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Divulgaciones Matem´
aticas Vol. 9 No. 1(2001), pp. 25–34
Testing Block Sphericity
of a Covariance Matrix
Prueba de Esfericidad por Bloques
para una Matriz de Covarianza
Liliam Carde˜
no (lacero@matematicas.udea.edu.co)
Daya K. Nagar (nagar@matematicas.udea.edu.co)
Departamento de Matem´aticas
Universidad de Antioquia, Medell´ın
A. A. 1226, Colombia
Abstract
This article deals with the problem of testing the hypothesis that q
p-variate normal distributions are independent and that their covariance
matrices are equal. The exact null distribution of the likelihood ratio
statistic when q = 2 is obtained using inverse Mellin transform and the
definition of Meijer’s G-function. Results for p = 2, 3, 4 and 5 are given
in computable series forms.
Key words and phrases: block sphericity, distribution, inverse
Mellin transform, Meijer’s G-function, residue theorem.
Resumen
Este art´ıculo trata el problema de probar la hiptesis de que q distribuciones normales p-variadas son independientes y con matrices de
covarianza son iguales. La distribuci´
on exacta nula del estad´ıstico de
raz´
on de verosimilitudes cuando q = 2 se obtiene usando la transformada inversa de Mellin y la definicin de la G-funcin de Meijer. Los
resultados para p = 2, 3, 4 y 5 se dan en forma de series calculables.
Palabras y frases clave: esfericidad por bloques, distribuci´
on, transformada inversa de Mellin, funci´
on G de Meijer, teorema del residuo.
Recibido 2000/09/18. Aceptado 2001/04/10.
MSC (2000): Primary 62H10; Secondary 62H15.
26
1
Liliam Carde˜
no, Daya K. Nagar
Introduction
Suppose that the pq × 1 random vector X has a multivariate normal distribution with mean vector µ and covariance matrix Σ and that X, µ and Σ
are partitioned as
¡
¢′
¡
¢′
µ = µ′1 µ′2 · · · µ′q
X = X ′1 X ′2 · · · X ′q ,
and
Σ11
Σ21
Σ= .
.
.
Σq1
Σ12
···
Σ22
···
Σq2
···
Σ1q
Σ2q
Σqq
where X i and µi are p × 1 and Σij is p × p, i, j = 1, . . . , q. Consider testing
the null hypothesis H that the subvectors X 1 , X 2 , . . . , X q are independent
and that the covariance matrices of these subvectors are equal, i.e.,
∆ 0 ··· 0
0 ∆ ··· 0
H:Σ=.
.
.
0
0
···
∆
against the alternative Ha that H is not true. In H the common covariance
matrix ∆ is unspecified. Olkin, while extending the circular symmetric model
to the case where the symmetries are exhibited in blocks, defined H and called
it block sphericity hypothesis (see [13]). Let A be the sample sum of squares
and product matrix formed from a sample of size N = n + 1. Partition A as
A = (Aij ) where Aij is p × p, i, j = 1, . . . , q. The likelihood ratio statistic for
testing H (see [17]) is given by
1
Λ=
det(A) 2 N
´ 12 qN .
³ P
q
det 1q i=1 Aii
The null hypothesis is rejected if Λ ≤ Λ0 where Λ0 is determined by the null
distribution and level of significance. When p = 1, the hypothesis of block
sphericity reduces to the Mauchly sphericity hypothesis H : Σ = σ 2 Iq which
has been studied extensively, e.g., see Sugira [15, 16], Khatri and Srivastava
Divulgaciones Matem´
aticas Vol. 9 No. 1 (2001), pp. 25–34
Testing Block Sphericity of a Covariance Matrix
27
[6, 7], Muirhead [10], Gupta [3], Nagar, Jain and Gupta [11] and Amey and
Gupta [1]. The hypothesis of block sphericity is useful in multivariate repeated
measures designs (see [17]). Since, A ∼ Wpq (n, Σ), the hth null moment of Λ
can be obtained by integrating over Wishart density. We have
Z
1
¢
¡
1
1
det(A) 2 N h
h
pqN h
2
2 (n−pq−1) etr − 1 Σ−1 A dA
E(Λ ) = q
cpq,n
q
2
¡P
¢ 12 qN h det(A)
Aii
A>0 det
i=1
=q
1
2 pqN h
cpq,n
Z
A>0
¶− 21 qN h
µX
q
¡
¢
1
det
Aii
det(A) 2 (N h+n−pq−1) etr − 12 Σ−1 A dA
i=1
where
cm,n = 2
1
2 nm
π
1
4 m(m−1)
m ·
Y
1
j=1
2
¸
(n − j + 1) det(Σ)
Consequently,
1
E(Λh ) = q 2 pqN h
cpq,n
E det
cpq,N h+n
à q
X
i=1
1
2n
−1
.
