The research was partially sponsored by NSF Grant No. GU-2059 and
the U.S. Air Force Grant No. AFOSR 68-1415.
This work was completed while the author was at the University of
North Carolina at Chapel Hill, Department of Statistics •
••
ASYMPTOTIC EXPANSIONS OF THE DISTRIBUTIONS OF
IS 1S;1 1
and
IS2(S1+S2)-11
FOR TESTING THE EQUALITY OF TWO COVARIANCE MATRICES
by
NARIAKI SUGIURA
Hiroshima University
Institute of Statistics Mimeo Series No. 653
University of North Carolina at Chapel Hill
DECEMBER 1969
Asymptotic Expansions of the Distributions of
152(Sl+5)-1\
1\ S;1 1
and
for Testing the Equality of Two Covariance Natrice/
by
t
Nariaki Sugiura
Hiroshina University
1.
I NTRODUCTI ON
I
p x 1 vectors
Let
X'1 l ' X'1 2 ' .•. , X'1 N•
be a ran-
1
dom sample from p-variate normal distribution with mean ~. and covariance
N.
1
= 1,2. Put 5. = E 1 (X. -X.)(X. -X.)I and
matrix E. for
1
1
a= 1 1 et
1
1 et
1
-1 N;
Then the statistic
X.1 = N Eet= 1 X.•
or
may
let
be used to test the null hypothesis
1 2 for unknown 1 against
one-sided alternatives K: y. > 1 and E.p 1 y; > p, where y. means the
1 =
1=
1
-1
;-th largest characteristic root of L: E . We can reject H, when the
1 2
observed value of 15 S;1\ (or 152(51+S2)-11-1) is larger than a pre1
assigned constant.
H:L: =L:
~.
The monotonicity of the power functions of these
tests with respect to
y.
among other tests, were proved by Anderson and
J
Das Gupta [2] from a more general point of view.
Putting
ni
=
N;-l
n1+n 2
and
=
n, we shall derive, in this paper,
asymptotic expansion of the non-null distribution of
A1 = In 1091(n2/n1)51S;1\,
as well as of the null distribution, according
to Box [3] and then asymptotic expansions of the distributions of
A2
= -In l091(n/n2)S2(S1+S2)-1\
both under the null hypothesis and local
alternatives, based on the hypergeometric function of matrix argument
due to Constantine [4], James [7] and used by Fujikoshi [5], Sugiura and
Fujikoshi [13], Sugiura [12].
The limiting distribution of
A2
under
* Research
partially sponsored by NSF Grant No. GU-2059 and the U.S. Air
Force Grant No. AFOSR 68-1415.
t This work was completed while the author was at the University of
North Carolina at Chapel Hill, Department of Statistics.
2
K is also derived.
The asymptotic expansions are given in terms of the
normal distribution function and its derivatives, which are different
from the expansions of the distributions of the likelihood ratio criteria
for similar problems in Sugiura [11], [12].
In fact, the null distri-
butions of the likelihood ratio criteria were expanded in terms of the
2
X
distributions with different degrees of freedom.
Ai
for the upper percentage points of
A
and
Asymptotic formulas
will also be given by
2
applying the general inverse expansion formula of Hill and Davis [6],
by which some numerical values of the power of these tests are computed
and compared with the exact value obtained by Pillai and Jayachandran
[9] for the test
A2 with
p=2.
In the following discussion, we shall always assume \lithout loss of
generality that, under
S2
and
has
hypothesis
and let
2.
n
Wp(n 2,I),
K, S1
where
H can be expressed
tend
has the Wishart distribution
Wp (n 1 ,r)
r = diag(y 1 , Y2, ... , Yp)' Then the null
by r=I. Put n = Pin with P1+P = 1
i
2
to infinity for fixed
AsYMPTOTIC DISTRIBUTIONS OF
the characteristic function of
A1
Pi
O.
>
IS 1S;1 1 •
It is easy to see that
= In logl(P2/P1)SlS;11
under
K,
is given by
(2.1)
(p / P )Inpit
1
where
Ir 1Ini t r p(~nl +lr1it) rp(.~·n2-;ni t) /[r p(~nl) rp(~n2) ]
rp(x) = n P(p-l)/4 rrj=l r(x-(j-l)/2).
formula of
logr(x+h)
for large
Ixl
Applying the asymptotic
with fixed
h,
to the above ex-
pression as in Box [3], Anderson ([1], p.204), Sugiura [12] and etc.,
3
we can rewrite (2.1) as
(2.2)
{2(itl2(r+1l-1(r+2l-1 [(-ll r
+ (p+1l[(-ll
r
p~r-1 + p;r-1 J
p~r + p;r]/(4r l}
()()
L (-2/n)r Br +1{r(r+l)}-1
r=l
+ 2itn-~)-r
Gr +1
where
of degree
Ko:
{(Pi
= Ej=l Br +1(-(j-l)/2) with the Bernoulli polynomial Br(h)
r. Specifying the sequence of alternatives K as
r = I + n-oe
(0
~
for a fixed diagonal matrix
0)
e
having non-
negative diagonal elements and .considering the first term of (2.2),
we easily get
THEOREM
2.1.
