Int_.rn. t. J. Math. & Math. Sci.
Vol. 8 No. 3 (1985) 555-562
555
DISTRIBUTION OF LRC FOR TESTING SPHERICITY
OF A COMPLEX MULTIVARIATE GAUSSIAN MODEL
D. K. NAGAR and S. K. JAIN
Dpartment of Statistics
University of Rajasthan
Jaipur, Rajasthan, India
ad
A. K. GUPTA
Department of Mathematics and Statistics
Bowling Green State University
Bowling Green, Ohio 43403, U.S.A.
(Received February 13, 1984)
In this paper, exact null distribution of the likelihood ratio criterion
for testing sphericity structure in a complex multivariate normal covariance matrix
is obtained in computable series form. The method of inverse Mellin transform and
contour integration has been used. Certain special cases are given explicitly in
ABSTRACT.
terms of the hypergeometric functions.
KEY WORDS AND PHRASES. Exact distribution, moments, inverse Mellin transform, series
representation, Meijer s G-function.
18B0 Mathematics Subject Classification Code. 62HI0
I.
INTRODUCTION.
Let _Z’
(Z;
_’
vector
let
t,j
) be distributed as a complex multivariate normal with mean
(_,...,) and Hermitian positive definite covariance matrix Z, and
_Zt
Pt xl,
be of order
I
o2Ig
Pl
0
0
vs.
2 > 0
0
is unknown,
t
+
P2
Z
0
and
_j)’}= ltj,
0
L
o
_t )(_Zj
Pl
0
where
E{(_Zt
Also, Let
q.
Consider testing the following hypothesis
1,...,q.
H:
1
t
+
Pq
Pt
0
o2I
q
A:
# H
Pc
is the identity matrix of order
Pt’
t
1
p"
It is easy to see that the likelihood ratio criterion for testing
!
is a one
,q
556
A. K. GUPTA, D. K. NAGAR, AND S. K. JAIN
to one function of
q
IAI/
U
(tr
t=l
Pt
Att/Pt
where
N
A
with
(Atj)
N
Z_t,
is the k-th independent observation on
Ztk
U
The h-th moment of
where
N-
n
k
1,...,N,
t
is given in Nagarsenker and Nagarsenker
q
E(uh)
NZt. k=fZ_tk,
Cztk Z-t’)(zj- k -_Zj.)
k r=l
Atj
(nPt)
t=ltPtn
P
h)
+
{(n
j=
[1]
as
j) + h
j)
/
r(n
q.
1
/
(1.3)
(> p), and Re(h) >-n- 1 + p.
q
1, the hypothesis defined in (1.1) is the well known
Note that when
Mauchly’s sphericity hypothesis.
The non-null distribution of sphericity criterion
[2] in series involving zonal polynomials and
was given by Pillai and Nagarsenker
[3] expanded G-functions in series involving Psi and Zeta
functions Nagarsenker and Das [4] derived the null distribution and gave percentage
points. Krishnaiah, Lee and Chang [5] approximated the distribution of certain power
G-functions.
Mathai
of the test criterion by Pearson’s type I distribution
Recently, Nagarsenker and Nagarsenker [1] derived the distribution of
testing (1.1) in terms of beta and chi-sqaure series.
U
for
They have also obtained
asymptotic results using Box’s method.
In this article the exact null distribution of the test statistic
U
for
testing (1.1) is derived in terms of generalized hypergeometric function as well as
in series form by using inverse Mellin transform, definition of G-function and
calculus of residues.
First in section 2 the density is expressed in Meijer’s
G-function and some special cases are given explicitly.
is given in series of the Psi and Zeta functions.
In section 3, this density
Some special cases for
p
< 5
have also been given
THE DENSITY IN MEIJER’S G-FUNCTION
Using inverse Mellin transform and the expression (1.3) the density of
given by
2.
f(u)
(2)-1f u -1-hE (uh)dh,
0 < u <
U
is
(2 1)
C
and C is the contour enclosing all the poles of the integrand.
(-1)
Simplifying (1.3) by using Gauss-Legendre multiplication formula (Luke [6]), we have
where
E(U
h)
q
P
K(p
pq;n)
p
II r(n
+
+
j=l
Pt -1
h)/ rI
It r(n + h +
t=l j=O
j/pt
(2.2)
where
K(Pl,
,pq;n)
(2u)
(p-q)/2/
p
n rCn
j=l
+ 1
q
j)) n
t=l
nPt-t
{rCnPt)/p t
(2.33
COMPLEX MULTIVARIATE GAUSSIAN MODEL
Now, substituting (2.2) in (2.1) and subsequently putting
557
p + h
n
a,
the
density is obtained as
pq;n)(2)-lun-p-lf
KfPl
f(u)
Pt -1
q
P
H r(a + j)/ H
H
t=l j=O
C j=l
1
rfa
+
p
+
j/pt u-ada,
0 < u < 1
is the changed contour and the constant
C1
where
K(p 1
(2.4)
pq;n)
is defined by
(2.3).