!− 12 qN h
Aii
(1.1)
(1.2)
where A ∼ Wpq (n + N h, Σ). Under H, AP
11 , . . . , Aqq are independent Wishart
q
matrices, Aii ∼ Wp (n+N h, ∆). Hence, i=1 Aii ∼ Wp ((n+N h)q, ∆). Using
Theorem 3.3.22 of [5], we obtain
" µ q
¶− 21 qN h #
X
E det
Aii
i=1
=2
− 21 pqN h
− 12 qN h
det(∆)
p
Y
(1.3)
Γ[ 21 (nq − i + 1)]
.
Γ[ 12 (nq + N qh − i + 1)]
i=1
Substituting (1.3) in (1.2) and using (1.1) we obtain the hth null moment of
Λ as
1
E(Λh ) = q 2 pqN h
pq
p
Y
Γ[ 21 (n + N h − j + 1)] Y
Γ[ 12 (nq − i + 1)]
. (1.4)
Γ[ 21 (n − j + 1)]
Γ[ 21 (nq + N qh − i + 1)]
j=1
i=1
The exact non-null distribution of Λ, for q = 2 and under two specific alternatives, has been derived by Gupta and Chao [4]. The asymptotic null
Divulgaciones Matem´
aticas Vol. 9 No. 1 (2001), pp. 25–34
28
Liliam Carde˜
no, Daya K. Nagar
distribution of −2 ln Λ is chi-square with 12 p(q − 1)(pq + p + 1) d.f. The null
2
distribution of Λ N , in series involving Bernoulli polynomials, is available in
[?, CG]
1
In this article we will derive the exact null distribution of V = Λ N by
using inverse Mellin transform and residue theorem (see [14, 12]).
2
Exact density of V
1
In order to obtain the exact density function f (v) of V = Λ N we start with
the null moment expression. Substituting q = 2 in (1.4) the hth moment of V
is simplified as
¾ p ½
¾
2p ½
Y
Γ[ 21 h + 12 (n − j + 1)] Y
Γ[n − 21 (j − 1)]
h
ph
E(V ) = 2
Γ[ 12 (n − j + 1)]
Γ[h + n − 21 (j − 1)]
j=1
j=1
This will be rewritten in a slightly different form by using duplication formula
for gamma function
¶
µ
1
22z−1
.
Γ(2z) = √ Γ(z)Γ z +
2
π
Then
E(V h ) = K(n, p)
¾
p ½
Y
Γ(h + n − 2p − 1 + 2j)
j=1
where
K(n, p) =
Γ[h + n − 12 (j − 1)]
¾
p ½
Y
Γ[n − 21 (j − 1)]
.
Γ(n − 2p − 1 + 2j)
j=1
(2.1)
(2.2)
1
Now the density function f (v) of V = Λ N is obtained by taking inverse Mellin
transform of E(V h ) as
Z
−1
E(V h )v −h−1 dh, 0 < v < 1,
(2.3)
f (v) = (2πι)
√
L
where ι = −1 and L is a suitable contour. Substituting (2.1) in (2.3) and
applying the transformation h + n − 2p = t, one obtains
¾
Z Y
p ½
Γ(t + 2j − 1)
n−2p−1
−1
v −t dt,
f (v) = K(n, p)v
(2πι)
1
L1 j=1 Γ[t + 2p − 2 (j − 1)]
0 < v < 1,
Divulgaciones Matem´
aticas Vol. 9 No. 1 (2001), pp. 25–34
(2.4)
Testing Block Sphericity of a Covariance Matrix
29
where L1 is the changed contour and K(n, p) is defined in (2.2). Now, using
the definition of Meijer’s G-function (see [8]), the density of V is written as
¸
· ¯
¯2p − 12 (j − 1), j = 1, . . . , p
, 0 < v < 1,
v
f (v) = K(n, p)v n−2p−1 Gp,0
¯
p,p
2j − 1, j = 1, . . . , p
where Gp,0
p,p [ ] is the Meijer’s G-function.
Explicit expressions for the density of V for particular values of p will
be obtained by evaluating the integral in (2.4) with the help of of residue
theorem.
From (2.4), the density for p = 2 is obtained as
f (v) = K(n, 2)v n−5 (2πι)−1
Z
L1
Γ(t + 1)
v −t dt, 0 < v < 1.
(t + 3)Γ(t + 72 )
(2.5)
The integrand has simple poles at t = −r − 1, r = 0, 1, 3, 4, . . ., and a pole of
order two at t = −3. The residue at t = −r − 1, r 6= 2, is
¸
(t + 1 + r)Γ(t + 1) −t
v
lim
t→−r−1
(t + 3)Γ(t + 72 )
·
¸
(t + 1 + r)(t + 1 + r − 1) · · · (t + 1)Γ(t + 1) −t
= lim
v
t→−r−1
(t + 1 + r − 1) · · · (t + 1)(t + 3)Γ(t + 27 )
¸
·
Γ(t + 1 + r + 1)
−t
v
= lim
t→−r−1 (t + 1 + r − 1) · · · (t + 1)(t + 3)Γ(t + 7 )
2
·
=
(−1)r+1
1
v r+1 .
r! (r − 2)Γ( 25 − r)
Simplifying Γ( 52 − r) in the above expression using the result Γ(β − j) =
(−1)j Γ(β)Γ(1−β)
Γ(1−β+j)
we obtain the residue at t = −r − 1 as
−
1 Γ(r − 23 ) r+1
v , r 6= 2.
