This research was supported by the National Science Foundation under
Grant No. GU-2059, the U.S. Air Force Office of Scientific Research under
Contract No. AFOSR-68-1415 and the Sakko-kai Foundation.
THE ASYMPTOTIC DISTRIBUTIONS OF THE STATISTICS BASED ON THE
COMPLEX GAUSSIAN DISTRIBUTION.
by
TAKESI HAYAKAWA
University of North Carolina and
Institute of Statistical Mathematics
Institute of Statistics Mimeo Series No. 654
Department of Statistics
University of North Carolina at Chapel Hill
DECEMBER 1970
ii
The asymptotic distributions of the statistics based on the
complex Gaussian distribution.
by
Takesi Hayakawa
University of North Carolina and
Institute of Statistical Mathematics
ABSTRACT
This paper considers the asymptotic distributions of the statistics
based on the complex Gaussian distribution.
To treat this problem, we
define the hermitian differential operator matrix and give the fundamental formulas of the zonal polynomials.
CLASSIFICATION NUMBER
40
A LIST OF KEY WORDS AND PHRASE.
(i)
a zonal polynomial.
CK (S).
(ii)
(iii)
(iv)
K: kappa
hermitian matrix, hermitian differential operator matrix.
unitary group.
0aS:
U(n), U(m ) , etc.
2
Kronecker's delta.
The asymptotic distributions of the statistics based on the
complex Gaussian distribution.
by
Takesi Hayakawa
University of North Carolina and
Institute of Statistical Mathematics
1.
Introduction and Notations.
Recently the asymptotic distribution of the statistics based on the
multivariate normal samples were derived by the use of the fundamental formulas of the series of the zonal polynomials [1], [2], [7].
The purpose
of this paper is to give the asymptotic distributions of the statistics
based on the complex multivariate Gaussian distribution which was developed
by Goodman, N.R. [3], James, A.T. [4] and Khatri, C.G. [5], [6].
To obtain
these distributions, we need also the fundamental formulas of the series of
the zonal polynomials of the positive definite hermitian matrix,
notice in this paper, we assume that all the matrices are
mxm
If we do not
positive
definite hermitian matrices.
Let
S
roots are
be a positive definite hermitian matrix whose characteristic
AI, ••• ,A
m
such that
Al
> ••• >
A
m
>
0
be a diagonal matrix whose diagonal elements are
order.
CK (S)
Let
partition
K
of
be a zonal polymonial of
k
into not more than
S,
m parts.
and
A
AI, ••• ,A
m
in a descending
which corresponds to the
It can be represented
This research was supported by the National Science Foundation under
Grant No. GU-2059, the U.S. Air Force Office of Scientific Research under
Contract No. AFOSR-68-1415 and the Sakko-kai Foundation.
2
by
cK (S)
where
X[K](l)
=
is the dimension of the representation
metric group and
X{K}(S)
[K]
of the sym-
is the character of the representation
{K}
of
the general linear group [4].
Let
00
C (S)C (T)
L [all K·"[a]
P K K ~ K
k=O K
[b ] ••• [b]
lK
,
k! C (I )
qK
K m
where
m
[ a]
=
K
IT
We denote as
Let
k
l
+
..
F ( ••• , ••• , S)
P q
Sand
(a)
a
a=l
k
,
(a-a+l)k
+ k
=
m'
=
x
k
l
~
...
a(a+l)
~
k
m
~
(m)
F
( ••• , " . , S, I)
P q
(a+x-l),
O.
if
T=1 .
m
R be a positive definite hermitian matrices and
T be
also a hermitian matrix, then
(1.1)
_ f
etr(-RS) (detR)a-mCK(RT)dR
~
r (a,K)(detS)
-a~-l
m
C (TS
K
),
R'=R>O
where
(1.2)
~rm( a,K ) = 'IT~(m-l) rrma=l r(a+k a -(a-I»,
_J
(detR)a-mdet(I-R)b-mCK(RS)dR =
I>R'=R>O
where
rm(a) =
and
'IT~(m-l) rr:=l r(a-(a-l»
r
r
(a,K) (b)
m
m
r (a+b,K)
m
r (a,K) /[a] •
K
m
C (S),
K
3
Let
X be an
nxn
on the unitary group
f
(1. 3)
arbitrary complex matrix and
U(n)
of order
etr(XU+U'X')d(U) =
n,
U be a unitary matrix
then
OF\ (n, XX'),
U(n)
where
d(U)
is the unitary invariant measure of the unitary group with
total volume unity.
