ASYMPTOTIC NON-NULL DISTRIBUTIONS OF THE LIKELIHOOD RATIO CRITERIA
FOR COVARIANCE MATRIX UNDER LOCAL ALTERNATIVES
by
NARIAKI SUGIURA
University of North Carolina and Hiroshima University
"
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics Mineo Series No. 609
JANUARY 1969
Research partially sponsored by NSF Grant No. GU-2059 and the U.S. Air Force
Grant No. AFOSR 68-1415 and the Sakko-kai Foundation.
1
ASYMPTOTIC NON-NULL DISTRIBUTIONS OF THE LIKELIHOOD RATIO CRITERIA
FOR COVARIANCE MATRIX UNDER LOCAL ALTERNATIVES
by
Nariaki Sugiura
University of North Carolina and Hiroshima University
1.
Introduction.
lihood ratio
(= LR)
Asymptotic expansions of the distributions of the likecriteria based on a random sample from a multivariate
normal population under fixed alternative hypothesis have been derived by
Sugiura [11], (1) for the equality of covariance matrix to a given matrix, (2)
for the equality of mean vector and covariance matrix to a given vector and a
given matrix, and also by Sugiura and Fujikoshi [12], (3) for testing the hypothesis of independence between two sets of variates.
distribution of the
LR
The limiting non-null
criterion (4) for the equality of several covariance
matrices has been obtained by Sugiura [11].
These limiting non-null distributions
always degenerate at the null hypothesis so that the asymptotic formulas do not
give good approximations when the alternative hypothesis is near to the null
hypothesis, as we have experienced in calculating the approximate powers of
Bartlett's test for homogeneity of variances in Sugiura and Nagao [14].
In this paper, we shall derive limiting non-null distributions of the
LR
criteria for the problems (1) and (2) under sequences of alternatives converging
to the null hypothesis with the rate of convergence
sample size, for arbitrary positive number
of the non-null distributions in the case of
two Sections.
-y
N
,
where
N means
and then asymptotic expansions
y
y
= -12
and
y
=
1
in the next
With the help of the hypergeometric function of matrix argument
due to Constantine [2], we shall derive an asymptotic expansion of the distribution for the problem (3) in the case y
Research partially sponsored by NSF Grant No.
AFOSR 68-1415 and the Sakko-kai Foundation.
= l in
G~2059
Section 4, and an
and U.S. Air Force Grant
2
asymptotic expansion of the distribution of the modified
LR
criterion, in
Sugiura and Nagao [13], for the equality of two covariance matrices under the
sequence of alternatives with
= 1 in Section 5.
y
The formulas in this paper
can be applied to compute the approximate power, when the alternative hypothesis
is near to the null hypothesis.
2.
Asymptotic distributions of the modified
2.1. Moments of the criterion. Let the
p
LR
1
x
criterion for
= EO.
~
vectors
be a random sample from a p-variate normal distribution with unknown mean vector
~.
and covariance matrix
instead of the
LR
We shall consider the modified
LR
criterion, for testing the hypothesis
positive definite matrix) against alternatives
~
K:
I
= EO
H: E
~O'
A* ,
1
criterion
(a given
which is given by
(~)np/2 ISE- 1 \n/2 etr{- 1 ~-l S},
(2.1)
2
nO·
0
N
where the symbol
N
E
x
X.
j=l
J
IN
and
etr
n
exp • tr
means
=
N-l.
and
S
X) ,
~
(X. - X) (X.
J
J
j=l
The unbiasedness of this modified
LR
with
criterion
was proved by Sugiura and Nagao [13], the monotonicity property of which was
established by Nagao [6].
The
h-th
Al*
moment of the statistic
under
K
can be found in Anderson [1, p.266] as
2
(~)
(2.2)
nph/2 r
n
where the function
2.2.
r p (x)
is defined by
Asymptotic distributions when
limiting distribution of the statistic
(~(l+h»
--"-p----
1
r p (2"n)
TI
1
II
p(p-1)/4
-1 zn(l+h)
+ hE ~O
P
IT
r(x -
'
I
j-1
2)
j=l
1
y < 2" •
- 2logA *
1
First we shall consider the
under the sequence of altern-
3
atives
K:
(= pd)
matrix
E E- 1 / 2 = I + n -y 8 with
y > 0
o
y
8.
(- 2logA *) /n 1/2-y
1
When
y <
and positive definite
1
2' we can write the characteristic function of
as
1
1
1
Y+2"
1
y+y+1
2
1
-:::n p+i t pn
y-2 r (-:::n-itn
)
2
2
(2e) - i t pn
---,,-P-=:._-::-_
_ (1 - 2itn 2)
n
1
rp(~)
(2.3)
1
1
2itn-1/2
11----81
.
- ~+ltn
y1-2
Y-1/2
1-2itn
Applying the asymptotic formula
logr(x+h)
= log/2"; + (x+h - i)logx - x + O(lxl- 1 )
to each gamma function in the first factor of (2.3), we can see that the first
factor is equal to
y 1 2
1 + O(n - / ),
which tends to one as
n
tends to infinity.
By the formula
- logl1 - n-
(2.4)
which is valid for large
metric matrix
n
-1
n
1
zl
such that all characteristic roots of the sym-
Z have absolute values less than one, the second factor of
(2.3) can be evaluated as
1
(2.5)
exp[it{/n tr 8 - n
y+t
Hence the characteristic function of
tends to
theorem.
exp[- t
2
tr
e2 ]
as
n
1
loglE E~
y 1 2
n - / [_
1
I} -
t
210gA~
tends to infinity,
2
2
tr 8 ] + O(n
1 y
- n - tr 8
y-2
).
+ n10giE
E~ll]
which implies the following
4
Theorem 2.1.
";',
Let
A
I
be the modified
Then under the sequence of alternatives
o
<
y
<
1
2'
statistic given by (2.1).
for
K :
y
the limiting distribution of the statistic
1
Y-2 [n
(2.6)
2logA
*
l
-1
- n{tr(E EO
is normal with mean zero and variance
2tr8
Noting that the asymptotic variance
2tr(~ ~~l - 1)2 • n 2y ,
2.1
LR
2
as
2tr8
2
1
- I) - loglE ~~ I}]
n
tends to infinity.
is equal to
we can see that the limiting distribution in Theorem
is of the same form as under fixed alternative hypothesis given by Sugiura
[11].
It may be noted that Theorem 2.1
statistic
2.3.
Al*
to the
LR
Asymptotic distributions when
C(t)
=
1
r p (?)
- 2logA *
l
C(t)
Y
for
~t.
We shall now consider the
1
Y => -2·
K
Y
From (2.2), we can get the
as
r p (l2n(1-2it»
npit
(2e)
The first factor of
K
y
under
characteristic function of
n
LR
statistic.
sequence of alternative hypothesis
(2.7)
holds if we replace the modified
tu
(1-2it)
PD - Zit )
II
II -
+
n- Y
2it
_y
I-2it n
-nit
811 ' - - - - -
81
.