Using the definition of G-function (Luke [6]), the density given in (2.4) can be
put in the form of the following theorem.
THEOREF4 2.1. The p.d.f, of the test statistic U defined by (1.2) for testing
the hypothesis (1.1), in the null case, is given by
n-p-1
pq;n)u
K(P
f(u)
(p, repeated q- times},{p
{j,j-1
p- 1}
p-l,p-1
+
j/pt,J=l
0 < u < 1
en
q
{2.5)
is given by (2.3).
pq;n)
K(p 1
where
q},
Pt-l,t=l
we get the distribution of the sphericity criterion as a corollary
1,
of the above theorem.
COROLLARY 2.1.
H"
o2I P
7.
The p.d.f, of the test statistic
IAl/(trA/p) p
for testing
in the complex normal population is given by
K(pn)un-p-16pP-l,p_ 1
f(u)
U
j/p
ipJ,
1
p-
P
0
U
1
(2.6)
I and Pl
K(p;n) is obtained from (2.3) by substituting q
The following special cases of (2.5) are stated below by using the results on
where the constant
G- function.
(i)
q
p
1,
(ii)
1,3/2)}-1un=2(1
{B(n
f(u)
where
2
B(a,b)
q
1, p
r (a) r {b) /r (a
2,
f{u)
(iv)
q
f{u)
2,
0 < u < 1
3
Pl
{r(2n)/15
{1
u),
0 < u < 1.
1
P2
{n- 1)u
Pl 1,
n-3
1)r{n- 2)}u
u)32Fl(4/3,5/3;4;1
(1
q
,
b).
{nr(3n)/33n+lr(n)F(n-
f(u)
(iii)
+
u)
n-2
P2
0 < u <
2
22n-4r{n-
n-3
1)r{n- 2)}u
u)S/22Fl{3/2,1;7/2;1
u),
0 < u < 1.
p"
A.K. GUPTA, D. K. NAGAR, AND S. K. JAIN
558
q
(v)
3,
f(u)
where
(n
I
PS
P2
Pl
l)2(n 2)un-3(l
u + u log u),
0 < u <
I,
is the Gauss’ hypergeometric function
2FI
THE DENSITY IN SERIES FORM.
Denoting the gamma products of the integrand in (2.4) by 5(a), and after
cancelling out the gammas occuring in the numerator with those of denominator, we get
(see Gupta and Nagar [7], and Nagar et. al. [8])
3.
p-q
n r(a
3=1
ACa)
p-1
(3.1)
Pt -1
q
(a + j) j-p+q
j=p-q+l
The poles of
j)
/
n
n r(a
+
p +
j/pt
t=1 j=l
x,
are available by equating to zero each factor of
a(a)
where the exponent
a
gives the order of the pole
i
-i,
a
N (a + i)
i=1
which can be seen to
be
q,
The integrand in (2.4) is
then the residue
Ri
1
R.
at
lira
A(a)u
-1
(i-
i)’"a
(i
im
1)! a+-i
-i
-a
(3.2)
+
and if
is the pole of order
-i
a
is given by
-i
a
p
i
(a + i)
x(a)u -a}
u-a
1
aa-.i
.-1
Z______.[ A.
i
-1
x
I), a
(i
lira
x
u -a I
r=0
-i
i-I )(-log
r
u)A x(r).
(3.3)
where
A.
i
(a + i)
p-q
a.
r x(a
n (a
+ j) j-p+q
+ i +
1)
N
r(a + j)
j=a.+l
/
j)
a.-1
x
Ca
j=l
jeS
q
a.
x
i-I
n Ca
/
n
Pt "I
n r (a
t=l j=l
j=a.
+ p +
j/pt
(.4)
S
{p
i
1,2,..,p
p
q + 1
I}
(3.s)
and
ai
q,
q
C3.63
q + 1
i
Now
A!I)
x
aA
---=
i
alog
Ai( a
A,)
A.B.
COMPLEX MULTIVARIATE GAUSSIAN MODEL
559
and
(r)
Ai
r-1 (AiBi)
r-I
Bar- I
m=0
(r
r.
l)A!r-l-m)B(m
I
i
m
where
3log A.
