IJMMS 30:4 (2002) 249–255
PII. S0161171202012838
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
NULL DISTRIBUTION OF MULTIPLE CORRELATION COEFFICIENT
UNDER MIXTURE NORMAL MODEL
HYDAR ALI and DAYA K. NAGAR
Received 14 April 2001
The multiple correlation coefficient is used in a large variety of statistical tests and regression problems. In this article, we derive the null distribution of the square of the sample
multiple correlation coefficient, R 2 , when a sample is drawn from a mixture of two multivariate Gaussian populations. The moments of 1 − R 2 and inverse Mellin transform have
been used to derive the density of R 2 .
2000 Mathematics Subject Classification: 62H10, 62H15.
1.Introduction.
µ(p×1), and Σ(p×p) > 0 are partitioned as
Supposethat x(p×1),
x1
µ1
σ11 σ21
,
µ
=
,
and
Σ
=
,
where
x(2) = (x2 , . . . , xp ) and µ(2) = (µ2 , . . . , µp )
x(2)
µ(2)
σ
Σ
x=
21
22
are (p − 1) × 1 and Σ22 is (p − 1) × (p − 1), so that Var(x1 ) = σ11 , Cov(x(2) ) = Σ22 , and
σ12 is the (p − 1) × 1 vector of covariances between x1 and x2 , . . . , xp . The multiple
correlation coefficient between x1 and x(2) , denoted by R̄1·2···p , is defined as
R̄1·2···p =
σ21 Σ−1
22 σ21
σ11
1/2
.
(1.1)
Let A be the sample sum of squares and products
matrix formed from N indepena
a
dent observations on x. Partition A as A = a 11 A21 , where A22 is (p −1)×(p −1). The
21
22
sample multiple correlation coefficient between x1 and x(2) is defined by
−1
a21 A22 a21 1/2
.
R=
a11
(1.2)
It is well known that, when the underlying population is normal, the random matrix A
has Wishart distribution with n = N − 1 degrees of freedom and parameter matrix Σ.
Further, R̄1·2···p = 0 if and only if x1 is independent of x(2) = (x2 , . . . , xp ). Furthermore,
when the population multiple correlation coefficient R̄1·2···p is zero, the distribution
of R 2 is beta with parameters (1/2)(p − 1) and (1/2)(N − p).
In practice, it is often the case that the random variables are not normally distributed. When such is the case, how would the departure from the normality affect
the conventional inference procedure? Specifically, one may wonder what would be
the sampling distributions of some commonly used statistics? For providing some
answers to the above questions, Srivastava and Awan [9] and Tan [11] derived the
distribution of the sample sum of squares and products matrix when sampling from
a mixture of two multivariate normal distributions. The normal mixture is defined as
follows:
(1.3)
f (x) = Np µ1 , Σ; x + (1 − )Np µ2 , Σ; x , x ∈ Rp ,
250
H. ALI AND D. K. NAGAR
where
Np (µ, Σ; x)
1
= (2π )−(1/2)p det(Σ)−1/2 exp − (x − µ) Σ−1 (x − µ) ,
2
x ∈ Rp , µ ∈ Rp , Σ > 0,
(1.4)
and 1 − is known as the degree of contamination. This model is very common in
medical, biological, and agricultural experiments (Titterington et al. [12]). For results
on the distribution theory and robustness studies of certain test statistics when sampling from a mixture normal model, see Srivastava [8], Srivastava and Awan [9, 10],
Kabe and Gupta [5], Amey and Gupta [2], and Nagar and Castañeda [7].
Srivastava [8], using certain transformations, derived the null distribution of multiple correlation coefficient when sampling from a mixture of two multivariate normal
distributions (see also Gupta and Kabe [3]). Amey [1] integrated the joint density of
a11 , a21 , and A22 suitably to derive the density of R 2 and studied its robustness.
In this article, we derive the null distribution of R 2 when sampling from a mixture of
two multivariate normal distributions. First, we derive the hth null moment of 1−R 2 .
Then, by using the inverse Mellin transform, the density of 1 − R 2 is obtained from
which the density of R 2 is deduced.