2"U(1-2it)
in (2.7) is simply the characteristic function of
under the null hypothesis, which was expanded asymptotically by
Sugiura [11] as
(2.8)
B
(1-2it)-P(p+1)/4[1 + ~{ 1
n 1-2it
6B
2
2
3B2
}
-3
l-2it +
2 + 4B 3 + O(n )],
where
p( 2p 2 + 3p - 1)/24
(2.9)
- p(p-1) (p+1) (p+2)/32
5
Applying the asymptotic formula (2.4) to the second factor of the characteristic
function (2.7), we can get
exp[- nit loglr + n- Y
(2.10)
81 - ~
8
(_
~ trY k-1
{
lk=2 n
k
expf
Zit
(l-Zit)loglr - I-Zit n- Y
(it~k}
l)kit + Zk-1
(l-Zit)k 1
81]
+ 0(n(l+1)Y-1)]
2
3
1-Zy it tr 8 + 1-3y tr 8 { I
3
exp[n
Z(l-Zit)
n
6
Z - I-Zit
(I-Zit)
+ n
l-4y tr 8 4 {
~-8~
1
4
6
l-Sy
3 - --"'---::-Z + I-Zit - 3} + 0 (n
)] .
(I-Zit)
(I-Zit)
Multiplying (2.8) and (Z.10), we can see that
C(t)
=
1
Z
(1_Zit)-P(p+1)/4 + 0(1)
when
y
Z
tr 8 } + (1)
(l -Z.1t )-p(p+1)/4 exp {it
Z(l-Zit)
0
when
1
y =Z
>-
(Z.l1)
which implies that the limiting distribution of
f
1
= p(p+l)/2 degrees of freedom when y
>
21
210gA *
1
is
X2
,
with
and noncentra1
1
= 2.
Further,
we can get the asymptotic expansion of the characteristic function of
- ZlogA *
1
degrees of freedom and noncentra1ity parameter
when
(Z.lZ)
y = 1
Z
tr 8 /4
when
y
as
C(t)
Z
2
3B Z-4B 3
1
6B
}
1 Z
3
Z + 3B Z + 4B
+ -Z E gZN(l-Zit)-o.]+ 0(n- ) ,
+-Z
2 - I-Zit
2
3
n 0.=0
6n { (1-2it)
u-
where
6
BZ
go
~
tr 8
BZ
~
tr 8
Z
1
3
1
Z Z
+ 3 tr 8 + 32 (tr 8 )
(Z.13)
=
g4
When
y
Z
3
1
Z Z
1
+ 6 tr 8 + 32 (tr 8) .
1
= 2'
we can get another asymptotic formula for
C(t)
from (Z.8) and
(Z .10) as
(Z.14)
C(t)
Z
3
. -f 1/Z
{it tr 8 }
tr 8
1
(l-21.t)
exp Z(l-Zit) • [1 +
{-----=---2
6fu (I-Zit)
where the coefficients
h
h
h
h
(Z .15)
(a = 0, ... , 4)
Za
h
h
B
O
are given by
3
3
4
1
- - tr 8 + (tr 8 )
18
Z 8
3
+ 4"
tr 8
4
1
3
- - (tr 8 )
6
Z
BZ
4
3 Z
1
- - tr 8 4 + ld (tr 8 )
Z
72
6
-1 tr 8 4 - 1- (tr 8 3 )
8
1Z
1
8
72
3
(tr 8 )
Z
Inverting the characterisitc function (Z.8), we have
(Z.16)
+ O(n-3 ),
2
2
Z
7
variate with
where
degrees of freedom.
It follows from
(2.11) and the characteristic functions (2.12), (2.14) that the following theorem
holds.
Theorem 2.2.
Under the sequence of alternative hypothesis
K :
is
(8
y
statistic
LR
freedom, when
y
Pd), tre lini.ti ng distributions of the modif ie d
21ogA * given by (2.1) is
with f = p(p+1)/2 degrees of
1
1
1
and noncentral
degrees of freedom and nonwith
> 2'
centrality parameter
2
tr 8 /4,
When
when
y
= 1,
1 tr 8 2 {P(X 2 +2 < z) - P(X 2 < z)}
PH(-2logA 1* < z) + 4n
f1
f1
2
+ ~2 ~~
n
P (-2logA *
H
1
where
(a
= 0,
< z)
a=O
g
2
2a P(X f +2a
1
z)
+ O(n- 3 ),
is given by (2.16) and the coefficients
are given by (2.13) with
1, 2)
<
g2a
in (2.9).
and
When
1
y
= 2'
we have
4
-3/2 ),
- 3P(X 2 +2(6 2 ) < z) + 2P(X f2 (6 2 ) < z)} + -1 L: h P(X f2 +2 (2)
6
< z) + O(n
f1
1
n a=O 2a
1 a
where the symbol
f
l
=
p(p+l)/2
and the
2.4.
h
2a
X2 (6 2 )
f
means the noncentra1
X2
variate with
l
degrees of freedom and noncentrality parameter
02
2
tr8 /4,
are given by(2.15).
Numerical examples.
It may be useful to note that applying the gen-
eral inverse expansion formula of Hill and Davis [4] to the asymptotic nu1ldistribution of
- 210gA *
1
given by Theorem 2.1
in Sugiura [11], we can get
8
an asymptotic formula of the 100a
lOOa
X2
point of the
%
2logA *
l
point of
%
distribution with
f
l
in terms of the
= p(p+l)/2 degrees of
freedom as
(2.19)
+
1
~
n
where
u
3
2
(u g3 + u g2 + ug l ) + O(n
P(X~
is so chosen that
-4
),
u) = a
>
and
4
2
3
2f B B + B )
2
3 (fIB 4 l 2 3
3f
1
gl
(2.20)
3 4
(fiB4 - 2f l B2B3 - 5B;)
3f (f +2)
l l
4
(fiB4 + 4f l B2B3 +
3f~(fl+2)(fl+4)
with
B2 , B3
.
~n
(2 • 9) an d
B4
= P (6 P4 + l5p3
4B~)
10p2 - 30p + 3)/480.
We shall examine the effectiveness of these formulas in the following
simple examples.
Example 2.1.
- 210gA *
l
When
p
= 1 and n = 10,
can be obtained from Table I
the exact
5
by Pachares [7] as
point of
%
3.9682.
The asymp-
totic formula (2.19) gives
points of
5 %
P
- 2logA *
l
= 1 and n = 10
P
2 and n = 100
3.84146
7.81473
0.12805
0.05644
-0.00060
0.00055
-0.00059
0.00001
approx. value
3.9683
7.87173
exact value
3.9682
first term
term of order
n
term of order
n
term of order
n
-1
-2
-3
9
Our asymptotic formula (2.19) gives good approximations for the percentage
points.