We use the following notations.
X be an
Let
mxn
(m~n)
matrix which has a complex Gaussian distribtuion with mean
variance matrix
E,
then we denote as
X
~
CN (M,E).
Let
m
tian matrix which has a complex Wishart distribution of
dom with a non-central matrix
2.
Q,
then we denote as
n
S
~
complex
M
mxn
S
and co-
be an hermi-
degrees of freeCW(E, n,
~).
The fundamental formulas of the sum of the zonal polynomials.
In this section, we consider only
matrices.
Let
mxm
E be hermitian matrix and
,I =
L
I'
E
where
operator matrix
positive definite hermitian
d
I
(0 as) ,
a,S
1,2,: .. ,m,
= -E I • We here define the hermitian differential
as follows.
(2.1)
and
1-6
=
=
and
=
2
as
4
ZR
From the symmetry of
a
R
R
= a
as
Sa
and
a
and the skew symmetry of
1
as
Hence
a
and
R
a
we can see
are a symmetric and a skew
1
SYmmetric differential operator matrices, respectively.
Let
f(Z)
belongs to
be a real valued function of an hermitian matrix
00
C,
then we have a Taylor series expansion of
Z = Za
neighborhood at
f(Z)
Z,
and it
in the
as follows.
f(L)
(2.2)
We can show easily that (2.2) is same as (2.3) if
=
f(5)
(2.3)
is an hermitian matrix.
5
The following lemmas are fundamental.
Lemma 1.
K
Let
=
Z:=l
K be a partition of
(k 1 ,··· ,km) ,
k a (k a -2a)
k = k1+"'+k ,
m
and
k
into not more than
k
l
~
•..
~
k
~
m
m
2
2 2
-a ()
2Z a=l k a (k a -3ak a +a ),
2 K =
0
m parts, i.e.,
and let
then
(2.4)
(2.5)
-2 (K) - 2a
- (K) + 6ka
- (K) - 6a
- (K) + 3k2
- (Z)
{3a
- 2k}C
K
Z
1
l
1
=
where
Proof.
(2.6)
[8(tr(Aa)3) + 3(tr(Aa)2)2]C (Z)l =A
K
z
A = diag(A1, •.. ,A )
m
is a diagonal matrix of latent roots of
From (1.1), we have
nmn
(n)(deU)n
m
r
_
J etr(-nL-1R) (detR)n-mCK(R)dR
R'=R>O
Z.
5
We can see easily that the L.H.S. of (2.6) is invariant under the transformation
R = UWU'
such that
L
diagonal latent roots matrix and
= UAD'
where
UEU(m).
A = diag(A1, ... ,A )
m
is a
Hence we can rewrite L.H.S. of
(2.6) as (2.7),
(2.7)
n
mn
~
r m (n)(detA)
n
-f
etr(-nA
-1
W)(detM
n-~
C
K
(W)dW.
W'=W>O
~
Here we expand
C (W)
K
into a Taylor series expansion in the neighborhood at
W = A by the use of (2.3).
llUl
(2.8)
n
(n) (detA)n
m
r
_
Then we have a following asymptotic expansion
f
etr(-nA-1W)(detW)n-metr«W-A)3)dWCK(L) IL=A
W'=W>O
On the other hand, R.H.S. of (2.6) also have an asymptotic expansion such
that
(2.9)
Hence by comparing with the both side of order
Lemma 1.
lin
and
2
lin ,
we have
6
Lemma 2.
00
I
I
k=r
K
(2.10)
xkC (l:)
K
(k-r) !
x
.
r
r
(trl:) etr(xl:)
(2.10) holds for all integers.
k~
00
I
I
k=O
K
(2.11)
k~
00
I
I
k=r
K
(2.12)
~
x a (K) C (L)
K
1
k!
2
(x trL 2-xtrL)etr(xL).
~
x a (K) C (L)
1
K
(k-r)!
=
{xr+2trL2(trL)r _ x r +1 (trL)r+1
+ 2rx
r+1
r-1
r
r
2
trL (trL)
- rx (trL)
+ r(r-1)x r trL 2 (trL) r-2 } etr(xL).
k~2
00
I
I
k=O
K
(2.13)
~
x a (K) C (l:)
1
K
k!
