ASYMPTOTIC DISTRIBUTIONS FOR HOTELLING'S T~ AND PILLAI'S STATISTIC
FOR TESTING INDEPENDENCE - A DIFFERENT APPROACH
by
D. J. de Waa1*, K. M. Portier**, L. Weidman
Department of Statistias
University of North Caro~ina
and
H. Majumdar
Department of Biostatistias
University of North Carolina
Institute of Statistics Mimeo Series #1001
May, 1975
* Visiting from the University cf
supported by the C.S.I.R.
·l·b~
Orange Free State.
Research partially
** Also in Department of Biostatistics, University of North Carolina.
Asymptotic Distributions for Hote11ing's T~ and Pi11ai's Statistic
for Testing Independence - A Different Approach
D. J. de Waal*, K. M.
P~rtier**,
L. Weidman
Department ,of ~tatis~i~s
and
H. Majumdar
Department of Biostatistics
University of North Carolina
.
1.
Introduction. Let A(p x p) be distributed Wishart with covariance matrix
E and n degrees of freedom.
A =
[All
(q x q)
AZI
Let
and
L
-1
-1
= EIlE12E2ZE21
-1
-1
R = AlIAl2A22A21
P
Partition A and
E as
A
IZ]
AZ2
define the generalized population correlation matrix
define the generalized sample correlation matrix.
Fujikoshi [2] derived the asymptotic distributions under the alternative of
the two test statistics
(i)
n tr R - Pillai's-statistic,
(ii)
m tr R(l - R)-l - Hotelling's T~,
m =n - p
+
q
HO: P = 0 against the alternative HI: P ~ 0 •
We shall derive the same two noncentral distributions in this paper but will
for testing the hypotheses
use a different approach.
The approach will be to consider A22
as fixed.
The asymptotic characteristic functions of the two test statistics conditional
on A22
is known from Fujikoshi [2] and by averaging out over A22
the
,
unconditional characteristic functions will be derived.
The asymptotic
distributions will be considered under the assumption that
pointed out by Sugiura [3].
P
=~
as
The following preliminary results will be
* Visiting from the University of the Orange Free State. Research partially
supported by the C.S.I.R.
** Also in Department of Biostatistics, University of North Carolina.
-Znecessary:
2." Pre1 iminary Results:
Lemma 2.1. Let A and L be partitioned as in Section 1 with A distributed
-1
W(E,n) ,then A11 • Z = All - A1ZA2ZA21 is distributed W(L II . Z' n - p + q)
independently of G = A1ZA;~AZ1 and G conditional on AZ2 is distributed
W(E 11 • Z' p - q, 0) , i.e. a noncentral Wishart with noncentra1ity parameter
o
-~
-~
= LII.Z6A2Z6'LI1.Z'
6
-1
= E1Z LZZ
•
Proof: The result is well known and can be proved from Anderson
[1], Theorem
4.3.2.
Lemma 2.2.
If A11 . Z is distributed W(L l1 • Z' n - p + q) and G conditional
is distributed W(L ll • Z' p - q, 0) , then conditional on A22 the
on A22
characteristic function of n tr R
=n
tr(A ll • Z
+
G)-lG
is given by
(Fuj ikoshi [2], Equation 5.17).