Example 2.2.
- 2logA *
l
of
When
p
=
2
and
=
n
is given by Example 2.1
100,
as
7.87173.
2
8 /4
K
82
the "distance" of
from the null hypothesis.
K
L~1/2 L L~1/2 =
as
1 (6
K
= 0.5)
I + 61.
in Theorem 2.2
Then
82
=
according to the
from the null hypothesis.
centrality parameter
= tr
3 (6
Let us specify the alternatives
n6 2 /2
and the following case
is computed by the formula (2.16) in Sugiura [11] with the normal
=
0.02)
6
82
=
p
0.5
12.5
2
6
=
and
0.1
n
=
100
6
=
0.5
0.02
0.05
0.1134
second term
0.0714
-0.0033
0.00229
third term
0.0052
0.0018
0.00001
approx. power
0.942
0.112
0.0523
Moments of the criterion.
L: (pxp, pd)
be given.
vector) and
L:
L: .; L '
O
=
(~
0.1)
0.02
0.8651
LR
= L: O
criterion for
~
(pxl)
We wish to test the hypothesis
(a given
pd
LR
criterion
N
from a
and covariance matrix
H:
~
=
matrix) against alternatives
The h-th moment of the
is given by Sugiura [11] as
= L: o and
L:
Let a random sample of size
p-variate normal population with mean vector
and/or
=
first term
Asymptotic distributions of the
3.1.
2
by the formulas (2.18) and (2.17) respectively.
approximate powers, when
3.
The non-
may be used as a measure of
distribution function and its derivatives, as well as the cases
and
point
The asymptotic powers
should be computed by the different formulas in Theorem 2.2
departure of the alternative hypothesis
%
the approximate 5
A2
~O
K:
(a given
11
I- ~O
for this problem
10
Nph/2 r «n+Nh)/2)
2
(~)
p
N
....J::..---:("""'n-:-/2-")-r
(3.1)
p
where
n
=
N-1.
LR
The unbiasedness of the
- 21ogA2
II
I
+ h E E~
criterion
cation was proved by Sugiura and Nagao [13].
distributions of
IE E~
1 Nh/2
1 N(1+h)/2
I
A2
without modifi-
The asymptotic expansions of the
both under the null hypothesis and a fixed a1ter-
native hypothesis have been derived by Sugiura [11].
3.2.
Asymptotic distributions when
sequence of alternative hypotheses as
"-1 /2 ~" ~O
,,-1/2
~O
= I
+ N- y
e,
y
K :
y
<
1
2.
~
-
Now we shall specify the
~O
N-
y
O
an d conSl. d er t h
e case
< yI
< 2·
y 1 2
(-21ogA2)N - /
can express the characteristic function of
r
1
(7N -
P 2
1
y-+z
itN
-
ana
E;/2 v
From (3.1) we
as
1
-)
2
1
1 1
r
1
1
p 2
2
(7N - -) (1 -
2itN
.
y---Np-ltpN
2) 2
Y+2
(3.2)
l.N+ itN
1
II -
2itN -2
-~~1
y--
1-2itN
1
2- y
x etr[itN
v'v - 2t
2
el
-2
2
1
y--
v' (I - 2itN
1
y+2
2 E~1/2 E E~1/2)
-1
(I + N- Y e)v].
11
The first factor and the second factor have the same form as the characteristic
function (2.3).
y l 2
1 + O(N - / ),
Hence we can see that the first factor is equal to
which tends to one as
by (2.5) after substituting
as
n
for
N
N.
(- ZlogA2)Ny-l/2
(3.3)
exp[it{lNtre - N
and the second factor is given
It follows that the characteristic
can be expressed as
1
YTz
00,
The third factor is easily evaluated
2
l 2 y
etr[itN / - v'v - 2t v'v] + 0(1).
function of
~
1
-1
10giE EO
I
+
2- Yv'v}
N
Z
- t (tre
2
+ 2v'v)] + 0(1) ,
which implies that the characteristic function of the statistic
(3.4)
tends to
2
2
exp[- t (tr e + 2v'v)]
as
Thus we can get the following
theorem.
Theorem 3.1.
Under the sequence of alternatives
and
for
o
<
Y <
1
2'
K :
Y
the limiting distribution of
the statistic (3.4) is normal with mean zero and variance
N
Ztr e
2
+ 4v'v
as
tends to infinity.
The limiting distribution in the above theorem is of the same form as
under fixed alternative hypothesis given by Sugiura [11].
3.3.
Asymptotic distributions when
under
asymptotic distribution of
the characteristic function of
(3.5)
(~)
Z
N
-Npit
1
> Y =2
- 2logA2
K
Y
_
r (IN- .!)(l-Zit)Np (1-2it) /2
2
when
Y ~
1
2·
From (3.1),
can be expressed as
r (IN(1-2it)- 1)
_ _--"'-p_Z:..-.Z
P 2
We shall now consider the
II
+ N- Y el
-Nit
--::N~(":""l---'Z:-:i-t-:-")"7/Z
2it N-Y el
II - l-2it
12
• etr{itNl-2y v'v _
2
2t Nl - 2y v'(I _ 2it
1-2it
1-2it
-e)
-1
(I
NY
The first factor is equal to the characteristic function of
e
+ -)v}
.
NY
- 21ogA2
under
the null hypothesis, which was expanded asymptotically by Sugiura [11] as
(3.6)
3B,2_ 4B ,
6B,2
_ 1} + __1__{ 2
3 2 + 3B,2 + 4B '} + 0(N- 3 )],
2
1-2it
2
3
6N (1-2it) 2
where
p(2p +9p+11)/24
B'
- p(p+1) (p+2) (p+3)/32 .
2
(3.7)
2
B'
3
The second factor in (3.5) can be expanded by (2.10) after substituting
N.
n
for
The third factor can be expanded asymptotically as
(3.8)
When
Y >
1
2'
follows from (3.6) that the characteristic function of
(1-2it)
-f /2
2
tribution of
where
f
- 21ogA2
2
=p
+ p(p+1)/2
is
X2 with
as
f
2
N
-+
00.