2 2 + 4x 3trL 3 - 2x 3 trLtrL 2
{x 4 (trL)
2
2
+ 3x (trL) - 4xtrL + xtrL} etr(xL).
k~
00
I
I
k=O
K
(2.14)
~
x a (K) C (L)
K
2
k!
=
322
2
{2x trL 3 + 3x (trL) - 3x trL 2
+ 2xtrL} etr(xL).
Proof.
Since
etr(xL) = Lk=O LK (xkCK(L)/kl),
we have (2.10) by differen-
tiation or integration on both sides, successively.
From Lemma 1, we know
=
Multiply
xk/kl
on both sides and sum form
k=O
to infinite, we have
7
k~
00
x a (K)
1
~
C 0:)
K
2
tr(Aa) etr(xI)
k!
k=O
1,"_,
L-d
xtrIetr(xE).
K
From the definition of
a,
the first term of R.H.S. becomes
m
m
2
tr(Aa) etr(xI)II=A
I
AaASaaSaasexp(x
a,S=l
°aa) II=A
a=l
etr(xL) I I=A
=
Hence we obtain (2.11).
(2.12) can be obtained by applying the Leibnitz
formula of differentiation to (2.11).
As we can show (2.13) and (2.14) by
the same way as one of [7], we will omit.
3.
The asymptotic distribution of the statistics based on the non-central
complex Wishart matrix.
Recently Fujikoffirl [1], [2] has obtained the asymptotic distributions
of a generalized variance and a trace of a non-central Wishart Matrix.
In
this section, we give the asymptotic distribution of these statistics based
on a complex non-central Wishart matrix by the completely same way as [1]
and [2].
Theorem 1.
~
Let
nS
be distributed with
CW(I,n,~)
is a constant matrix with respect to
n.
Put
and let's assume that
8
In/m log{detS/detE}.
(3.1)
Then we have
~(x) +
Pr{A~x}
(3.2)
G
1 +
&"
G
-l +
mn
G
3
mn&"
+ O(1/n 2 ).
where
G
l
=
2
- ¢' (x) _ ~ ¢ (2) (x) +..!. ~(3) (x)
2
6
G
2
=
¢(2)(x) {m2 (m 2+2)
¢ (3) (x) 2
+ (trS1)2} +
m trS1
2
4
2
+
G
3
where
=
¢(4)(x)
2
(m +1-2trS1) + 7~ ~(6)(x)
12
~'l(;) {m 2 (2m 2_l) + 6mtrS1 2 }
+
¢ (3) (x)
22222
.
{m (m +2)(m +4) - 6m (m +2)trS1 - 8(trS1)3}
48
+
¢ (5) (x)
4
2
{5m + 20m + 12 - 10(m2+l)trS1 + 10 (trS1) 2}
240
+
<1>(9) (x)
~ (7) (x)
2
(m + 2 - 2trS1) +
1296
144
~(k)(x)
denotes the k-th derivative of the standard normal dis-
tribution function
Proof.
(3.3)
~(x).
We can easily obtain the characteristic function of
(..!.)it&
etr( -'"')
ab
n
rm(n+it/n/m)
rm (n)
A as follows:
~ ( n+itvntm,
r-;-/ n, S1 ) .
lF
l
Hence by expanding (3.3) as the series of order l/';~ and by applying Lemma
9
2, we have the asymptotic expansion of
¢(t).
Therefore, by inverting this
series, we obtain the result (3.2).
Let
Case 1.
is a constant matrix with respect to
~
~
Put
=
nS
CW(Z,n,~).
Theorem 2.
be distributed with
(~/T)(trS-trZ),
<1>(x) -
T2 = trZ 2 ,
where
.1:_
~
T
+
1
n.
1.
n
T
then
+ O(1/n 2) ,
__1_ T
2
n~
3
where
Case 2.
~
+
<1>(4) (x)
4
4
<1>(6) (x) (t 'C3)2 •
{3trZ + 3trHrZ } + - - - : ?6- ' r l.>
12T 2
18T
+
<p (7) (x)
10T 7
+
<1>(9) (x)
(trZ 3)3 •
162T 9
= ne, where
e
{.L.
12
1
3 2
trL:4trZ 3 + 18
trL:(trZ ) }
is a constant matrix.