Under
K,
the statistics
Ai'
Ai -
tre
and
/rll og Ir I, have asymptotically the same normal distribution with
2
-1-1
mean zero and variance T = 2p(P
1 + P2 ), according as o > ~, o = ~
Ai -
and
0
< ~
respectively.
There is no singularity of the limiting distribution at the null
hypothesis as was the case with the likelihood ratio tests for covariance
4
matrix in Sugiura [11], [12].
We can write the first four terms of (2.2)
as
3I b . (it) 2J. + n-3 /25I c.
+ n-1
j=l
2
T = 2p (-1
P1 + P-1)
2
where
ai
b
b
2J
j=l
2J-1
and
= ~p(p+1)(_p~1 + p;i),
2 (-2
-2)
a 3 = 3P
-Pi + P2
2
(
)( -2
-2)
2 = ~ai + ~p p+1 Pi + P2 '
6
'lJ ,
(it) 2J-
b
,
2 (-3
-3
4 = a i a 3 + 3P Pi + P2 )
= ~a32 '
(2.4)
.
e
C =
1
c
3
1 ( 2p 2 +
"f2P
3p - 1 )( -P 1-2 + P-2)
2 '
{.( -P -3 + P-3) + ~ (-2
1 3
= P( p+1 ) {3
P1 + P-2
2 )a i }+ ~1
1
2
C5
4 ( -Pi-4+P -4) + ~p (
) (-2
2 (-3 -3)
2
= 5P
p+1
Pi +P -2)
2
2 a 3 + 3P Pi +P 2 a i + ~a3al
C7
=
1 3
C9 = ~3 .
2 + 3P
2 (-3
P1 + P-3)
2 a3 ,
~a1a3
Inverting the characteristic function (2.3) yields
Then under alternatives
(2.5)
P({A 1
In
K,
we have
log!rl}/T
<
Z)
=
~(z) - __
1 {a T-3~(3)(z)
In
3
I
+ a T-1 ~ (1) ( z)} + 1
b . T- 2j ~ ( 2j ) (z)
1
n j=l 2J
-~ I
n/n j=l
C2j_1T-2j+l~(2j-l)(z)
+ O(n- 2 ) ,
5
are given by (2.4) and
where the coefficients
means the j-th derivative of the standard normal distrihution
function
<p(z).
H is true, the sal~ll:' forr.1l11a
If the null hypothesis
(2.5) holds by simply putting
10gl!1 =
o.
Applying the general inverse expansion formula of Hill and Davis
[6) to (2.5) with
10glrl = 0,
for the upper
point of
a%
\ole get tile following asymptotic formula
AliT,
in terms of the upper
~%
point
U
of the standard normal distribution function.
,
(2.6)
+ _1
U
Iil
{a
b2j H2j + (a 3H3 + al)u{a3(~H3-2) + ~:al}J
H
3 3
+ _1_
nlil
j
-j
, b. = b'T -j , c. = C.T- and HI. means the deriva· = a.T
J
J
J
J
J
J
J
ative of Hermite polynomial H. = Hj(U) defined by ¢(j)(u) = H. ( U) ~l 1) (u ) .
J
~
~
where
(
j = 1 , 2,
For
J
... ,
9
(2.7)
Hi
=
1,
H
=
u
H
=
u
H
=
u8
s
7
9
e
4
6
=
Hz
6u
2
15u
28u
- u,
+ 3,
4
6
2
+ 45u - 15,
+ 210u
4
- 420u
2
1,
H4
+ 10u
3
H3
=
H6
=
u
H
=
U + 21u
8
2
+ 105.
u
5
7
5
u
=
3
+ 3u,
- 15u,
105u
3
+ 105u ,
6
3.
AsYMPTOTIC DISTRIBUTION OF
Unlike the previous
section, we cannot get a simple expression for the characteristic function
lemma in Sugiura [11], which is a direct extension of Siotani and
Hayakawa [10], to the case of several Wishart matrices, to
f(5 1 ,5 2)
1.