(1)
gl (t IA 22 ) = (1 - Zit) -J;q (p-q)
etr{1_~~t~O}
[ 1 -
~n
{q (p - q)(p + 1)
(1 - 2(1 - 2it)-1) + (q(p - q)(p + 1) -4(p +
(1 - 2it)-2
~ {q(P
96n""
O(n-3) ]
where
+
4(p
+
1)(1 -
2it)-3tr~ +4(1
- q)(h O - h1 (1 - Zit)-1)
+
-
l)tr~- 4tr(~)2)
2it)-4tr(~)2}
r8_ z A (4~)(1
a-
a
+
- Zit)-a} +
-3-
AZ(~)
= q(p
- q)h Z - 2h1tr(~) - 24q(p - q)(p
A3(~) = -q(p - q)h 3
+
4h2tr(~)
4(p - q + 1) }tr(~) 2
A4(~)
= q(p
As(~)
= 8h4tr (2n) -
+
48{(p - q)q2
+
1)tr(~)2
+
((p - q)2
+
(p - q)
+
4)q +
128tr(~)3
+
- q)h4 - 6h 3tr(¥) + 48{q2 + 2(p - q + 1)q + (p - q)2 + 2(p - q) - l'
(t~12 - 96(p + 2)tr(¥)2 + 96(p + 1)tr(¥)tr(¥)2 + 48(tr(~)2)2
96{q 2 + 2(p - q + 1)q + (p - q) 2 + 2(p - q)
48{(p - q) q2 + ((p - q) 2 + (p _ q) + 12) q + 4 (3 (p _ q)
+
n) 2 3} ( trr
+
4) hr (~) 2 -
96(p + 1)(tr n)tr(¥)2 _ 384tr(g~ 3
A6(~) =
48{q 2 .... 2(p - q
+
1)q
((p - q)~:+ (p - q)
+
+
(p - q) 2
20)q
+
+
2(p - q)
4(S(p - q)
+
n) 2
+ 7} ( tr2"
+ 24{(p - q)q 2
8) }tr(~)2 - 96(p +
1)
(tr n)
n) 2 - I28tr (n)
n 2) 2
tr (2")
2) 3 - 96 (tr(2")
and
h = 3(p - q)q3 + 2(3(p _ q)2 + 3(p _ q) _ 4)q2 + 3(p _ q + 1)((p _ q)2 +
O
(p _ q) _ 4)q - 4(2(p - q)2 + 3(p - q) - 1)
2
hI
= 12q(p
- q)(p
h
= 6{3(p
- q)q3 + 2(3(p _ q)2
2
+ 1)
3 (p - q) + 16) q +
8 (p _ q
+
+
3(p _ q)
~) 2}
+
4)q2
+
(p _ q
+
1) (3(p _ q)2
+
+
-4-
h
3
= 4{3(p
- q)q3
(p - q)
h
4
= 3{(p
+
Lerrma 2.3.
1Z)q
- q)q3
(p - q)
+
Z(3(p _ q)2
+
+
+
4(4(p - q)2
Z((p _ q)Z
ZO)q
+
3(p _ q)
+
+
+
9(p - q)
(p _ q)
4(Z(p - q)2
+
+
+
8)q2
+
4)q2
5(p - q)
+
+
3(p _ q
+
1)((p _ q)2
+
7)l
+
(p _ q
+
l)((p _ q)2
+
~T
S)l
If All . 2 is distributed W(E ll . 2 , n - p
+
q)
and G conditiona.l
on A22 is distributed W(E 11 . 2 , p - q, n) , n = S'Eii.2SA22' S = E12L;~ ,
then conditional on A22 the characteristic function of m tr R(I - R)-l =
-1
m tr All . 2G,
(2)
g2(tIA22 )
m= n - p + q
= (1
is given by (Fujikoshi [2], Equation 6.16).
2it)- q(p-q)
-
etr{l_~~t n} ~ 1 + ~m
{q(p - q)
(p - 2q - 1) - 2(p - q)(q(p - q) - tr(n))(1 - 2it)-1 +
(q(p - q)(p
2((p
+
~
2(-q
+ 1) -
+
2p
+
l)tr(n)
l)tr(n) - tr(n)2) (1 - 2it)-3
r Ba(~)(l
+
+
tr(n) 2) (1 - 2it)-2
tr(n)2(1 - 2it)-4} +
- 2it)-a + OCm- 3) ]
96m a=O
where
= q(p - q)t o
Bl(~) = -t 1 (q(p - q) - tr(n))
B2(.~) = q(p - q)~2 - l~l + 2t2)t~(n)
BO(~)
+
12(p - q)2 tr 2 cn)
- 6(p
- q)