-
21og A2
+ __1__
2
-+
00
It
tends to
Thus the limiting dis-
degrees of freedom.
get the asymptotic formula of the characteristic function when
6N
N
both the second and the third factors tend to one as
Moreover, we can
y
=
1
as
6B,2
2
_---:;;..2_ + 3B' 2 + 4B'} + J:..- L gZa (1-2it)-a + 0(N- 3 ),
1-2it
2
3
N2 a=O
13
where
Bi
g'
0
= -
4
B'
2
g'
2
(3.10)
trO
-2
2
1
3
1
2
+ - trO + -(trO)
3
32
trO
2
2
v'v 1
2 ,
1 ,
(v'v)
+ -(-trO +B ) + -v Ov +
2 4
2
2
8
1
1 _ 2
3
trO - -(trO )
2
16
2
Bi
2
1
3
1
2 2
= - trO + - trO + --(trO) +
g'
4
4
6
32
-
1
2
v'v(4 trO +Bi)
-
v'Ov -
2
(v'v)
4
2
'1
2
1
(v,\)2
~(-trO +B') + - v'Ov + -'-'---'--<-2 4
2
2
o
LJ
When
N
->- co
1
the second factor in (3.5) tends to
= 2'
y
exp[~
and the third factor tends to exp[itv'v/(1-2it)] as
it tr 02/(1-2it)]
N
-'>-
as
Hence from
co.
-f /2
(3.6) the characteristic function of
{~
• exp [it
tr 0
2
+ v' v 11 (l-2it)]
distribution is noncentra1
parameter
62
=l
4
tr 0
2
+
l2
X2
as
(1-2it)
• [1
N
with
v'v.
f
2
->
co
tends to
(1-2it)
2
which implies that the limiting
degrees of freedom and noncentra1ity
Further, we can get an asymptotic formula
for the characteristic function of
(3.11)
- 21ogA2
- 21ogA2
as
-f /2
it (V + l tr 8 2 )]
2
exp [ l-2it v v
2
3
3
t 0 +3 v 'e-v _ 3tre +6v'8v + 2 tr 0 3 + 3v'8v}
+1- {r2
1-2i t
6/N
(l-2it)
4
I:
0'.=0
h' (1-2it) -a. + 0 (N- 3 / 2 )]
20.
'
where
(3.12)
h'
0
1
3
- B' - 8"3 tr o4 + -(tr8
)
2
18
h'
2
1
3
B' + 1 tr 8 4 - -(tr8 )
6
2
4
h'
4
4
3
1
tr 0 + ld(tr0 )
2
72
h'
6
1
3
4
1
tr 0 - -(tr8 )
12
8
2
2
2
, 2
2
(v'Ov) +1.
v Ov
+
v' Ov
6
2
8
2
+1
2
, 2
v 0 V
-
1(,
)2
V Ov
2
-
2
3
v'O v + -3 (V' Ov ) 2
4
2
+1 v'8 2 v - -l (v, Ov ) 2
2
2
7
-12
+1
4
v'ev
.
v' 8v
5
v'Ov
12
. tr
.
0
. tr
tr 0
tr 0
3
3
3
0
3
14
Inverting the characteristic function (3.6) under the null hypothesis
H,
we have
B'
P(x~ < z) + N2 {P(x~ +2 < z) - P(x~ < z)}
2 2 2
(3.13)
< z)}
+ O(N-3 ).
From the asymptotic formulas (3.9) and (3.11) for the characteristic function,
we can get the following theorem.
Theorem 3.2.
Under the sequence of alternatives
'i.- 1/2 'i. 'i.- 1/2
and
o
I + N- Y 8 ,
0
1
2
x2 with
and noncentra1
> -
the limiting distribution of the
p + p(p+1)/2
statistic
Y
centrality parameter
P (-2logA2 < z)
02 =
PH(xi
Y
1.4
<
2
f
2
tr 8 2 +
z) +
1. {.!
N 4
%v' v,
tr 8
2
when
When
L:
go' gz and
=
y
g4
degrees of freedom, when
Y=
1
2.
When
Y = 1,
we have
1
+ - v'v} {P(x~ +2 < z) - P(X 2 < z)}
2
f
2
Z
g2a P (X~ +2a < z) + O(N
N a=O
2
+.!2
LR
= P + p(p+l)/2 degrees of freedom and non-
2
where
K :
Y
-3
),
are given by (3.10) and
< z)
is given by (3.13).
1
2' we have
(3.15)
- (3tr8
3
+ 6v'8v)P(X 2
(0 2)
f +2
2
< z)
+ (2tr8
4
'i.
a=O
where
h
2a
(a
0, ... , 4)
are given by (3.12).
3
+ 3v'8v)P(x~ (0 2 )
2
< z)}
15
3.4.
If we consider the sequences
More general sequences of alternatives.
of alternatives
and
the asymptotic distribution of
- 210g A2
can be
obtained by the same argument as above.
Theorem 3.3.
Under the sequences of alternatives
K y (Y1 f Y2)'
y
the
1 2
limiting distributions of the
LR
statistic
- 210gA2
are given in the fo1-
lowing.
(1)
When
(3.16)
1
1
and
the statistic
[-210g A2
"2- Y2
N
is distributed asymptotically according mthe normal distribution with mean zero
2
2tr 0 .
and variance
If further
21 ~
Y1'
the term
in (3.16) may be omitted.
(2)
has asymptotically the noncentra1
When
distribution with
parameter
f
2
=
P + p(p+1)/2
2
tr 0 /4.
has asymptotically the
(3)
f
tribution with
(4)
degrees of freedom and noncentra1ity
2
=
p + p(p+1)/2
and
When
X2
dis-
degrees of freedom.
the statistic
has asymptotically the normal distribution with mean zero and variance
If further
omitted.
1
-
< y
2 =
2'
the term
tr(L La-1 - I) - log IL La-11
4v'v.
in (3.17) may be
16
has asymptotically the noncentral
When
(5)
distribution with
f
v'v/2.
degrees of freedom and noncentrality parameter
2
The above theorem shows that the limiting distributions are normal when
1
min(y l , Y2) <
min(y l , Y2)
3.5.
- 2logA2
2'
noncentral
when
1
> -
2·
Numerical examples.
Since the asymptotic null-distribution of
has the same form as that in the previous section as was shown by
Sugiura [11],
pe~aE~
the asymptotic formula (2.19) for fue
also, in this case after changing
(3.7) with
and
when
4= p(6p
B
n
to
Nand
B
ex
432
+ 45p + 110p + 90p + 3)/480
to
point can be used,
B' (ex = 2, 3, 4)
ex
in
and putting
p(p+l)/2 + p.
Examp Ie 3. 1.
imations to the 5
When
%
N
=
100
and
p
2,
we have the following approx-
point.
first term
11.0705
-1
N
-2
N
-3
N
term of order
term of order
term of order
0.1365
0.0024
0.0000
approx. value
For the alternative hypothesis
E~1/2 (~ - ~O) = (0.05, 0.05)',
K:
11.2094
-1/2
EO
-1/2
E EO
.
= d1ag(1.1,
0.9)
and
the following approximate powers are computed
by the formulas (3.15) and (3.14), based on the above 5 %
point.
17
approximate power
4.