2
under
= 10g1521 - 10gI51+521,
we get the limiting distribution of
K.
THEOREM
3.1.
Under alternatives
K,
the statistic
2 - In 10glp1r + p}1 has asymptotically normal distribution with mean
-1)-2
zero and variance 2( P1-1 + P2-1) tr ( I + P2P-1
1 r
1.
-e
Now we shall consider the sequence of alternatives
r = I + n-Oe(o ~ 0),
Ko:
1.
2
under which the characteristic function of
can be expressed for large
diagonal matrix
n-oe
n
such that all positive elements of
are less than one, as
(3.1)
The above formula (3.1) can be obtained by the similar argument as in
Sugiura ([12], Section 5), based on the hypergeometric function of matrix
argument due to Constantine [4] and James [7].
the characteristic function under
H,
The first factor gives
which can be expanded asymp-
totically in a similar way as in Section 2, having
7
(3.2)
I
exp[p(p;l - 1)t 2 +
L-
(2it)r pn -r/2 {2(it)2(r+1)-1(r+2)-1(p;r-1 - 1)
r=l
00
+ (P+1)(p;r - 1)/(4r)} -
I
r=l
(1-2itn-"l-r +
{1 -
(-2/n)r Br +1r-l(r+1)-1
(P2- 2itn -"l-r
- p;r l]
•
the first four terms of which is expressed by (2.3) after putting
10glrl
=0
and replacing
a~, b~, C~
J
(3.3)
J
a' =
1
2
e
aj ,
bj ,
respectively, given by
J
b'
-
T,
-1
~(p+1)(P2
= ~a,2
1
c'1
= ~(2p2
CI
= p(p+1){f(p;3
C
I
5
2 ( -2
P2
=3?
b'
= ka 3,2
-1),
+ ~p(p+1)(p-2 _ 1),
2
2 (-3
b ' -- a I a I + 3P
P2 - 1) ,
1 3
4
3
a3
l
- 1),
6
2
'
+ 3p - 1)(p;2 _ 1),
4 (-4 )
=sP
P2 -1
- 1) +
~(p;2
- l)al} +
tal
3
,2
2
-3
-2
a3'
+ 3P(P2 -l)al + ~p(p+l)(p2 -l)a~ + ka
'
2 1
_ 1
2
C' -_ ~ a I a 1 + 2 (-3
P2 - 1) a 31 ,
7
1 3
3P
13
C' - sa3
9
.
Inverting the characteristic function, we get
THEOREM 3.2. Under the null hypothesis H, the distribution of
1
A = -In T I - 109!(n/n2)S2(Sl+S2)-11 is expanded asymptotically by the
2
k
formula (2.5) after replacing T, a j , b , Cj with T' = {2p ( P2-1 -1 ) }2
j
respectively.
8
Asymptotic formula for the upper a% point of
aj , bj , Cj
by (2.6) after replacing
C~
J
=
C~/Tlj
J
Hhen
A /T
2
1j
aj = aj/T
with
1
,
is also given
bj
• T Ij ,
= b '/
J
respectively.
<5
=
~,
the second factor of (3.1) is written by
co
I
(3.4)
I
k=O (K)
(-Irli t)
(~n1)
C (-n -~8)/{(~n - /nit) k!}
KKK
By Fujikoshi [5] and Sugiura [12], we know already that
(3.5)
( -Irlit)
(~n -Irli t)
K
the right-hand side of (3.4) is equal to
I
(3.6)
I
k=O (K)
c (P1iteHkl)-1
K
~
+ _1 {2itk - a1 (K)/2it}
Irl
It follows from the lemma by Sugiura and Fujikoshi [13] that
(3.7)
/1
=
{etr(P1itenr1 + _1 {2P 1(it)2 tre Iii"
L
t p~(it)tre2}
K
9
Multiplying (3.7) with the first factor of (3.1) given by (2.3) with
b l. , c I. , T I instead of a· , b ., c . , T, we finally get the asymptotic
J
J
J
J
J
formula of the characteristic function of 11
under Ko(o = ~) as
2
a~,
J
(3.8)
where
(3.9)
and
ajI
,
bJI.
91
=
33
- -1
3 p1tre
92
=
b2l + P1 ( 1 - 3p 1
93
=
p tre(p2 tre 2 - 2a' - 4)
1
1
1
94
=
2p2(tre)2
1
95
=
2a~P1tre
96
=
b6l
Ko(o
=
~): r
tribution of
=
11
Hence we can get
Under the sequence of alternatives
I + n-~e,
2
2 + "8
1 4(
p1 t re 2) 2
ka l P2tre 2 + b'
2 3 1
4
are given by (3.3).