(q2 _ q(~ _ q _ 1) _ 4)tr(n)2
B3(~)
= -q(p
"'I
- q)t s + (Zt z
p _ q -}-
2)tr2U~)
+
3t 3) tr Cn) - 24(q(p - q) + 2(p - q)~ +
12fq2 (p - q) - (2(p _ q)2 - P + q + 4)q +
l
8(2(p - q) + 1) ]tr(n)2 + 12(p - q)tr(n)tr(n)2
+
16t1.'(n)3
+
-5-
B4(~) = q(p
- q)t 4 - (St s + 4~4)tr(Q) + 12(q2 + 2(S(p - q)
S(2(p - q)2 + 2(p _ q) + 5l)tr 2{Q) + 12(S((P _ q)2
4(9(p - q)
3tr 2 (Q)2
B5(~) = 4t 4tr(Q)
8)Jtr(Q)2 - 12(q
+
- 24(q2
12(q2(p _ q)
36(q
B6(~)
=
+
= 3(p
p
~1
+
t
3
+
2(p - q)
4
+
+
q(p _ q)
+
20q
+
4(S(p _ q)
s)tr(Q)tr(Q)2 - 160tr(Q)3
= S{q 3(p
(p _ q)
2
7)tr (Q)
+
g))tr(Q)2 - 12(3q
18tr2 (Q)2
4)2
q
+
+
3q(p - q - 1) (
(p 2
- q) -
+
2
- q) (q
- q)2
- q)
+
+
1)
+
2q2 - Sq((p - q)2
+
4)q2
+
3q((p _ q)3
+
3(p - q)
+
2(
(p 2
- q)
20)q
+
+
1) - 4(2(p - q)
+
(p _ q)2
+
+
+
1))
8(p - q)
+
4)
+
2)}
+
(p - q)
4(2(p - q)2
+
+
4)2
q
5(p - q)
+
+
(p - q
5)}
+
+
6(q2(p - q)
4) - 4(2(p - q)2 - 3(p - q) - 1)
- q)(qS
= 4{(3(p
+
+
+
+
1) (
(p 2
- q)
+
2
9))tr (O)'-
1)q + (p _ q)2
- q)q 3 - 2 (
3(p 2
- q) - 3(p - q)
+
+
+
q)2
q
3(p - q)
2(p - q
+
24)q
+
+
+
(2(p _ q)2 + P _ q
+
1)tr(Q)tr(Q)2 - 96tr(Q)S
2(p - q)2
+
6)q
1)tr(D)trCQ)2
q(p
8(2(p - q)2
t
2)
+
;
+
+
= -12q(p
~2 = -6(p
+
+
+
2(p - q)
4(p - q)
~O
q(3(p - q)
4(p - q)
1)q
8(4(p - q) + 5) )tr(Q)2
192tr(Q)3 - 12tr2 (Q)2
12(q2
+
+
+
+
+
+
-6-
Lemma 2.4.
-1
e = ~12r.22'
(3)
E
Let
A22 (P - q
E11 . 2
= Ell
x
be distributed
p - q)
-1
- Ll2L22L21
and
P
W(L 22 , n) , then for
-1
-1
1
= LIILl2L22E21
=~
etr{l_~it e'Ei~.2BA22} = etr{l~~:t} ~ 1 + *(~re2(l
~re
- 2it)jl - 1) t
2 ( (1 - 2it) -2 - 2(1 - 2it) - 1
+ )
1 ] +
~ [~re3«1 - 2it)-1 - 1)
+
~re3«1 _ 2it)-2 .
n
2
3 ( (1 - 2it) -3 2(1 - 2it) -1 + 1) + ~re
3(1 - 2it)-Z + 3(1 - 2it)-1 - 1) +
81 ( (1
{
) 2 2
- 2it) -2 - 2(1 - 2it) - 1
+ 1 tr a +
1 Ze Z( 1 - 21t
")-3 - 3 ( 1 - Z"1t )-2 +
str
"-1 - 1) + 32tr
1
Ze-2 « 1 - 2")-4
3(1 - 21t)
1t
-
" -3 + 6(1 - Zit) -2 - 4(1 - Zit) -1 +
4(1 - 2lt)
(4)
l
it
,-1
(L' -1
) _
{ ita } [1.+ a 1 I eZ
E etr{ I-Zit 6 El1.28A22Jtr ~e Lll.26AZ2 - etr 1-2it 7~r + 2n"tr
+
-7-
- 2it)-1 -
( 7)
it
I -1
}
2(L' -1
) _
{ ita } [~ 2
E etr{ I-Zit B Ll1.Z8AZ2 tr ~8 LI1.2SA22 - etr 1-2it ~~r a
(8)
E etr{ I-Zit
(9)
it
,-1
}
2(k' -1
)2 _
{"ta} 1 2 2
E etr{ 1-2it B Lll.28A22 tr 28 LII.28A22
- etr 1~2it [16tr e
+
O(n
-1
1)
+
)]
it
+
O(n
-1 ,
)!