π r! (r − 2)
Divulgaciones Matem´
aticas Vol. 9 No. 1 (2001), pp. 25–34
30
Liliam Carde˜
no, Daya K. Nagar
The residue at t = −3 is derived as
¸
¸
·
·
∂
Γ(t + 4)
∂ (t + 3)2 Γ(t + 1) −t
−t
=
lim
v
v
lim
t→−3 ∂t (t + 1)(t + 2)Γ(t + 7 )
t→−3 ∂t
(t + 3)Γ(t + 72 )
2
½
¸
¾
·
∂
Γ(t + 4)
Γ(t + 4)
= lim
− ln v
v −t
ln
7
t→−3 ∂t
(t + 1)(t + 2)Γ(t + 2 )
(t + 1)(t + 2)Γ(t + 27 )
½
¾
X
Γ(t + 4)
−t
(t + i)−1 − ψ(t + 72 ) − ln v
= lim ψ(t + 4) −
7 v
t→−3
)
(t
+
1)(t
+
2)Γ(t
+
2
i=1,2
½
¾
1
1
1
= ψ(1) + + 1 − ψ( 2 ) − ln v
v3
2
(−2)(−1)Γ( 12 )
¾
½
1
3
+ 2 ln 2 − ln v √ v 3
=
2
2 π
where ψ( ) is the psi function (ver [9]). Now applying the residue theorem to
the right hand side of (2.5), the density f (v) is obtained as
·
½
¸
∞
³v´ ¾ 1
1 X Γ(r − 23 ) r+1
3
√ v3
f (v) = K(n, 2)v n−5 −
v
+
− ln
π
r! (r − 2)
2
4
2 π
r=0r6=2
for 0 < v < 1. For p = 3, (2.4) simplifies to
Z
Γ(t + 1)
f (v) = K(n, 3)v n−7 (2πι)−1
L1 (t + 5)(t + 4)(t + 3)Γ(t +
11 v
2 )
−t
dt
(2.6)
The integrand has simple poles at t = −r − 1, r = 0, 1, 5, 6, . . ., and poles of
order two at t = −r − 1, r = 2, 3, 4. Evaluating residues at these poles and
using residue theorem, the density in this case is obtained (for 0 < v < 1) as
·
∞
Γ(r − 72 )
K(n, 3) n−7 2
4 2
1 X
√
√
f (v) =
v− v −
v r+1
v
315
45
π
π r=5 r! (r − 2)(r − 3)(r − 4)
¸
1³8
1 ³ 43
v ´ 3 1³1
v´ 4
v´ 5
+
v −
v +
v .
− ln
+ ln
− ln
3 3
4
3 6
4
48 12
4
For p = 4, (2.4) reduces to
f (v) = K(n, 4)v n−9 (2πι)−1
Z
L1
Γ(t + 1)Γ(t + 3)
7
Q
j=5
(t + j)Γ(t +
15
2 )Γ(t
v −t dt.
+
13
2 )
Divulgaciones Matem´
aticas Vol. 9 No. 1 (2001), pp. 25–34
(2.7)
Testing Block Sphericity of a Covariance Matrix
31
The integrand has simple poles at t = −1, −2, poles of order two at t = −1−r,
r = 2, 3, 7, 8, . . ., and poles of order three at t = −1 − r, r = 4, 5, 6. Evaluating
residues at these poles and using the residue theorem, we get the density for
p = 4 as
f (v) = K(n, 4)v n−9
−
+
·
1
1
v−
v2
2( 9 )
660 Γ2 ( 11
270
Γ
)
2
2
³
1
v´ 3
1
2521
v −
+
ln
7
420
2
16
168 Γ ( 2 )
90 Γ2 ( 52 )
µ
∞ ½
7
X
1 X
1
− ψ(r −
ψ(r
+
1)
+
ψ(r
−
1)
+
π 2 r=7
r
−
j
j=4
71
v
+ ln
15
16
11
2 )
¶
v4
− ψ(r − 92 ) − ln v
¾
½³
¾
Γ(r − 92 )Γ(r − 11
1
13 ´2 1781 2π 2 v 5
v
2 )
v r+1 +
+
−
+
ln
r!(r − 2)!(r − 4)(r − 5)(r − 6)
12π
16
4
144
3
2
½³
¾
127
v ´2 31769 2π 2 v 6
1
− ln
−
+
−
360π
60
16
3600
3
2
½³
¾
¸
v
121 ´2
2π 2 v 7
33
15
, 0 < v < 1.