(3.15)
(3.14)
first term
0.1217
0.0500
second term
0
0.0618
third term
0.0036
0.0150
approx. power
0.125
0.127
Asymptotic distributions of the
4.1.
LR
criterion for independence.
Sugiura and Fujikoshi [12] have proved that the mo-
Preliminaries.
ments of the
LR
statistic
A3
sets of variates
for testing the independence between
in a p-variate
ulation, based on a random sample of size
N,
PI
and
normal popcould be expressed, under alter-
native hypothesis, by using the hypergeometric function with matrix argument
due to Constantine [2] as
~
E[A~ ] =
(4.1)
1
1
2 F 1 (2"(N-l), 2"(N-l);
2
h
1
+ 2"(N-l) ;
= diag (pi, P~' ... , p 2 )
and Pa (a = 1,2, ... , PI) are the papPI
ulation canonical correlations. The following lemma for zonal polynomials due
where
p
to Fujikoshi [3] and Sugiura and Fujikoshi [12] will be used later.
Lemma 4.1.
Let
C (Z)
K
be a zonal polynomial corresponding to the parand
>
=
k
p
>
O.
Putting
18
p
=
al(K)
L:
a=l
k (k - a)
a a
(4.2)
P
2
k (4k - 6ak + 3( 2)
a
a=l a
a
a (K)
2
L:
,
then the following equalities hold.
00
(4.3)
L:
k=l
C (Z)/(k-l)!
(K) K
(trZ)
L:
l
etr Z,
l = 0, 1, 2, ...
for
00
(4.4)
L:
k=O
L:
(K)
al(K) CK(Z)/k!
2
(trZ )etr Z
=
00
(4.5)
L:
L:
k=l
(K)
00
(4.6)
L:
L:
k=O
(K)
(2trZ
a1(K) CK(Z)/(k-l)!
a (K)2C (Z)/k!
K
l
=
{( trz 2 )
00
L:
(4.7)
L:
k=O (K)
a (K)C (Z)/k!
K
2
{4trZ
2
2
+ trZ trZ) etr Z
2
+ 4trZ
322
+ trZ +(trZ) }etr Z
322
+ 3trZ + 3(trZ) + trZ} etr Z.
It is also convenient to note the following lemma, the proof of which
comes by considering
k
log (a) ,
where
K
K
of
and
(a)
(4.8)
K
p
IT
ka
a=l
j=l
IT
1
(a - Z(a+l) + j) .
Lenuna 4.2.
(4.9)
(am
+
b)
K
(am)
k
[1
+
1
am {kb
1
+Z
a (K)}
l
19
holds, where
4.2.
a (K)
1
a (K)
2
and
Asymptotic distribution when
asymptotic distribution of the
of alternatives
by
p
are defined by (4.2).
K:
P
y
LR
y
=
i.
statistic
We shall now consider the
- 2p1ogA3
under the sequence
= m- 1 / 2 8, where the correction factor
= 1 - (PI + P2 + 3)/2N and m = pN
characteristic function of
- 2p1ogA3
p
(Anderson [1, p.239]).
under
K
Y
is given
The
can be written from (4.1)
as
r
(.k
PI 2
1
+ -4 (P1- P 2+1) )r
1
PI 2
(7IIl (1-2it) + 1:::.)
(4.10)
00
E
k!
k=O
with the abbreviated notation
characteristic function of
I:::.
= (PI + P 2 + 1)/4.
- 2plogA3
The first factor gives the
under the null hypothesis
(8
= 0),
which can be expanded asymptotically as
(4.11)
The second factor in (4.10) can be expanded by the formula (2.4) as
(4.12)
1
II - -;
+
1
2 zm+1:::.
G I
1 1
2
1
4 2
-z
{Z(l:::.trG + 4 trG ) m
1
Z
I:::.
4
1
6
-3
tr G - 6 tr G } + O(m )] •
The third factor in (4.10) can be expressed by Lemma 4.2
as
20
00
(4.13)
L:
k=O
11
12
21
1
+ m2{2(2 - 1-2it) (2k~ + a 1 (K)) + 3(1 - 2(1_2it)2)(k - 12~2k - 12~a1(K) - a 2 (K))}
+0(m- 3)],
which can be simplified by Lemma 4.1
2
trEJ
ex p [2(1-2it) ]
[1 +
as
l
242
+ trEJ - 2~trEJ
1-21t
2(1-2it)2
{2~tr~
m
4
trEJ
}
4(1-2it)3
(4.14)
6
+.l2
m
L:
g2a(1-2it)
-a-3
+ O(m )],
a=l
where
(4.15)
g2
2~2 tr EJ2
g4
(3~
+ Z)tr EJ4 +
(2~2
+
t)
(trEJ 2) 2 -
4~2
tr EJ2
g6
2
564
214
12
2
= - trEJ + ~trEJ trEJ - (4~ + 2)trEJ - (2~2 + 2) (trEJ ) + 2~2trEJ
g8
2
4
2
3
141
1
2 2
14
6
= -( trEJ ) - trEJ - ~trEJ tr8 + (2~ + "4)tr8 + (2~2 + "4) (tr8 )
8
6
14
g10 = - a(tr8)
1
2
1
6
1
+ 3 tr 8 + 4
~
tr 8
4
tr 8
2
4 2
g12 = 32 (tr8 ) .
It follows from (4.11), (4.12) and (4.14) that the characteristic function
(4.10) of
- 2p1ogA3
-p p /2
(4.16)
+
(1-2it)
1 2
under
K
y
can be expanded asymptotically as
2
it tr8
2~tr82
1
1
4
2
etr[ 1-2it ] • [1 + ; {- 4 trEJ - ~tr8 + 1-2it
1
6
tr8 4 - 2~tr82 _
trEJ 4
}
1 P1 P2 2 2
a
+ -- {----(p +p -5)(
- 1) + L: h 2N (1-2it)-}
2
2(1-2it)2
4(1-2it)3
m
48
1 2
(1-2it)2
a=O
~
+ 0 (m-3 ) ] ,
21
h
2
114
= -(-trG
+ Il trG )
O 2 4
2
4
(4.17)
h
2
14
= - S(trG)
1
6
4 2
= 16(trG)
+ 21l 2 trG
+
6)
are given by
1
6
1
4
- -trG - -lltrG
6
2
1422
- ZlltrG trG - 21l 2 (trG)
h
... ,
h 2a. (a. = 1, 2,
where the coefficients
2
+ 21l 2 trG
2
Il
4
2
1
4
12
- 4 trG trG + OIl + 4) trG + OIl 2 + 4) (trG )
45
4
IltrG trG
2
2
5
6
1
4
12
+ (;trG - (41l + Z)trG - (21l 2 + Z)(trG )
2
2
8, 10, 12).
Inverting this characterisitc function, we can conclude the following theorem.
Theorem 4.1.