Proof: Since the zonal polynomial CK(~8'Eii.2SA22) can be expressed in terms
of sums of powers of the characteristic roots of the matrix and since the
inverse is also true, the key integral will be
-8-
But
(11)
aiel
= nP = a
Ot] tre 2
(1~2it
(12)
it 12
3
2 ( 1-2it] tra
+
]3 3
it
34(1-2it
tra
~q
a CK) -- Li=1
k C4k 2 - 6k
2
i
i
i
+
+ 3)
+
, then it follows that
it 12tra 2}
(I-Zit]
(it)Z Z
~ 1-2it tr a
•
2
+
Use was made
+
0l (it
1
3
n 2 1_2it]tra
(it)3 2 2
1-2it tr a
0
+
f t h e blnomial
°
expansion
into a series and
C~)K
= C~) k (
I +ln«lCK)
+
6n1 2 ( k - a 2 CK)
+
3a 2l CK) )..+ OCn -3 ) ]
3. Asymptotic Distributions.
Thr.orem 3.1. The
P
= .!e
,
n
as~ptotic
is given by
+
distribution of Pillai's statistic n tr R if
-9(13)
Pen tr R
<
p[ Xf2(Q2)
z] •
<
z] _ 1
p[ Xf+2a
1 J\,8
b
~La=O a
where
a
a
(16) and
,
0
a = 0, ... ,4
2
= tr~e
distribution.
f
and
J~4
4n1La~O
b
2
a
,
a
= 0,
a
a
2
p[ xf+2a
(0 2 ) < z ]} + O(n- 3)
... ,8
are given in (15) and
, the noncentrality parameter of the noncentra1
= q(p
2
Xf+2a
- q) .
Proof: Applying Lemma 2.4 on Lemma 2.2, the unconditional characteristic
function of n tr R is given by
(14)
_ 2it)
-a},
+
where
(15)
a
O
a,...
= q(p = -2q(p
q)(p + 1) + 4tr(~6)2
- q)(p + 1)
az
= q(p - q)(p
a3
= 4(p
+
1) - 4(p
f
l)tr~e
-
8tr(~)
2
+ l)tr~e
a 4 = 4tr·(fi6)
2
and
(16)
b
O
bI
= q(p - q)h O + 24q(p - q)(p + l)tr(~e)? - 128tr(~e)3
= -q(p - q)h 1 - 48q(p - q)(p + l)tr(~e) 2
+ 48tT2(~e)2
b = q(p - q)h - 2hltr(~ ) + 96tr2(~e) - 24(qrp - q)(p + 1) - 4)tr(~e)2 _.
2
2
96(p + 1)tr~etr(~e)2 - 192tr2(~e)2
-10b3
= -q(p ~~q)h3 +
4(p - q
b4
+
4h2tr~e + 96[(p - q)q2
+
q(p _ q)2 + q(p - q) + 4q +
1)]tr(~e)2 + 96(p + 1)tr~etr(~e)2 + 640tr(~e)3
= q(p - q)h4 - 6h3tr(~e) + 48(q2 + 2(p - q + l)q + (p - q)2 + 2(p - q) - 3)
tr2(~e)
- 24((P - q)q2 + ((p _ q)2 + (p _ q) + 12)q + 4(3(p _ q) +
5))
tr(~e)2 + 192(p + 1)tr(3;e)tr(3;e)2 + 288tr2(~e)2
bS
- 96(q2 + 2(p - q + l)q + (p - q)2 + 2(p - q) + 3)tr 2 (3;e) -
= 8h4tr~e
48((p - q)q2 + q(p _ q)2 + q(p _ q) + 12q + 12(p _ q) + 16)tr(3;8)2 _
192(p + 1)tr(3;e)tr(3;e)2 - 768tr(~e)3
b6
= 43 (q2
)
+ 2(p - q + l)q +(p - q) 2
+ 2(p - q)
+ 7 tr 2 (~e) + 24 (2
q (p - q)
q(p - q) 2 + q(p - q)
+
20q + 20(p - q) + 32 ) tr(3;e) 2 - 96(p +
l)tr(~e)
tr(3;e)2 - 128tr(~e)3 - 192tr2(~e)2
b
7
= 96(p
and
h. ,
1
+
1)tr(~e)tr(~e)2
+
384tr(~e)3
given in Lenma 2. 2 .
i = 1, ..• ,4
Theorem 3.2. The asymptotic distribution of Hotel1ing s-T; statistic
i
m tr R(I - R)-l,
(17)
where
c
m =n - p
P[m tr ReI - R)-l
02
= O•...