Put
~
a
Then we have
(trZ - tr(I+2e)Z),
=
10
=
(3.4)
~(x)
M1
M2
M3
2
- - - + -- - --- + O(l/n ),
~
n
n~
where
M
1
=
M
2
=
M
3
=
~(3) (x)
~(4)(x)
Case 1.
tr(I+58)L:4 +
40 4
~(5)(x)
~ (9) (x)
As
~(6)(x)
tr(I+58)E 5 + ~
505
+
Proof.
tr(I+38)L: :3
30 3
540 9
{tr(I+38)E3}2
180 6
(7)
(x) tr(I+38)E 3tr(I+48)E 4
120 7
(tr(I+38)L3)3 .
the characteristic function of
we expand this as the series of order
det(I-cA)
1/~
~
is given by
by using the formulas such that
2
2
etr ( c dA){l + c 2 d trA +. . . } ,
-d
(I_X)-l
=
2
I + X+ X +
Hence by inverting this series, we obtain (3.3).
Case 2.
4.
This is obtained by the same way as Case 1.
The likelihood ratio criterion in a linear model.
In the linear hypothesis we have a following canonical model.
each column vectors of
Y
=
[Y1""
,~_]
"'-tl
mxn
Let the
be independently distributed
11
with the complex normal distribution with the same covariance matrix
The hypothesis
(4.1)
H and the alternative
H:
E (Ya) = Q,
K:
E(L)
K are specified by
a = 1,2,···,q1 and q2+1 , •.. ,N.
for some a (1 :5; a :5; q1) and
for a = Q2+ 1 , .•• ,N.
:1 fl,
E (Ya) = 0
L.
(q1
::;
q2) .
The likelihood ratio test for this model is expressed by
{detA/det(A+B)}N,
(4.2)
A = EN
Y y'
a=Q2+1 -a-a
where
and
B = L
q1
Y Y'
a=l -a-a
Under
buted according to the complex Wishart matrix with
and
N-q
2
A is distridegrees of freedom
B is distributed with according to the non-central complex Wishart
distribution with
Q= r
K,
r'
Put
function
(4.3)
-1
L
,
degrees of freedom and non-centrality parameter
ql
where
r
= E[Yl'''· '~l]·
A = -p logA,
where
¢(t)
is given by
of
A
pN = 2n = 2N-2q2+ql-m.
rm(n~(ql+m»rm(n(1-2it)-~(ql-m»
rm(n-~(ql-m»rm(n(1-2it)~(Ql+m»
Then the characteristic
etr(-Q) •
Note that (4.3) is not the same form as (1.26) in [7], because we can't use
the Kummer's formula since we have not a Kummer's formula for a hermitian
matrix.
We expand (4.3) as before.
12
I
=
q m
(1-2it) 1 {
mq 1 (5m 2+ 2_ ) (
1
2
q1 6
. 2
48n
(1-21t)
1
rt 1
1 { trrt
etr ( 1-2itj[1 + 2n (1-2it)3 +
2
6
+ ~2
24n
A
L
(q +m)trrt - trrt 2 _ ( q +m)trrt}
1
1
1-2it
(l-2it) 2
+ O(l/n 3»),
a
a=1 (1-2it)a
where
(4.4)
Al
=
2
-8B trrt
A2
=
2
2
2
-8B trrt - (6B+7)(trrt) + (4B + I4B + 5)trrt 2
2
2
I6B trrt + 2trrt 3 + 2 (6B+I) (trrt) - 2(2B+I)(2B+5)trrt 2
A
3
+ 6Btrrttrrt 2 •
A
4
=
-(6B-5) (trrt)
2
+ (4B+5) (B+I)trrt 2 - 12trrt 3
-
12Btrrttrrt 2
+ 3(trrt 2 )2.
B
I
= "2
(ql
+
m).
Therefore, we obtain the asymptotic expansion of
order
(4.5)
lIn
as follows.
I
--~---
q m
(1-2it) I
etr ( 2it rt)
1-2it
~(t)
with respect to the
13
2
{
2
(q l +m)trO-tr0
ql+m
}
, I +..l..
trO
+ ------- . trO
_
2n (1-2it)3
(1-2it)2
1-21t
6
+
Z
A
24~2 { a~l -(-l--Z-:-t-)-a +
Since we know that
function of the
Z
2q l m(5m +ql-6)
(l-Zit)Z
exp«2it/I-2it)trQ) (l/(l-Zit)
X2 variable with
non-central parameter
0 2 = trO,
2 (qlm+ S)
Ql m+S
is a characteristic
)
degrees of freedom and with
by inverting (4.5) we have a following
theorem of the asymptotic distribution of the likelihood ratio criterion
as follows.