Under the sequence of alternatives
distribution of the
LR
and
between
statistic
- 2plogA3
y
P
2
= m-1
2
G,
the
for testing the independence
sets of variates
m = pN = N - (Pl+P2+3)/2N
for large
K :
can be expressed asymptotically
as
P(-2plogA3 < z)
(4.18)
where the symbol
X2 (0 2 )
f
means the noncentral
3
degrees of freedom and noncentrality parameter 0 2
h a.
2
(a.
=
4.3.
0, 1, ... ,6)
are given by (4.17) with
Limiting distributions when
distributions of the
LR
statistic
1
Y~Z·
- 2plogA3
x2 variate with
1
2
= ztrG
~
=
f3
= PlP2
and the coefficients
(Pl+P2+l)/4.
We shall now consider the limiting
under the sequences of altern-
22
atives
P
K:
y
= m- Y El
1
2)·
(y <
Noting that the moments (4.1) can be rewritten
by the Kummer transformation formula as in Sugiura and Fujikoshi [12],
k
E [A~ ]
( 4.19)
r
1
1
(-(N-p -l»r (h + -(N-l»
Pl 2
2
Pl
2
h
• Ir
2 h
-
I
P
1
+ 2(N-l);
1
and considering the characteristic function of
2- Y
(-2plogA3)/m
based on the
above moments, we can easily see that the first four gamma products tend to one
and the hypergeometric function
2Fl
tends to
2
2
exp[-2t trEl J
as
1
gives the first part of the following theorem.
Also, when
Y > -2'
from (4.1) that the characteristic function of
- 2plog A3
tends to
m~oo
This
we can see
-p p /2
(1-2it)
1 2
Theorem 4.2.
as
Hence, we can imply the following theorem.
Under the sequence of alternatives
limiting distributions of the
LR
statistic
P = m- Y El
K :
Y
- 2plogA3
the
for the hypothesis
of independence are given in the following.
( 1)
When
o
1
< Y <
the statistic
2'
(4.20)
m
1~2-Y
[-2plog A3 - m loglr - p
2
1]
has asymptotically the normal distribution with mean zero and variance
(2)
When
distribution with
4.4.
1
Y >
2'
P1 P 2
the statistic
- 2p1ogA3
has asymptotically the
degrees of freedom.
Numerical example.
Computing the one more term in the asymptotic
formula of the characteristic function in the null case given in (4.11), we
can get
2
4trEl .
23
(4.Zl)
P(X
2
f3
<
w2
2
2
z) + -Z{P(X f +4 < z) - P(X
< z)}
m
3
f3
2
-5
2
+ 4;[W4 {p (X~ +8 < z) - P(X f < z)} - w2 {P(X~ +4 < z) - P(X f2 < z)}] + O(m ),
3
m
3
3
3
where
Z Z
P1PZ (Pl+PZ-5)
w2 =
48
(4.ZZ)
P1P Z
2
1
Z
Z
4
4
Z Z
W4 = -Z w2 + 19Z0 OP1 + 3P Z - 50 P1 - 50 P z + 10 P1P Z + l59}
Since the above formula has the same form as in the case of multivariate linear
hypothesis given in Anderson [1, p.Z08], we can use the asymptotic formula for
the percentalJ! paint of the
LR
statistic for multivariate linear hypothesis
given by Hill and Davis [4] for our present purpose, that is, the 100a %
point of
ZplogA3
u +
is given by
1
-Z
m
Zu W2
+
f(f+Z) (u + f + Z)
Zu w4
4 [f(f+2) (f+4) (f+6)
-l.
m
2
w2 u
3
Z
+ (f+6) (f+4)u + (f+6)(f+4)(f+2)} {u + (f-Z)u
f 2 (f+Z)Z
(4.Z3)
+ (f-6) (f+Z)u + (f-Z)(f+Z)2}] + 0(m- 5 ),
where
f
=
Example 4. L
and
u
When
P(X~
is determined such that
N = 87
and
P1 = 2,
mula (4. ZZ) gives the following approximate 5 %
term of order
approx. value
3,
u)
-Z
m
-4
m
a.
the asymptotic for-
point.
lZ.59l6
first term
term of order
P2
>
0.0016
0.0000
12.5932
24
Pi11ai and Jayachandran [8] gave the exact 5
%
point as
(computed from their Table 8. Lower 5% points of
For the alternative hypotheses
2
P2
= 0.1,
{w(2)}1/2
12.593156
m= 0
and
n = 40).
K :
1
the following approximate powers are computed by the formula (4.18)
in this section and (2.1Z) in Sugiura and Fujikoshi [12] respectively.
approximate power
first term
K
1
0.28505
0.5231
second term
0.00680
0.2871
third term
0.00023
-0.0038
approxo power
0.Z921
0.806
K
2
Pi11ai and Jayachandran [8] gave the exact power when the alternative hypothesis
is
K
1
as
0.2919
in their Table 9.
Hence our approximate power
0.2921
shows good approximation to the exact value.
5.
Asymptotic distribution
5.1.
vectors
the
~
Xal ' Xa2 ' ... , XaN
~a
+L2
*
n.
J
N.-1
J
L
for
a
L =
1
H:
N·J
=
n
n
1
n
n
(X. - X.) (X. - X.) ,
Ja
J
Ja
J
a=l
L
and
n = n
1 + nZo
p/2
n
ln 1
J
= L
Z
.
Let the
p x 1
a
L
Z
= 1, 2.
The modified
LR
against alternatives
with unknown mean vectors, is given by
A4
S.
l
be a random sample form a normal population with
and covariance matrix
(5.1)
where
L
a
criterion for testing the hypothesis
L
1
criterion for
Moments of the criterion under local alternatives.
mean vector
K:
LR
ISl'
n /2
l
n /2
2
Is 2 1
IS 1 + S2 1n /
2
2
and
2
-1 Nj
L X.
X. = N.
J
J a=l Ja
with
It is easy to see that the non-null distribution
25
A~
of
depends only on
this modified
LR
- 2logA4*
tained by Sugiura [11].
p x p
matrix
8
is
(5.2)
pd
and
m
K :
n
=
pn
L:
-1/2 6~ 6~-1/2
2
1 2
=I +
2
1 _ 2p +3p-l
6(p+l)
=
L:- l
2
/ 2 L:
L:- l
1
2
*
A4
moment of
/2
Wp(n l , r)
Since
.