Note that R 2 is a function of the elements of sample sum of squares and products
matrix A. Therefore, in our derivation, we use the distribution of A when sampling
from the above model. Srivastava and Awan [9] and Tan [11] have shown that the
density of A, when sampling from (1.3), is a binomial sum of linear noncentral Wishart
densities:
N γ
(1 − )N−γ Wp n, Σ, cγ2 Σ−1 νν ; A ,
f (A) =
γ
γ=0
N
(1.5)
where n = N − 1, cγ2 = γ(N − γ)/N, and ν = (µ1 − µ2 ). Here Wp (n, Σ, cγ2 Σ−1 νν ; A) represents the noncentral Wishart density with n degrees of freedom and noncentrality
parameter matrix cγ2 Σ−1 νν defined by
1
1
(p) 1
n; cγ2 Σ−1 AΣ−1 νν ,
Kp (n, Σ, ν) etr − Σ−1 A det(A)(1/2)(n−p−1) 0 F1
2
2 4
(1.6)
where
−1
1
1
n det(Σ)(1/2)n
etr − cγ2 Σ−1 νν
Kp (n, Σ, ν) = 2(1/2)pn Γp
2
2
and Γm (a) = π (1/4)m(m−1)
(1.7)
m
j=1 Γ (a − (1/2)(j − 1)).
2. Null moments of 1 − R 2 . In this section, we derive moments of 1 − R 2 when
R̄1·2···p = 0 (or equivalently σ21 = 0). Let Σ0 = σ011 Σ022 and U = 1 − R 2 . Since a11 is
scalar, then
a A−1 a21
det(A)
.
=
U = 1 − R 2 = 1 − 21 22
(2.1)
a11
a11 det A22
NULL DISTRIBUTION OF MULTIPLE CORRELATION COEFFICIENT . . .
251
The hth null moment of U is given by
N γ
(1 − )N−γ Eγ U h ,
γ
γ=0
N
E Uh =
Eγ U h = Kp n, Σ0 , ν
(2.2)
−h
1
etr − Σ−1
A
a−h
0
11 det A22
2
A>0
(2.3)
1
(p) 1
−1
n; cγ2 Σ−1
× det(A)(1/2)(n−p−1)+h 0 F1
AΣ
νν
dA.
0
0
2 4
−h
by their integral representations, namely
Replacing a−h
11 and det(A22 )
∞
1
1
a−h
=
exp
−
y
y1h−1 dy1 , Re(h) > 0,
a
11
1
11
2h Γ (h) 0
2
−h
1
1
= (p−1)h
etr − A22 Y22
det A22
2
Γp−1 (h) Y22 >0
2
h−(1/2)(p−1+1)
dY22 ,
× det Y22
Re(h) >
1
(p − 2),
2
respectively, in (2.3) and integrating A, the moment expression is rewritten as
2(1/2)np Kp n, Σ0 , ν Γp (1/2)n + h
Eγ U h =
Γ (h)Γp−1 (h)
∞
h−(1/2)p
−(1/2)n−h
×
y1h−1
det Y22
det Σ−1
0 +Y
0
(p)
× 1 F1
where Y =
y1
0 0 Y22
(2.4)
(2.5)
Y22 >0
−1 −1 −1
1 1
1
Σ0 νν dy1 dY22 ,
n + h; n; cγ2 Σ−1
Σ0 + Y
0
2
2 2
(p)
and 1 F1
is the confluent hypergeometric function of matrix argu-
−1
−1 −1
ment (Gupta and Nagar [4]). Since rank(Σ−1
0 (Σ0 + Y ) Σ0 νν ) = 1, the only nonzero
−1
−1
−1 −1
−1 −1
−1
characteristic root of the matrix Σ0 (Σ0 + Y ) Σ0 νν is tr((Σ−1
0 + Y ) Σ0 νν Σ0 )
and therefore,
−1 −1 −1
1 1
(p) 1
n + h; n; cγ2 Σ−1
Σ0 νν
Σ0 + Y
1 F1
0
2
2 2
(2.6)
−1 −1 −1 1 1
1
= 1 F1
n + h; n; cγ2 tr Σ−1
+
Y
Σ
νν
Σ
,
0
0
0
2
2 2
where 1 F1 is the confluent hypergeometric function of scalar argument (see [6]). Substituting (2.6) in (2.5) and expanding 1 F1 in series form, the moment expression simplifies
to
∞ 2 t cγ
2(1/2)np Kp n, Σ0 , ν Γp (1/2)n + h (1/2)n + h t
Eγ U h =
Γ (h)Γp−1 (h)
2
(1/2)n t t!