− ln
+
−
+
−
2(6!)2 π
16
30
200
3
2
For p = 5, (2.4) slides to
f (v) =
K(n, 5)v
n−11
(2πι)
−1
Z
L1
Γ(t + 1)Γ(t + 3)
Q9
aj
j=5 (t + j) Γ(t +
17
2 )Γ(t
+
19
2 )
v −t dt,
0 < v < 1, (2.8)
where a5 = a6 = a8 = a9 = 1 and a7 = 2. The integrand has simple poles at
t = −1, −2, poles of order 2 at t = −1 − r, r = 2, 3, 9, 10, . . ., poles of order 3
at t = −5, −6, −8, −9 and a pole of order 4 at t = −7. Evaluating residues at
Divulgaciones Matem´
aticas Vol. 9 No. 1 (2001), pp. 25–34
32
Liliam Carde˜
no, Daya K. Nagar
these poles and using residue theorem, the density is derived as
f (v) = K(n, 5)v
n−11
(0)
·
(0)
(0)
(0)
(0)
A1 v + A2 v 2 + (− ln v + B3 )A3 v 3
(0)
+ (− ln v + B4 )A4 v 4 +
o
j
X n
(0) v
(0)
(1)
Aj
(− ln v + Bj )2 + Bj
2
j=5,6,8
o
n
7
(0) v
(0)
(1)
(0) (1)
(2)
A7
+ (− ln v + B7 )3 + 3(− ln v)B7 + 3B7 B7 + B7
3!
¸
∞
X
r
+
(− ln v + Br(0) )A(0)
r v , 0 < v < 1.
r=9
where
(0)
A1 =
(0)
B3
12
,
(10)!Γ2 ( 15
2 )
=−
(0)
A5 =
(0)
A6 =
(0)
(0)
5219
+ 2 ln 4,
693
B5
2
,
15(6)!Γ2 ( 25 )
B6
1
,
3(5)!(7)!π
B8
7
,
2(9)!Γ2 ( 23 )
A3 =
2
,
3(6)!Γ2 ( 92 )
B4
(0)
1
,
44(6)!Γ2 ( 11
2 )
1691
+ 2 ln 4
252
(0)
=−
(1)
=
1202849 2π 2
−
,
88200
3
(1)
=
10969 2π 2
−
,
720
3
(1)
=
911681 2π 2
−
,
88200
3
(1)
=
24947 2π 2
−
,
1800
3
=−
569
+ 2 ln 4,
105
B5
(0)
=−
69
+ 2 ln 4,
20
B6
(0)
=
989
+ 2 ln 4,
210
B8
(0)
=−
2
+ 2 ln 4,
15
B7
656261
+ 2π 2 − 4ζ(3),
54000
15
Γ(r − 13
2 )Γ(r − 2 )
=−
Q
9
π 2 r!(r − 2)! j=5 (r + 1 − j)aj
(2)
A(0)
r
4
,
5(7)!Γ2 ( 13
2 )
(0)
B7
B7
and
(0)
A4 =
5
,
(8)!Γ2 ( 27 )
A8 = −
A7 =
(0)
A2 =
=−
Br(0) = ψ(r + 1) + ψ(r − 1) +
9
X
j=5
aj (r + 1 − j)−1 − ψ(r −
15
13
) − ψ(r − ).
2
2
Divulgaciones Matem´
aticas Vol. 9 No. 1 (2001), pp. 25–34
Testing Block Sphericity of a Covariance Matrix
33
The method employed above can also be used to derive the exact density
for p ≥ 6. However, because of the higher order of poles, the expressions for
the density will involve generalized Riemann zeta function (see [9]).
Acknowledgment
This research was supported by the Comit´e para el Desarrollo de la Investigaci´on, Universidad de Antioquia research grant no. IN358CE.
References
[1] Amey, A. K. A., Gupta, A. K. Testing sphericity under a mixture model,
Australian Journal of Statistics, 34 (1992), 451–460.
[2] Chao, C. C., Gupta, A. K. Testing of homogeneity of diagonal blocks
with blockwise independence, Communications in Statistics–Theory and
Methods, 20(7) (1991), 1957–1969.
[3] Gupta, A. K. On the distribution of sphericity test criterion in the multivariate Gaussian distribution, Australian Journal of Statistics, 19 (1977),
202–205.
[4] Gupta, A. K., Chao, C. C. Nonnull distribution of the likelihood ratio
criterion for testing the equality of diagonal blocks with blockwise independence, Acta Mathematica Scientia, 14(2) (1994), 195–203.
[5] Gupta, A. K., Nagar, D. K. Matrix Variate Distributions, Chapman &
Hall/CRC Press, Boca Raton, 1999.
[6] Khatri, C. G., Srivastava, M. S. On exact nonnull distribution of likelihood ratio criteria for sphericity test and equality of two covariance
matrices, Sankhy¯
a, A33 (1971), 201–206.
[7] Khatri, C. G., Srivastava, M. S. Asymptotic expansion of the nonnull
distribution of likelihood ratio criteria for covariance matrices, Annals of
Statistics, 2 (1974), 109–117.
[8] Luke, Y. L. The Special Functions and Their Approximations, Vol. I,
Academic Press, New York, 1969.
[9] Magnus, W., Oberhettinger, F., Soni, R. P. Formulas and Theorems for
the Special Functions of Mathematical Physics, 52, Springer-Verlag, 1966.