Sl
and
(-l.-
and
has
S2
-1 8,
n
+
l
_ 1.)
n
l
n
2
W (n , I) ,
p 2
Sl
has the
where
are independent, we can write the
S2
h-th
as
nl(l+h)-p-l
*h
E[A4 IK ]
n
where
m
with the correction factor
Without loss of generality, we may assume that the statistic
f
K was ob-
We shall now consider the moments of the statistic
p
Wishart distribution
The limit-
under fixed alternative hypothesis
under the sequence of alternatives
the
Unbiasedness of
criterion was proved by Sugiura and Nagao [13].
ing distribution of
A4*
characteristic roots of
p
n
=
n
n
l
l
ph/2
n
1
n
n
2
2np/2fp(nl/2)fp(n2/2)
f
2
Isll
If I
2
n /2
l
Is 2 1
n (1+h)-p-l
2
2
nh/2
lSI + S21
(5.3)
where the range of integration is such that the two
Sl
and
S2
are
pd.
We can expand the last part
p
p
x
symmetric matrices
etr{t(I - r-l)Sl}
in the
above integration to infinite series by zonal polynomials as
00
L:
k=O
Transforming the variables
U
2
= U- l / 2
1
S
2
U- l / 2
1
L:
CK(i(I - f-l)Sl)/kl
(K)
(Sl' S2)
is chosen
to
pd)
(U l , U2 )
by
Ul
= Sl and
with the Jacobian
la(Sl'S2)/a(U l
,u 2) I
26
= lu l(p+1)/2
and integrating out with respect to
1
U
1
by the formula due to
Constantine [2]
(5.4)
)
{etr(_Rs)}\sl t -(p+1)/2 C (ST)dS
r
S
>
K
a
which holds for any
number
t
p
x
satisfying
p
pd
R(t)
>
matrix
R,
(p-1)/2,
symmetric matrix
T
and complex
we can rewrite the moment (5.3) as
n (1+h)-p-1
2
ph/2
r
(5.5)
1
(-:"11 )
ju21
00
p 2
1::
k=O
Putting
= (I
V
+ U )-1
2
(I
J
lau /avl
2
=
n /2
1
(in\
II +u 2\ n (1+h) / 2
p 1
Ivl- -
and integrating
V by the formula due to Constantine [2]
out with respect to
(5.6)
with the Jacobian
,
) Ir I
2
II -
jsl t -(p+1)/2
sl u-(p+1)/2 CK(RS)dS
r (t) r (u) (t)
P
P
K
()
r (t+u)(t+u)
CK R ,
P
a
which holds for any
p
x
p
pd
matrix
R,
we can express the
K
h-th
moment
(5.3) as
Ph/2
1
nn
n2J
n1
n
1
n
2
r
1
(-::n)
P 2
-""""'(1:---"-)=r-(""""'1,...----)
r p?l p?2
(5.7)
1
1
1
2F1 (?' ?1 (1+h); ?(1+h);
I
-
r
-1
).
Z7
5.Z.
Asymptotic distribution under
b',
mation formula
Z) =
K.
n
II - zi
Applying the Kummer trans for-a Z
-1
• ZF (b-a , a ; b; -Z(I-Z) )
1
1
Z
*
- Zp1ogA4
in James [5] to (5.7), we can write the characteristic function of
under
K
n
as
f p (¥L1)f p (~1 (1-Zit)+L1 1 )f p (~z(1-2it)+L1z)
m
m
f p (~1+L11) f P (~2+L12)r P (~(l-Zit)+L1)
(5.8)
II
+ -m
81
-m it
1
00
(-mit) K(~1 (1-2it)+L1 1 ) K
C (-8)
k
(tm(l-Zit)+L1)K m
k!
and
where
L1 = L1
1
1
+ L1 •
2
function of
K
and
The first factor in the above expression gives the characteristic
- Zp1ogA4*
ptotica11y for large
under the null hypothesis, which can be expanded asym-
m with fixed
p
a
= ma 1m
(a
= 1, Z)
in the usual way
as in the previous sections, giving (Anderson [1,p.Z55])
-f
(l-Zit)
(5.9)
where
(5.10)
f
4
p(p+1)/Z
4
IZ
and
1
Z
1
1
1
2
w2 = 48P (P -1) (p+Z) (-=---z + -=---z - 1) - -;;1'Z (p+1)L1 •
PI
P2
The second factor in (5.8) can be expanded easily by the formula (Z.4) as
-m1it
II
+
~I
etr(-PIit8) • [1 +
~Iit
• tr8 2
(5.11)
Z
1.
3 + -1 (
1 )] .
+ - 1 {- ~lit·tr8
P.
it·tr8 Z)} + 0 (-3
I
Z
3
8
m
m
28
The third factor in (5.8) can be
written~
~
by Lemma 4.2
as
00
l:
k=O
l:
(K)
k!
(5.12)
which can be simplified, by Lemma 4.1
~
as
(5,13)
3tr8
1
l-p j 2
2 2
+ 8(-1 + I-Zit) ([pj:i:t·tt8]
3
2
2
+ 3[tr8] )
Z
Z
+ 4pjit·tr8 + tr8 + [tr8] )} +
1
0(3)]'
m
Multiplying the three factors
(5.9)~
terms according to the power of
(l-Zit)
/Z
-f
(5.14)
4
m
,
we can get the asymptotic formula
Kn
as
Pj P 2
Z
1
1 Z
[1 + ~ tr8 {I-Zit - I} + mZ a:O gZa(1-2it)-a
1
w2
+ 2" {
-1
(5.13) together and arranging the
- Zp1ogA4* under
for the characteristic function of
(I-Zit)
(5.11)~
(1-2it)
1
2 - I} + 0 (3) ~
m
where
(5.15)
P 1 P2
2 2
PIP2{~(tr8)
go
=
g
2
= P 1 P2{-
4
1 2
= PIP2{~(tr8)
g
PIP2
P l -2
3
~
Z
- ---6- tr8 - ztr8 }
2 Z
~(tre)
P P
Z 2
+
P2
3
1
Z
1
2
- :rtr8 + (~ + 4)tre + 4(tr8) }
1-2p
3
6 Itr8 -
1
2
1
Z
(% + 4)tre
- 4(tr8) }
29
Inverting the characteristic function (5.14), we can get the following theorem.
Theorem 5.1.
-1
0,
I + m
Under the sequence of alternatives
LR
the distribution of the modified
be expanded asymptotically for large
m = pn
K : E;1/2 E E;1/2 =
n
l
criterion given by (5.1) can
with fixed
p
= na /n
a
= 1, 2)
(a
as
P(-2plog/..4*
P(X 2
f
z)
<
PIP2
z) + --tr8
4m
<
4
2
{P(X~ +2
<
4
z) - P(X 2
f
<
z)}
4
(5.16)
+12
m
where
f
4
[W2
{P(X~ +4
W2
4
= p(P+l)/2,
coefficients
<
z) - P(X 2
f
<
z)} +
4
2
E g20.
0.=0
and the correction factor
= 0,
(a
p
P(X~ +20.
4
1
z)] + 0(3)
,
m
is given by (5.9).
are given by (5.15) with
1, 2)
<
~
The
= (n-m)/2
and
is given by (5.10).