= ~re,
,8
f
<
+
q , if P
= *6
, is given by
z]
= q(p
- q)
and
given in (19) and (20).
C
a
J
a = 0, ... , 4 and
d- ,
a
+
-11-
Proof: Applying Lemma 2.4 on
Len~a
function of m tr R(I - R)-l
is given by
(18)
2.3, the unconditional characteristic
q2 (t)
2it)-~q(p-q) [1
- 2it) -a )~
J.
+
1 f\4
4m\.l·a=0
c. (1 _ 2it) -a }
~
+
Oem -3 ) ]
+
where:
Co = q(p
(19)
- q)(p - 2q - 1) - 4tr(~e)2
cl
= -2(p - q)(q(p - q) -
c2
= q(p - q)(p
c3
= 4((p + l)tr~e
= 4tr(~e) 2
c4
+
2tr(~e))
1) - 4(2p - q
+
l)tr(~e)
+
8tr(~e)2
- 2tr(~e)2)
and
(20)
2 2 3
- q)t o + 24(p - q){q - q(p - q - 1) + 4}tr(~e) - 128tr(~6)
48tr 2 (lf6)2
dO
= q(p
d
= q(p
2
- q)t 2 - 2(~1 + 222)tr(~e) + 48((p - q)2 + 2)tr2(~e) 24{3(p - q)q2 _ (p _ q)(p _ q _ 3)q - 4(3(p - q) + 1)}tr(~e)2
96(2p - q
+
1)tr(~e)tr(~e)2 - 192tr2(~e)2
+
+
-12d3
= -q(p
- q)t 3 + 2(Zt 2 + 3t3)tr(~e) - 96((p - q)q
tr2(~e)
+
2(p - q)2 + P - q
4)
+
48{(p - q)q2 _ (3(p _ q)2 - P + q + 8)q - 8(3(p - q) + 2)}
tr(~e)2 -96(2q-p+l)tr(~e)tr(~e)2 + 640tr(~e)3
d4
+
+
192tr2(~e)2
. . 2
= q(p
2
- q)t 4 - 2(3t S + 4t 4 )tr( 0) + 48(q + 2q(3p - 3q + 1) + 6(p - q)
6(p - q) + 17}tr 2 (~e) + 24{q 2 (p - q) + (
7(p 2
- q) + (p - q) + 44 ) q +
4(20(p - q)
+
19)tr(~e)2 - 192(q
+
3(p - q)
+
+
1)tr(~e)tr(~e)2 -
1536tr(~e)3 + 96tr2(~e)2
d
S
= 8i4tr(~e)
- 96{q2 + q(3p - 3q + 2)
+
2(p - q)2 + 3(p - q) + 9}tr2(~e) -
P_q
+
24) + 8(4p - 4q
48{q2(p _ q) + q(Z(p - q)2
96(4q
d
6
= 48{q 2
+
7p - 7q
+ 2q(p - q
q((p - q)2
+
4)tr(~ )tr(~e)2
+
+
+ P _ q
1)
+
+ 20)
= 96{(p
+
l)tr(~e)tr(~e)
Acknowledgement:
2
5)tr(~e)2
+
1920tr(~e)3 - 384tr2(~e)2
2 2
(p - q) 2
+ 2(p - q) + 7}tr (~e) + 24{q (p - q) +
+
tr(~e)tr(~e)2 - 1280tr(~e)3
d7
+
+
+
4(Sp - Sq
+
+
8)}tr(~e)Z - 96(3q
384tr2(~e)2
4tr(~e)
322
- 2tr (te) }
Part of this paper lvas assisted by Yi Tsang.
+
4p - 4q
+
3)
-13-
References
AnaZysis~
[1]
Anderson, T. W. (1958): Introduation to MUZtivariate StatistiaaZ
Wiley, New York.
[2]
Fujikoshi, Y. (1970): Asymptotic expansions of the distributions of test
statistics in multivariate analysis. J. Sai. Hiroshima Univ. Sera
A-l~ 34, pp. 73-144.
[3]
Sugiura, N. (1969): Asymptotic non-null distributions of the likelihood
ratio criteria for covariance matrix under local alternatives.
f~imeo Series No. 609, Institute of Statistics, University of North
Carolina.