Theorem 4.
In the linear statistical testing hypothesis model (4.1), we
have the asymptotic distribution of
A under the alternative
follows.
(4.6)
pr{ -p logAsx}
-
(q1+m)trOpr{X~qlm+2(02)
6
+~{ I
24n
a=l
s
X}}
K as
A
14
pN
where
and
2n = 2N - 2q2 + ql - m,
A
a
and
B are given in
(4.4).
5.
The likelihood ratio test for the independence.
Let
into
m
l
CWO:: ,N)
be distributed with
S
m
2
and
rows and columns
~ll SlJ'
=
S
8
8i2
and let's partition
(m ::::m )
l 2
l:
=
The likelihood ratio test for the independence
ternatives
K: l:12 ; 0
and
l:
as
~11
l:i2
22
S
E.
1J
l:22
H: l:12 = 0
against all al-
is given by
(5.1)
Lemma 5.
Under the alternative
~
r
(5.2)
the moment of
h
A
is expressed as
~
m
1
fm
1
Proof.
K,
(N)r
m
(N-m +Nh)
2
1
(N-m
2
)fm
(N+Nh)
1
The proof of this lemma is different from [7].
let's partition
D as
D
Let
l:
-1
D
and
15
h
A
Then the expectation of
~
1
f
N
fm(N) (detE)
is written as follows
etr(-E -1 S)(detS) N-m
S'=S>O
det S
de tS
[ detS
u
then
Let
Nh
22
dS
]
Hence
(5.3)
Since
det(I-WW')
is invariant under the transformation
and we integrate
Therefore, by using (1.3),
WU, UEU(m 2 ),
into the space of
we first project
¢(WW')
W to
¢(WW')
on the whole space such that
WW' > O.
16
Hence by applying the Hsu's lemma in a complex case and (1.2), the above
integral can be written as
(S.4)
~
=
mlm2
r
ml
(N-m 2+Nh)
-~-'-----
1T
r m (N+Nh)
l
Thus inserting (S.4) into (S.3), we have the moments of
with respect to
S11
Since
det L
rm(N)
we have (S.2).
=
and
S22
as follows
A by integration
17
Theorem 5.
The power function of the likelihood ratio criterion for the
testing of independence between two sets of variates is expressed asymptotically in the following way.
A= -(phI;;) {logA
Let
(5.1) ,
2n
is
= pN = N-(m1+m2 )
Then
Pr{A~X}
Proof.
6.
=
~(x)
- __1__ {m l m2
31;;
T
~'(x)
+
~(2)(x)
+ 16
~(3)(x)
3
trp 4} + O(l/n) •
T
The proof is done completely same way as before.
References.
[1].
Fujikoshi, Y. (1968). "Asymptotic expansion of the distribution of
the generalized variance in the non-central case." J. SCI.
Hiroshima Univ. Ser. A-I. 32, 293-299.
[2].
Fujikoshi, Y. (1968). "Asymptotic expansions of the distributions
of some statistics in multivariate analysis." Report of the
symposium on multivariate analysis at the Research Institute
for Mathematical Sciences (Kyoto).
[3].
Goodman, N.R. (1963). "Statistical analysis based on a certain
multivariate complex Gaussian distribution (An Introduction)."
Ann. Math. Statist. 34, 152-176.
[4].
James, A.T. (1964). "Distributions of matrix variates and latent
roots derived from normal samples." Ann. Math. Statist. 35,
475-501.
[5].
Khatri, C.G. (1965). "Classical statistical analysis based on a
certain multivariate complex Gaussian distribution." Ann.
Math. Statist. 36, 98-114.
[6].
Khatri, C.G. (1966). "On certain distribution problems based on
positive definite quadratic functions in normal vectors."
Ann. Math. Statist. 37, 468-479.
[7].
Sugiura, N. and Fujikoshi, Y. (1969). "Asymptotic expansions of the
non-central distributions of the likelihood ratio criteria for
multivariate linear hypothesis and independence. Ann. Math.
Statist. 40, 942-952.