Proof of Theorem 1.
.
k
with respect to the
C (Z)
K
p
is a homogeneous
characteristic roots
it is sufficient to prove the equalities in Theorem 1 and Theorem 2,
Z,
when
is assumed to have characteristic roots
Since the zonal polynomial
symmetric polynomial of degree
of
-1
2
Z
is a diagonal matrix
Y
= diag(Y1' Y2' ... , yp).
Fujikoshi [2] has
shown, in the proof of Theorem 2, that the following differential relations
hold:
(2.1)
3
(2.2)
where the symbol
(~(l
o =
(0 ..
+
0
means a matrix of differential operators given by
o.1J. )0/00 1J
.. ),
operating on a positive definite matrix
is a Kronecker delta.)
1J
~
= (0 .. ),
The proof of this formula is based on the asymp-
totic expression of the equality (1.22) in Sugiura and Fujikoshi [5].
formula
P
~
al(K)CK(Y) =
(K)
(2.4)
~
(K)
By the
(tr~)k in James [3], we have from (2.1)
~(K) CK(~)
(2.3)
~
1J
a=l
y2(02/ 00 2 )(tr~)
a
aa
k
I"
k 2
Y = k(k-l)try 2 (trY) -
~=
a (K)2C (Y)
K
l
P
k(k-l) {~ y2(02/ 00 2) + ~
~
a=l a
aa
a <
y~yo(02/00~0)} tr~2(tr~)k-21,,=y
~ ~
~~
~
{2try2(tr~)k-2 + 4(k-2)try3(tr~)k-3 + (k-2)(k-3)(try2)2(tr~)k-4
= k(k-l)
+
(3
~
2
a
<
which imply equalities (1.2) and (1.3) respectively.
From (2.2) we have
(2.5)
~
(K)
a (K)C (Y)
2
K
+ 2
a
< (3
which yields (1.4) of Theorem 1.
It is interesting to note that the first two equations (1.2) and (1.3) in
Theorem 1 can also be obtained from the following linear partial differential
equation of second degree derived by James [4]:
(2.6)
~
a=l
y2(02/ oy 2)C (Y) +
a
a K
~
a~(3
y2(y - yo)-l(%y )C (Y)
a a
~
a K
Summing both sides of the above formula with respect to
equation (1.2).
Operating
~(K)al (K)
K for fixed
k,
yields
on both sides of (2.6) yields equation (1.3).
4
Alternative proof of Theorem 2.
3.
00
(3.1)
l:
l:
k=O (K)
Noting that
(b) C (2)
KK
k!
b
II - 21- ,
when all characteristic roots of positive definite matrix
2
are less than one
(James [3]), we can get from (2.6)
(3.2)
(b) al(K)C (Y)
00
l:
k=O
K
l:
(K)
K
k!
- (p-l) (d/dt) II - tY I-b
2
= b II - Y I
{(b+l)trW +
where
W = Y(I_y)-l.
by noting
l:
ex~S
y 2 (y
ex
b
It=l
- yS)
-1-1
(1 -
ex
YN)
- (p-l)trW}
u.
The second term in the above equation can be simplified
(I_y)-l = I + W as
(3.3)
ex
<
S
= ~{2trW tr(I + W) - (trW)2 - 2trW(I + W) + trW 2 }
=
~{2(p-l)trW
+ (trW)2 - tr W2 } ,
which implies the first equation (1.6) in Theorem 2.
From the differential
equation (2.6), we have
(3.4)
00
y2
(b) al(K) 2C (Y)
K
K
l:
ex~S
k!
- (p-l) (d/dt) II - tYIwhere
fey) = (b/2) {(2b + l)tr
w2
+ (trW)2}
~} fey) II _ YI- b
ex
Yex-YS
Yex
b
with
f(tY) It=l '
W = Y(I_y)-l.
The first
term in the right hand side of (3.4) can be verified after some computation as
5
(3.5)
~ II -
yl-b {b(b+l) (2b+l) (tiW 2)2 + b(b+l)trW 2 (trW)2 + 8(b+l)2 trWLf
+ 4(b+l)trW trW 3 + 4(2b 2+5b+3)trW 3 + 4(b+l)trW trW 2 + 4(b+l)trW2}
The second term in (3.4) can be written as
(3.6)
b
I-b
I
-1
-3
2
-1
-2
3
Ya(Ya-y~) (l- Ya)
+ 2trW a~~ Ya(Ya-y~) (l-Ya)
[2(2b+l) a~S
2 ,I - Y
+ b{(2b+l)trW 2 + (trW)2}
2
y (y -y)
a a 13
-1
(I-v)
-a
-1
].
Noting that
(3.7)
y3
_ _ _.=.a
L:
a~13 (y -y ) (l-y )3
a S
a
y2
=
1,
a < 13
_ _ _....:a~_ _
+
(l-Ya) 3 (l-y 13)
y~YQ
"'...:..1-''--_ _
(1-Ya)2(1-YI3)2
+
y2
. .:. 13
_ }
(l-Ya)(l-yS) 3
(3.8)
L: y2(y _y )-l(l_y )-2
a~S a a 13
a
trW 3 + trW 2trW + (trW)2 + (p-2)trW2 + (p-l)trW,
we can simplify the second term in (3.4) as
(3.9)
%II -
yl-b [(b 2+2)trW2(trW)2 +
~b(trW)4
+
(1-~b)(2b+l)(trW2)2
- 3(2b+l)trW 4 + 4btrW 3 trW + {8b + 2 + (2b 2 + b + 2)(p-l)} trW 2 trW
+ {b(p-l) + 2}(trW)3 + 2(p-3) (2b+l)trW 3 + (2b+l)(2p-3)trW2 + {2b + 1 +
2(p-l)}(trW)2].
The third term in (3.4) can be written as
6
(3.10)
(d/dt)
11-
tyl-b f(ty)lt=l = ~blI - yl-b {2(2b+1)trl.;r3 + (2b 2+b+2)trH 2 trH
+ b(trW)3 + 2(trW)2 + 2(2b+1)tr"l\,2}.
Substituting (3.5), (3.9), and (3.10) for the right hand side of (3.4), we
can derive the second equality (1.7) in Theorem 2.
Fujikoshi [2] obtained a futher formula concerning
based on the equality (2.2).
proof from the formula (2.6).
It
see~s
difficult, however, to give an alternative
7
REFERENCES
[1]
Fujikoshi, Y. (1968).
Asymptotic expansion of tlle distribution of the
generalized variance in the non-central case.
University. 32.
[2]
FUjikoshi, Y. (1968).
J. Sci. Hiroshima
In press.
Asymptotic expansions of the distributions of
some statistics in multivariate analysis.
(Reported in the
symposium on multivariate analysis at the Research Institute
for Mathematical S dences.
[3]
James, A.T. (1964).
(Kyoto)
August, 1968)
The distribution of matrix variates and latent
roots derived from normal samples.
Ann. Math. Statist. 35
475-501.
[4]
James, A.T.
(1968).
Calculation of zonal polynomial coefficients by
use of the Laplace-Beltrami operator.
Ann. Math. Statist. 39
1711-1718.
[5]
Sugiura, N. and Fujikoshi, Y.
(1969).
Asymptotic expansions of the non-
null distributions of the likelihood ratio criteria for mu1tivariate linear hypothesis and independence.
40
[6]
Sugiura, N.
No.3.
(1969)
Ann. Math. Statist.
To appear.
Asymptotic distributions of the likelihood ratio
criteria for covariance matrix under local alternatives.
To appear.