1
B.
p-q
+ i +
ai(a
1) +
(a
l
r (j
+ j)
j=a. +I
a.-1
Z
j( + j)
i-1
-1
Z
a.
i
p
q)(a’+ j)
-1
jS
(a
j)
+
. Pt
q
-1
-1
Z (a + p
t--I j=l
j=a.
i
j/pt
and
m+llog
mB.
B (m).
A.
m+l
p-q
(-l)m+lm![ai(m
1,c
a.-I
i
+
Z j(a + j)
j=l
-l-m
Z
(m + 1,a + j)
j=a. +I
I)
i
i-I
+ a.
(a
l
+ j)
q
Z
-l-m
r. (j-p+q)(a+j) -1-m
jES
Pt -I
Z (m
t=l j=l
j=a.
x
a +
p
j/pt)].
(3.9)
From (3.3) we can write
R
i
u
i
(i
m
-I
r
1)! r=O
i
A!
A (r).
(-log u)
r
6 -1-r
i
tr
A iO
(3.1o)
where
i
r-1
r
m=O
(r
at
-i
a
-m 1)A
-1-m)
B(m)i0
(5.11)
with
Aio A.x
at
a
-i
p-q
n
r(j
i)
j=a.+l
l
(j
i)
j-p+q
js
x
H (j
j=l
a.
i-1
(j
i)
j=a.
i)
x
q
n
=t
n r(-i
t=l j=l
p
j/pt
(3.12)
Bio B.I
at
a
-i
A.K. GUPTA, D. K. NAGAR, AND S. K. JAIN
560
p-q
r.
(j
=a. +I
1
+
ai(l
r
CJ
i)
p
+ q)(j
i)
-1
jS
a.-1
i
r. (j
i)
-1
i-I
E
j=a.
a.
I
j=l
CJ
i)
q
-1
z
Pt -1
r. (-i
+
p
(3.13)
j/pt
/
t=l j=l
and
B(m)
i
B(m)
iO
at
-i
a
p-q
(-1)m+lm![ai(m
a.-1
x
Z jfj
j=l
+
i)
1,1)
+
+
E
(m + 1,j
j=a.+l
i-1
-1-m
r.
+ a.
(j
i)
i)
p
+ q)(j
i)
r.
(m
+ 1,-i +
p
+
t=l j=l
j=a.
i
-1 -m
Pt -1
q
r.
-1-m
Z (j
jS
+,
j/pt)].
(3.14)
Note that (-) and (.,-) respectively are the Psi and Zeta functions (Abraowitz
and Stegun [9]). Using residue theorem, the integral in (2.4) is evaluated as a sum
of residues and the density of
U
is given in the following theorem.
The p.d.f, of the test statistic
THEOREM 3.1.
defined by (1.2) for testing
0
the hypothesis (1.1) is given by
K(p I
where
u
z
I),
i=1 C i
pq;n)u
..,pq;n)
6
A (r)
and
i
5.-1
i
n-p-I
K(Pl
f(u)
6.-1
I
r.
r=O
)(-log u)
r
i-l-rA(r)
i0
(3 1S)
are given by (2.3), (3.2) and (3.12)-(3.14)
i0
respectively.
The cumulative distribution function
F(u)
of
U
can be obtained by straight
forward integration of (5.15).
q
When
we get the distribution of the sphericity criterion as a corollary of
1,
the above theorem.
COROLLARY 3.1.
The p.d.f, of the test statistics
U
for testing
H"
Z
in the complex normal population is given by
f(u)
K(p;n)u n-p-1
+
.
[p.l
Li=
u
i
i
u
=
i-1
1),.
(i
r0(
p-2
K (p;n)
q= I,
and
r
z)(_og
z
(-log u)
i-l-r
A
p-z-)
r
(p- 2)! r
A (r) are obtained from (2.3) and (.12)-(.14) by substituting
i0
i=p
where
1
i
o2I P
pl=p.
e following special cases of (3.15) are stated below which have been simplified
by using the results on ga, Psi and Zeta functions.
2, Pl
3
(i) q
1, P2
f(u)
K(1
3;n)un-5[k2r ( O/a)u r 1/3)
u
2
4r(4/)r(s/) (-og u)
+
u2(-log
u + 3log 3
r (7/a) r
(-2og u)(log
2)
117/20)
2
+
27
+ (sog
s
2) 2)
COMPLEX MULTIVARIATE GAUSSIAN MODEL
C213) (113) u
u4
E
rCll3)rC2i3) j=o
2,
q
K(2,2;n)u
f(u)
u
3){Cj + 2).’}
+ 1)Cj +
1/3)},
(j +
2
(2Cj
33
/
CJ
+
+
13
)
}]
-1
-1
0 < u < 1.