5.3.
Numerical examples.
Evaluating the asymptotic formula (5.9) for
the characteristic function under
P (-2p log/..4*
H more precisely, we can get
W
<
z) = P(X 2
f
<
4
z) +--2 {P(X~ +4
2
m
4
<
z) - P(X 2
f
<
z)}
4
W
+_3 {P(X 2
f +6
3
m
4
(5.17)
-
<
z) - P(X 2
f
<
4
2{
P(X 2 +4 < z) - P(X 2
2
f
f
W
4
where
w2
z)} +-1:...
4
m
is defined by (5.10) and
<
4
[W4
{P(X~ +8
4
<
z)}] + O(m-5 ) ,
z) - P(X 2
f
<
4
z)}
30
(5.18)
= ~~ + ~(p-1)(2p4 + 8p3 + 3p2 - 17p - 14)(p~4 + p;4 - 1)
W4
.-..E....4
3
2
-3-3
- 120(6p + 15p - lOp - 30p + 3)6(P
+ P
1
n
+ 4(P
2
2
-2
- 1) (p+2)6 (PI
-2
+ P2
-
2
-
1)
1) - 3p(p+1)6 4 •
Applying the inverse expansion formula due to Hill and Davis [4] to the above
expression, we get the following asymptotic formula for the
2w u(u+f+2)
100a% point of
2w u
2
3
{u + (f+4)u + (f+4)(f+2)}
3
m f(f+2) (f+4)
1
2w 4 u
3
2
+ ~ [f(f+2) (f+4) (f+6) {u + (f+6)u + (f+6) (f+4)u + (f+6) (f+4) (f+2)}
2
u +
(5.19)
2
m f(f+2)
m
+
2
w2 u
where
u
is defined such that
Example 5.1.
P(x~ > u)
and
f
p(p+l) /2.
2p1ogA4*
The approximate 5% points of the statistic
given
by (5.1) are computed by the formula (5.19) in the following two cases,
p = 2
p = 1
n
first term
1
4,
n
2
3.8415
20
n
1
13,
n
2
7.81473
-2
m
-3
m
-0.0389
0.00777
-0.0045
-0.00010
-4
term of order m
0.0030
0.00001
approx. value
3.801
7.82241
term of order
term of order
63
The exact 5% point in the first case according to Table 743 in Ramachandran
[10] is
In the
3.80,
Thus our approximate value is accurate to 2 decimal palces.
uni~riate
case, an asymptotic expansion of the distribution of
- 2p1ogA *
4
31
under the fixed alternative Mypothesis has been derived by Sugiura and Nagao
[14], by which approximate power, when alternative hypothesis is not near to
the null hypothesis, can be computed.
Example 5.2.
,,-1/2 = d.
"-1/2,,
L.
2
L.
Specifying the alternative hypothesis
l
L.
~ag
2
(J:
J:
vI' v 2'
J:
••• , v p
)
K as
and using the 5% points obtained in
Example 5.1 , we can get the approximate values of the power of the modified
LR
test from the formula (5.16).
= 1
4, n = 20
2
01 = 0.5
p
n
1
=
p
n
1
= 13,
2
= 63
n
2
01 = 02 = 1.05
0.049828
first term
0.0512
second term
0.0443
0.001479
third term
0.0083
-0.000008
approx. power
0.104
0.05130
From Table 744a in Ramachandran [10], we can see that the exact power in the
first case is
0.113.
and Jayachandran [9]
In the second case, we can see from Table 2 in Pillai
(m
= 5,
= 30)
n
that the powers of the other tests are
given by
Roy's largest root criterion (root of
0.067021
Lawley-Hotelling's trace criterion
0.070087
Pillai's criterion
Wilks' criterion
(tr Sl(Sl + S2)
(tr
-1
0.070338
)
(II + Sl s;ll)
0.070273.
However, this does not mean that the modified
LR
criterion is worse.
the above four test criteria are to test the null hypothesis
against the alternatives
P
I:
a=l
oa
>
1,
K:
°a=
>
1
for
a
= 1, 2, ••• , p
H:
I:
1
=
Because
I:
2
and
which is an extension of the one-sided test and the modified
LR
32
criterion is against all alternatives
the two-sided test.
K: E
l
+ E2 ,
which is an extension of
33
REFERENCES
[1]
Anderson, T.W. (1958).
Analysis.
[2]
An Introduction to Multivariate Statistical
Wiley, New York.
Constantine, A.G. (1963).
Some non-central distribution problems in mul-
tivariate analysis.
[3]
Fujikoshi, Y. (1968).
Ann. Math. Statist. 34, 1270-1285.
Asymptotic expansion of the distribution of the
generalized variance in the non-central case.
Dniv.
[4]
Ser. A-I, 32.
[5]
In press.
Hill, G.W. and Davis, A.W. (1968).
Cornish-Fisher type.
James, A.T. (1964).
Generalized asymptotic expansions of
Ann. Math. Statist.
Nagao, H. (1967).
Pachares, James. (1961).
normal population.
[8]
Pi11ai, K.C.
1264-1273.
Ann. Math. Statist.
12,
475-501.
Monotonicity of the modified likelihood ratio test for
a covariance matrix.
[7]
~,
Distributions of matrix variates and latent roots
derived from normal samples.
[6]
J. Sci. Hiroshima
J. Sci. Hiroshima Dniv.
Ser. A-I 31, 147-150.
Tables for unbiased tests on the variance of a
Ann. Math. Statist.
1l,
84-87.
Sreedharan and Jayachandran, Kanta. (1967). Power comparisons
of tests of two multivariate hypotheses based on four criteria.
Biometrika~,
[9]
195-210.
Pi11ai, K.C. Sreedharan and Jayachandran, Kanta. (1968).
Power comparisons
of tests of equality of tow covariance matrices based on four criteria.
Biometrika~,
[10]
335-342.
Ramachandran, K.V. (1958).
A test of variances.
J. Amer. Stat. Asso.
21,
741-747.
[11]
Sugiura, N. (1968).
Asymptotic expansions of the distributions of the
likelihood ratio criteria for covariance matrix.
Submitted to Ann.
Math. Statist.
[12]
Sugiura, N. and Fujikoshi, Y. (1969).
Asymptotic expansions of the non-
null distributions of the likelihood ratio criteria for multivariate
linear hypothesis and independence.
to appear.
Ann. Math. Statist. 40,
34
[13]
Sugiura, N. and Nagao, H. (1968).
Unbiasedness of some test criteria for
the equality of one or two covariance matrices.
Ann. Math. Statist.
39, 1686-1692.
[14]
Sugiura, N. and Nagao, H. (1968).
for homogeneity of variances.
On Bartlett's test and Lehmann's test
Submitted to Ann. Math. Statist.