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aticas Vol. 9 No. 1 (2001), pp. 25–34
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Liliam Carde˜
no, Daya K. Nagar
[10] Muirhead, R. J. Forms for distribution for ellipticity statistic for bivariate
sphericity, Communications in Statistics, A5 (1976), 413–420.
[11] Nagar, D. K., Jain, S. K., Gupta, A. K. Distribution of LRC for testing sphericity structure of a covariance matrix in multivariate normal
distribution, Metron, 49 (1991), 435–457.
[12] Nagar, D. K., Jain, S. K., Gupta, A. K. On testing circular stationarity
and related models, Journal of Statistical Computation and Simulation,
29 (1988), 225–239.
[13] Olkin, I. Testing and estimation for structures which are circular symmetric in blocks, in Multivariate Statistical Inference (D. G. Kabe and R.
P. Gupta, eds.), North-Holland, 1973, 183–195.
[14] Springer, M. D. Algebra of Random Variables, John Wiley & Sons, New
York, 1979.
[15] Sugira, N. Asymptotic expansion of the distribution of likelihood ratio
criteria for covariance matrices, Annals of Mathematical Statistics, 42
(1969), 764–767.
[16] Sugira, N. Exact nonnull distributions of sphericity tests for trivariate
normal population with power comparison, American Journal of Mathematical and Management Sciences, 15 (3&4) (1995), 355–374.
[17] Thomas, D. Roland. Univariate repeated measures techniques applied to
multivariate data, Psychometrika, 48(3) (1983), 451–464.
Divulgaciones Matem´
aticas Vol. 9 No. 1 (2001), pp. 25–34
aticas Vol. 9 No. 1(2001), pp. 25–34
Testing Block Sphericity
of a Covariance Matrix
Prueba de Esfericidad por Bloques
para una Matriz de Covarianza
Liliam Carde˜
no (lacero@matematicas.udea.edu.co)
Daya K. Nagar (nagar@matematicas.udea.edu.co)
Departamento de Matem´aticas
Universidad de Antioquia, Medell´ın
A. A. 1226, Colombia
Abstract
This article deals with the problem of testing the hypothesis that q
p-variate normal distributions are independent and that their covariance
matrices are equal. The exact null distribution of the likelihood ratio
statistic when q = 2 is obtained using inverse Mellin transform and the
definition of Meijer’s G-function. Results for p = 2, 3, 4 and 5 are given
in computable series forms.
Key words and phrases: block sphericity, distribution, inverse
Mellin transform, Meijer’s G-function, residue theorem.
Resumen
Este art´ıculo trata el problema de probar la hiptesis de que q distribuciones normales p-variadas son independientes y con matrices de
covarianza son iguales. La distribuci´
on exacta nula del estad´ıstico de
raz´
on de verosimilitudes cuando q = 2 se obtiene usando la transformada inversa de Mellin y la definicin de la G-funcin de Meijer. Los
resultados para p = 2, 3, 4 y 5 se dan en forma de series calculables.
Palabras y frases clave: esfericidad por bloques, distribuci´
on, transformada inversa de Mellin, funci´
on G de Meijer, teorema del residuo.
Recibido 2000/09/18. Aceptado 2001/04/10.
MSC (2000): Primary 62H10; Secondary 62H15.
26
1
Liliam Carde˜
no, Daya K. Nagar
Introduction
Suppose that the pq × 1 random vector X has a multivariate normal distribution with mean vector µ and covariance matrix Σ and that X, µ and Σ
are partitioned as
¡
¢′
¡
¢′
µ = µ′1 µ′2 · · · µ′q
X = X ′1 X ′2 · · · X ′q ,
and
Σ11
Σ21
Σ= .
.
.
Σq1
Σ12
···
Σ22
···
Σq2
···
Σ1q
Σ2q
Σqq
where X i and µi are p × 1 and Σij is p × p, i, j = 1, . . . , q. Consider testing
the null hypothesis H that the subvectors X 1 , X 2 , . . . , X q are independent
and that the covariance matrices of these subvectors are equal, i.e.,
∆ 0 ··· 0
0 ∆ ··· 0
H:Σ=.
.
.
0
0
···
∆
against the alternative Ha that H is not true. In H the common covariance
matrix ∆ is unspecified. Olkin, while extending the circular symmetric model
to the case where the symmetries are exhibited in blocks, defined H and called
it block sphericity hypothesis (see [13]). Let A be the sample sum of squares
and product matrix formed from a sample of size N = n + 1. Partition A as
A = (Aij ) where Aij is p × p, i, j = 1, . . . , q. The likelihood ratio statistic for
testing H (see [17]) is given by
1
Λ=
det(A) 2 N
´ 12 qN .