THEOREM 3.3.
, )
1 1
~ a p tre
=
the following asymptotic expansion of the dis-
-In l091(n/n2)S2(S1+S2)-11
holds, for large
that all positive elements of the diagonal matrix
-k
n 2e
n such
are less than one.
10
+ __
1 {_a , ,·-3¢(3)(z)
= ¢(z)
(3.10)
+
2PltrG,·-2¢(2)(z)
+
In
3
(~p;trG2 - a~),'-l¢(l)(z)}
6
+
,•
where
~ j=l
I
= {2p (-1
P
2
k
- 1 ) }2
From (3.7) and (3.5), it is easy to see that the limiting distribution of
,,2.
A
2
under
Ko(o
>~)
is normal with mean zero and variance
Hence the limiting distributions of
ko (0 >~)
are the same.
Even when
A /T
1
0 =~,
and
A /"
2
under
they are the same, because
their asymptotic means given Theorem 2.1 and Theorem 3.3 are equal.
power of the test
alternative
A will be almost the same as that of
1
K is near to the null hypothesis.
A ,
2
The
when
This is verified numer-
ica11y in the following special cases.
4.
NUMERICAL EXAMPLES,
We shall consider two cases, namely,
p=5 and n1 =50,
Case 2:
n2=150.
Asymptotic formula (2.6) for the upper percentage points gives
5% points of
first term
e
A1IT •
5% points of
Case 1
Case 2
Case 1
Case 2
1. 6449
1.6449
1.6449
1.6449
term of
O(n-~)
-0.3576
-0.4020
0.2365
0.2450
term of
O(n- 1 )
0.0791
0.0409
0.0206
0.0272
term of
O(n- 3 / 2 )
-0.0262
-0.0195
0.0044
0.0051
1.340
1.264
1.906
1.922
approx. value
A2/T' •
12
All these examples show that the powers of the Ai-test and the A -test
2
are almost the same when alternative is near to the null hypothesis.
REFERENCES,
[1]
Anderson, T.W. (1958). An Introduction to Multivariate Statistical
Analysis. Wiley, New York.
[2]
Anderson, T.W. and Das Gupta, S. (1964). A mono tonicity property
of the power functions of some tests of the equality of two
covariance matrices. Ann.
35, 1059-1063.
- Math. Statist. '\/\,
[3]
Box, G.E.P. (1949). A general distribution theory for a class of
likelihood criteria. Biometrika 36, 317-346.
'v"
[4]
Constantine, A.G. (1963). Some non-central distribution problems
in multivariate analysis. Ann. Math. Statist. 34, 1270-1285.
--
[5]
FUjikoshi, Y. (1968). Asymptotic expansion of the distribution of
the generalized variance in the non-central case.
J. Sci. Hiroshima Univ. Sere A-I, 32, 293-299.
-
[6]
'v"
--
---
'v"
Hill, G.W. and Davis, A.W. (1968). Generalized asymptotic expansions
of Cornish-Fisher type. Ann. Math. Statist. 39, 1264-1273.
-- ---
'v"
[7]
James, A. T. (1964). Distributions of matrix variates and latent
roots derived from normal samples. Ann. Math. Statist.
~, 475-501.
[8]
Pillai, K.C. Sreedharan and Jayachandran, Kanta (1967). Power
comparisons of tests of two multivariate hypotheses based on
four criteria. Biometrika~, 195-210.
[9 ]
Pillai, K.C. Sreedharan and Jayachandran, Kanta (1968). Power
comparison of tests of equality of two covariance matrices
based on four criteria. Biometrika~, 335-342.
[10] Siotani, M. and Hayakawa, T. (1964). Asymptotic distribution of
functions of Wishart matrix. The Proc. lnst. Statist. Math.
12
(in Japanese), 191-198.
tV"
[11] Sugiura, N. (1969). Asymptotic expansions of the distributions of
the likelihood ratio criteria for covariance matrix.
Ann. Math. Statist. To appear in December issue.
13
[12] Sugiura, N. (1969). Asymptotic non-null distributions of the likelihood ratio criteria for covariance matrix under local
alternatives. Submitted to Ann. Math. Statist.
-- ---
[13] Sugiura, N. and Fujikoshi, Y. (1969). Asymptotic expansions of the
non-null distributions of the likelihood ratio criteria for
multivariate linear hypothesis and independence.
Ann. Math. Statist. To appear in June issue.