2
P2
Pl
CJ
oCj + 2/3)
log u
(ii)
561
u)2
{(-log
3/2) 2}
+ (41og 2 +
/
16/3)
4log 2
22(5/2)
2(7/2)
3
42(3/2)
u (-log u
u
n-5
uir 2(i
7
+
log u + 2(i- 1) + (i- 3)
-1
)
4 +
1){ (i
3) (i
i=4 (i
}2r4()
2)!
1)
+ (i
22
3
3/2) +’41/4
+ (-21og u)(41og 2 +
-I
2(i -4 +
O<u<l.
q
(iii)
3,
,
K(1
f(u)
1,
P2
Pl
2
P3
r
u2(-log
u
2;n) un-5|4r(7/2)
-
21og 2- 11/3)
(5/2)
u +
h
2
3
2
u
41"C3/2’) {C-lg u)
i=4 (i- 2)
4,
q
(iv)
1) 3(n
(n
fCu)
2)
"53 2
2u2{(-log
2
u + 5/2) +
9/4}],
1
P4
8
+ CZ*og 2 +
2Cn
3) [I
25
0<u< I.
3)2(i-
Ci
P3
P2
Pl
i)u i
4 +
r(i
z
+
+-
3
+ C-tog u)(21og 2 +
8u(2 + log u)
+
0 <u < 1.
(v)
p
q
5,
f(u)
4,
K(2,1 1 1;n)u
+
{(-log
u)2
-u 3
(rCs/2)) +
+ (21og 2
u
4
C12r(32))*
(vi)
p
2,
Pl
S,
Pl
P3
P2
u
n-6
08r(9/2)
7
5) 2)
u
+
3
+
P5
-6;(8r(7/2))
259
36
+
35S
+--
+ 3
3
7
+-)
+
+ (21og 2-
3(-log u)
2
(.137
18
(j
3)(j *
2)2(j
3) C21og
*
-u 2
2
()2_
5
jO
19
2--)
2
3C-log u) (2log 2
12C3)
167.
(-log u + 2log 2
+
+ (-21og u)(21og
(-log u)
1
P4
1)3(j
2 +
7
)
167.2
--)
(13718
2
3
+ (2log 2 +
O<u<l.
* 1)
7 3
)
562
A.K. GUPTA, D. K. NAGAR, AND S. K. JAIN
f(u)
u
3)2(n 2)3(n 1)%n-6
4)(n
(n
3
u
{(-log u)
/-
3
u
2
-i-(-log
u
4)
21
3(-log u) +
2(-log u)
4
u
}
/
13( log u)
2
403.
/--(-log
u)
/
1214}
9
0 < u < 1.
REFERENCES
I. NAGARSENKER, B. N. and NAGARSENKER, P. B. Distribution. of the likelihood ratio
statistic for testing sphericity structure for a complex normal covariance
matrix, Sankhya, B43 (1981), 3S2-3S9.
PILLAI, K. C. S. and NAGARSENKER, B. N. On the distribution of sphericity test
criterion in classical and complex normal populations having unknown
covariance matrices, Ann. Math. Statist., 42 (1971), 764-767.
3. MATHAI, A. M. A few remarks about some recent articles on the exact distribution
of multivariate test criteria: I, Ann. Inst. Statist. Math., 25 (1973),
2.
566-577.
4.
NAGARSENKER, B. N. and DAS, M. M.
5.
KRISHNAIAH, P. R., LEE, J. C. and CHANG, T. C.
6.
LUKE, Y. L.
Press,
7.
GUPTA, A. K. and NAGAR, D. K.
Exact distribution of sphericity criterion in
the complex case and its percentage points, Comm. Statist., 4 {1975), 363-375.
The distribution of the likelihood
ratio statistics for tests of certain covariance structures of complex
multivariate normal populations, Biometrika, 63 (1976), 513-519.
H"
The Special Functions and Their Approximations, Vol. I, Academic
York, 1969.
Nw
0;
Z
o2I
Distribution of LR statistic for testing
in multivariate complex Gaussian distribution,
Department of Mathematics and Statistics, Bowling Green State University,
T.R. No. 83-03, 1983.
8.
NAGAR, D. K., JAIN, S. K. and GUPTA, A. K.
9.
ABRAMOWITZ, M. and STEGUN, I.A. Handbook of Mathematical Functions, Dover
publications, New York, 1965.
Distribution of LRC for testing
sphericity of a complex multivariate Gaussian model, Department of Mathematics
and Statistics, Bowling Green State University, T.R. No. 83-09, 1983.