³ P
q
det 1q i=1 Aii
The null hypothesis is rejected if Λ ≤ Λ0 where Λ0 is determined by the null
distribution and level of significance. When p = 1, the hypothesis of block
sphericity reduces to the Mauchly sphericity hypothesis H : Σ = σ 2 Iq which
has been studied extensively, e.g., see Sugira [15, 16], Khatri and Srivastava
Divulgaciones Matem´
aticas Vol. 9 No. 1 (2001), pp. 25–34
Testing Block Sphericity of a Covariance Matrix
27
[6, 7], Muirhead [10], Gupta [3], Nagar, Jain and Gupta [11] and Amey and
Gupta [1]. The hypothesis of block sphericity is useful in multivariate repeated
measures designs (see [17]). Since, A ∼ Wpq (n, Σ), the hth null moment of Λ
can be obtained by integrating over Wishart density. We have
Z
1
¢
¡
1
1
det(A) 2 N h
h
pqN h
2
2 (n−pq−1) etr − 1 Σ−1 A dA
E(Λ ) = q
cpq,n
q
2
¡P
¢ 12 qN h det(A)
Aii
A>0 det
i=1
=q
1
2 pqN h
cpq,n
Z
A>0
¶− 21 qN h
µX
q
¡
¢
1
det
Aii
det(A) 2 (N h+n−pq−1) etr − 12 Σ−1 A dA
i=1
where
cm,n = 2
1
2 nm
π
1
4 m(m−1)
m ·
Y
1
j=1
2
¸
(n − j + 1) det(Σ)
Consequently,
1
E(Λh ) = q 2 pqN h
cpq,n
E det
cpq,N h+n
à q
X
i=1
1
2n
−1
.
!− 12 qN h
Aii
(1.1)
(1.2)
where A ∼ Wpq (n + N h, Σ). Under H, AP
11 , . . . , Aqq are independent Wishart
q
matrices, Aii ∼ Wp (n+N h, ∆). Hence, i=1 Aii ∼ Wp ((n+N h)q, ∆). Using
Theorem 3.3.22 of [5], we obtain
" µ q
¶− 21 qN h #
X
E det
Aii
i=1
=2
− 21 pqN h
− 12 qN h
det(∆)
p
Y
(1.3)
Γ[ 21 (nq − i + 1)]
.
Γ[ 12 (nq + N qh − i + 1)]
i=1
Substituting (1.3) in (1.2) and using (1.1) we obtain the hth null moment of
Λ as
1
E(Λh ) = q 2 pqN h
pq
p
Y
Γ[ 21 (n + N h − j + 1)] Y
Γ[ 12 (nq − i + 1)]
. (1.4)
Γ[ 21 (n − j + 1)]
Γ[ 21 (nq + N qh − i + 1)]
j=1
i=1
The exact non-null distribution of Λ, for q = 2 and under two specific alternatives, has been derived by Gupta and Chao [4]. The asymptotic null
Divulgaciones Matem´
aticas Vol. 9 No. 1 (2001), pp. 25–34
28
Liliam Carde˜
no, Daya K. Nagar
distribution of −2 ln Λ is chi-square with 12 p(q − 1)(pq + p + 1) d.f. The null
2
distribution of Λ N , in series involving Bernoulli polynomials, is available in
[?, CG]
1
In this article we will derive the exact null distribution of V = Λ N by
using inverse Mellin transform and residue theorem (see [14, 12]).
2
Exact density of V
1
In order to obtain the exact density function f (v) of V = Λ N we start with
the null moment expression. Substituting q = 2 in (1.4) the hth moment of V
is simplified as
¾ p ½
¾
2p ½
Y
Γ[ 21 h + 12 (n − j + 1)] Y
Γ[n − 21 (j − 1)]
h
ph
E(V ) = 2
Γ[ 12 (n − j + 1)]
Γ[h + n − 21 (j − 1)]
j=1
j=1
This will be rewritten in a slightly different form by using duplication formula
for gamma function
¶
µ
1
22z−1
.
Γ(2z) = √ Γ(z)Γ z +
2
π
Then
E(V h ) = K(n, p)
¾
p ½
Y
Γ(h + n − 2p − 1 + 2j)
j=1
where
K(n, p) =
Γ[h + n − 12 (j − 1)]
¾
p ½
Y
Γ[n − 21 (j − 1)]
.
Γ(n − 2p − 1 + 2j)
j=1
(2.1)
(2.2)
1
Now the density function f (v) of V = Λ N is obtained by taking inverse Mellin
transform of E(V h ) as
Z
−1
E(V h )v −h−1 dh, 0 < v < 1,
(2.3)
f (v) = (2πι)
√
L
where ι = −1 and L is a suitable contour. Substituting (2.1) in (2.3) and
applying the transformation h + n − 2p = t, one obtains
¾
Z Y
p ½
Γ(t + 2j − 1)
n−2p−1
−1
v −t dt,
f (v) = K(n, p)v
(2πι)
1
L1 j=1 Γ[t + 2p − 2 (j − 1)]
0 < v < 1,
Divulgaciones Matem´
aticas Vol. 9 No. 1 (2001), pp. 25–34
(2.4)
Testing Block Sphericity of a Covariance Matrix
29
where L1 is the changed contour and K(n, p) is defined in (2.2). Now, using
the definition of Meijer’s G-function (see [8]), the density of V is written as
¸
· ¯
¯2p − 12 (j − 1), j = 1, . . . , p
, 0 < v < 1,
v
f (v) = K(n, p)v n−2p−1 Gp,0
¯
p,p
2j − 1, j = 1, . . . , p
where Gp,0
p,p [ ] is the Meijer’s G-function.
Explicit expressions for the density of V for particular values of p will
be obtained by evaluating the integral in (2.4) with the help of of residue
theorem.
From (2.4), the density for p = 2 is obtained as
f (v) = K(n, 2)v n−5 (2πι)−1
Z
L1
Γ(t + 1)
v −t dt, 0 < v < 1.
(t + 3)Γ(t + 72 )
(2.5)
The integrand has simple poles at t = −r − 1, r = 0, 1, 3, 4, . . ., and a pole of
order two at t = −3. The residue at t = −r − 1, r 6= 2, is
¸
(t + 1 + r)Γ(t + 1) −t
v
lim
t→−r−1
(t + 3)Γ(t + 72 )
·
¸
(t + 1 + r)(t + 1 + r − 1) · · · (t + 1)Γ(t + 1) −t
= lim
v
t→−r−1
(t + 1 + r − 1) · · · (t + 1)(t + 3)Γ(t + 27 )
¸
·
Γ(t + 1 + r + 1)
−t
v
= lim
t→−r−1 (t + 1 + r − 1) · · · (t + 1)(t + 3)Γ(t + 7 )
2
·
=
(−1)r+1
1
v r+1 .
r! (r − 2)Γ( 25 − r)
Simplifying Γ( 52 − r) in the above expression using the result Γ(β − j) =
(−1)j Γ(β)Γ(1−β)
Γ(1−β+j)
we obtain the residue at t = −r − 1 as
−
1 Γ(r − 23 ) r+1
v , r 6= 2.
π r! (r − 2)
Divulgaciones Matem´
aticas Vol. 9 No. 1 (2001), pp. 25–34
30
Liliam Carde˜
no, Daya K. Nagar
The residue at t = −3 is derived as
¸
¸
·
·
∂
Γ(t + 4)
∂ (t + 3)2 Γ(t + 1) −t
−t
=
lim
v
v
lim
t→−3 ∂t (t + 1)(t + 2)Γ(t + 7 )
t→−3 ∂t
(t + 3)Γ(t + 72 )
2
½
¸
¾
·
∂
Γ(t + 4)
Γ(t + 4)
= lim
− ln v
v −t
ln
7
t→−3 ∂t
(t + 1)(t + 2)Γ(t + 2 )
(t + 1)(t + 2)Γ(t + 27 )
½
¾
X
Γ(t + 4)
−t
(t + i)−1 − ψ(t + 72 ) − ln v
= lim ψ(t + 4) −
7 v
t→−3
)
(t
+
1)(t
+
2)Γ(t
+
2
i=1,2
½
¾
1
1
1
= ψ(1) + + 1 − ψ( 2 ) − ln v
v3
2
(−2)(−1)Γ( 12 )
¾
½
1
3
+ 2 ln 2 − ln v √ v 3
=
2
2 π
where ψ( ) is the psi function (ver [9]). Now applying the residue theorem to
the right hand side of (2.5), the density f (v) is obtained as
·
½
¸
∞
³v´ ¾ 1
1 X Γ(r − 23 ) r+1
3
√ v3
f (v) = K(n, 2)v n−5 −
v
+
− ln
π
r! (r − 2)
2
4
2 π
r=0r6=2
for 0 < v < 1. For p = 3, (2.4) simplifies to
Z
Γ(t + 1)
f (v) = K(n, 3)v n−7 (2πι)−1
L1 (t + 5)(t + 4)(t + 3)Γ(t +
11 v
2 )
−t
dt
(2.6)
The integrand has simple poles at t = −r − 1, r = 0, 1, 5, 6, . . ., and poles of
order two at t = −r − 1, r = 2, 3, 4. Evaluating residues at these poles and
using residue theorem, the density in this case is obtained (for 0 < v < 1) as
·
∞
Γ(r − 72 )
K(n, 3) n−7 2
4 2
1 X
√
√
f (v) =
v− v −
v r+1
v
315
45
π
π r=5 r! (r − 2)(r − 3)(r − 4)
¸
1³8
1 ³ 43
v ´ 3 1³1
v´ 4
v´ 5
+
v −
v +
v .
− ln
+ ln
− ln
3 3
4
3 6
4
48 12
4
For p = 4, (2.4) reduces to
f (v) = K(n, 4)v n−9 (2πι)−1
Z
L1
Γ(t + 1)Γ(t + 3)
7
Q
j=5
(t + j)Γ(t +
15
2 )Γ(t
v −t dt.
+
13
2 )
Divulgaciones Matem´
aticas Vol. 9 No. 1 (2001), pp. 25–34
(2.7)
Testing Block Sphericity of a Covariance Matrix
31
The integrand has simple poles at t = −1, −2, poles of order two at t = −1−r,
r = 2, 3, 7, 8, . . ., and poles of order three at t = −1 − r, r = 4, 5, 6. Evaluating
residues at these poles and using the residue theorem, we get the density for
p = 4 as
f (v) = K(n, 4)v n−9
−
+
·
1
1
v−
v2
2( 9 )
660 Γ2 ( 11
270
Γ
)
2
2
³
1
v´ 3
1
2521
v −
+
ln
7
420
2
16
168 Γ ( 2 )
90 Γ2 ( 52 )
µ
∞ ½
7
X
1 X
1
− ψ(r −
ψ(r
+
1)
+
ψ(r
−
1)
+
π 2 r=7
r
−
j
j=4
71
v
+ ln
15
16
11
2 )
¶
v4
− ψ(r − 92 ) − ln v
¾
½³
¾
Γ(r − 92 )Γ(r − 11
1
13 ´2 1781 2π 2 v 5
v
2 )
v r+1 +
+
−
+
ln
r!(r − 2)!(r − 4)(r − 5)(r − 6)
12π
16
4
144
3
2
½³
¾
127
v ´2 31769 2π 2 v 6
1
− ln
−
+
−
360π
60
16
3600
3
2
½³
¾
¸
v
121 ´2
2π 2 v 7
33
15
, 0 < v < 1.
− ln
+
−
+
−
2(6!)2 π
16
30
200
3
2
For p = 5, (2.4) slides to
f (v) =
K(n, 5)v
n−11
(2πι)
−1
Z
L1
Γ(t + 1)Γ(t + 3)
Q9
aj
j=5 (t + j) Γ(t +
17
2 )Γ(t
+
19
2 )
v −t dt,
0 < v < 1, (2.8)
where a5 = a6 = a8 = a9 = 1 and a7 = 2. The integrand has simple poles at
t = −1, −2, poles of order 2 at t = −1 − r, r = 2, 3, 9, 10, . . ., poles of order 3
at t = −5, −6, −8, −9 and a pole of order 4 at t = −7. Evaluating residues at
Divulgaciones Matem´
aticas Vol. 9 No. 1 (2001), pp. 25–34
32
Liliam Carde˜
no, Daya K. Nagar
these poles and using residue theorem, the density is derived as
f (v) = K(n, 5)v
n−11
(0)
·
(0)
(0)
(0)
(0)
A1 v + A2 v 2 + (− ln v + B3 )A3 v 3
(0)
+ (− ln v + B4 )A4 v 4 +
o
j
X n
(0) v
(0)
(1)
Aj
(− ln v + Bj )2 + Bj
2
j=5,6,8
o
n
7
(0) v
(0)
(1)
(0) (1)
(2)
A7
+ (− ln v + B7 )3 + 3(− ln v)B7 + 3B7 B7 + B7
3!
¸
∞
X
r
+
(− ln v + Br(0) )A(0)
r v , 0 < v < 1.
r=9
where
(0)
A1 =
(0)
B3
12
,
(10)!Γ2 ( 15
2 )
=−
(0)
A5 =
(0)
A6 =
(0)
(0)
5219
+ 2 ln 4,
693
B5
2
,
15(6)!Γ2 ( 25 )
B6
1
,
3(5)!(7)!π
B8
7
,
2(9)!Γ2 ( 23 )
A3 =
2
,
3(6)!Γ2 ( 92 )
B4
(0)
1
,
44(6)!Γ2 ( 11
2 )
1691
+ 2 ln 4
252
(0)
=−
(1)
=
1202849 2π 2
−
,
88200
3
(1)
=
10969 2π 2
−
,
720
3
(1)
=
911681 2π 2
−
,
88200
3
(1)
=
24947 2π 2
−
,
1800
3
=−
569
+ 2 ln 4,
105
B5
(0)
=−
69
+ 2 ln 4,
20
B6
(0)
=
989
+ 2 ln 4,
210
B8
(0)
=−
2
+ 2 ln 4,
15
B7
656261
+ 2π 2 − 4ζ(3),
54000
15
Γ(r − 13
2 )Γ(r − 2 )
=−
Q
9
π 2 r!(r − 2)! j=5 (r + 1 − j)aj
(2)
A(0)
r
4
,
5(7)!Γ2 ( 13
2 )
(0)
B7
B7
and
(0)
A4 =
5
,
(8)!Γ2 ( 27 )
A8 = −
A7 =
(0)
A2 =
=−
Br(0) = ψ(r + 1) + ψ(r − 1) +
9
X
j=5
aj (r + 1 − j)−1 − ψ(r −
15
13
) − ψ(r − ).
2
2
Divulgaciones Matem´
aticas Vol. 9 No. 1 (2001), pp. 25–34
Testing Block Sphericity of a Covariance Matrix
33
The method employed above can also be used to derive the exact density
for p ≥ 6. However, because of the higher order of poles, the expressions for
the density will involve generalized Riemann zeta function (see [9]).
Acknowledgment
This research was supported by the Comit´e para el Desarrollo de la Investigaci´on, Universidad de Antioquia research grant no. IN358CE.
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