t=0
×
∞
0
y1h−1
Y22 >0
−(1/2)n−h
h−(1/2)p
det Y22
det Σ−1
0 +Y
−1 −1 t
−1
× ν Σ−1
Σ0 ν dy1 dY22 ,
Σ0 + Y
0
(2.7)
252
H. ALI AND D. K. NAGAR
where (a)r = a(a + 1) · · · (a + r − 1) and (a)0 = 1. Noting that Σ0 is a block diagonal
matrix, we obtain
−1 −1 t
−1 −1
Σ0 ν
ν Σ0 Σ0 + Y
−1
−1
−1 −1 t
−1
= ν12 σ11
+ ν2 Σ−1
Σ22 ν2
1 + σ11 y1
22 Σ22 + Y2
=
k+=t
(2.8)
−1 k −1 −1
−1 −1 t! 2 −1 ν1 σ11 1 + σ11 y1
ν2 Σ22 Σ22 + Y22
Σ22 ν2 ,
k!!
−1 −1
det Σ−1
1 + σ11 y1 det Σ22
det Ip−1 + Σ22 Y22 .
0 + Y = σ11
Now substituting (2.8) in (2.7), we have
2(1/2)np Kp n, Σ0 , ν Γp ((1/2)n + h)
Eγ U h =
det(Σ0 )(1/2)n+h Γ (h)Γp−1 (h)
2 k
∞ 2 t cγ
(1/2)n + h t 1
ν1
×
2
k!! σ11
(1/2)n t
t=0
k+=t
∞
−((1/2)n+h+k)
×
y1h−1 1 + σ11 y1
dy1
×
(2.9)
0
Y22 >0
h−(1/2)p
−(1/2)n−h
det Y22
det Ip−1 + Σ22 Y22
−1
−1 −1 × ν2 Σ−1
Σ22 ν2 dY22 .
22 Σ22 + Y22
1/2
1/2
Substituting Z = (Ip−1 + Σ22 Y22 Σ22 )−1 , the integral involving Y22 is evaluated as
Y22 >0
h−(1/2)p
−(1/2)n−h −1 −1
−1 −1 det Y22
det Ip−1 + Σ22 Y22
Σ22 ν2 dY22
ν2 Σ22 Σ22 + Y22
−h
= det Σ22
det(Z)(1/2)(n−p)
0<Z<Ip−1
h−(1/2)p −1/2 −1/2 × det Ip−1 − Z
ν2 Σ22 ZΣ22 ν2 dZ
−h
= det Σ22
∂
∂η
η=0
0<Z<Ip−1
h−(1/2)p
det(Z)(1/2)(n−p) det Ip−1 − Z
−1/2
−1/2
× etr ην2 Σ22 ZΣ22 ν2 dZ
−h Γp−1 (1/2)n Γp−1 (h) ∂ 1
(p−1) 1
−1
n;
n
+
h;
ηΣ
F
ν
ν
= det Σ22
1
22 2 2
Γp−1 (1/2)n + h
∂η η=0 1
2 2
−h Γp−1 (1/2)n Γp−1 (h)
(1/2)n −1 ν Σ ν2 ,
= det Σ22
Γp−1 (1/2)n + h
(1/2)n + h 2 22
(2.10)
(p−1)
where 1 F1
is the confluent hypergeometric function of matrix argument (see [4]).
253
NULL DISTRIBUTION OF MULTIPLE CORRELATION COEFFICIENT . . .
Collecting terms containing y1 and integrating, we obtain
∞
−((1/2)n+h+k)
(1/2)n k
h−1
−h Γ (1/2)n Γ (h)
.
y1
dy1 = σ11
1 + σ11 y1
Γ (1/2)n + h (1/2)n + h k
0
(2.11)
Substituting (2.10), (2.11), and (1.7) in (2.9) and simplifying the resulting expression
using results on gamma function, we get
∞
cγ2 t (1/2)n (1/2)n
h
Γ (1/2)n
1 2 −1
k
Eγ U = exp − cγ ν Σ0 ν 2
Γ (1/2)(n − p + 1) t=0
2
(1/2)n t
+k=t
k −1 2
ν2 Σ22 ν2 Γ (1/2)(n − p + 1) + h Γ (1/2)n + t + h
ν1 /σ11
k!!
Γ (1/2)n + k + h Γ (1/2)n + + h
∞ 2
Γ (1/2)n Γ (1/2)(n − p + 1) + h cγ k
1 2 −1
= exp − cγ ν Σ0 ν 2
Γ (1/2)n + h Γ (1/2)(n − p + 1) k=0 2
×
2
k
ν1 /σ11
1
1 1
1
1
×
n + h + k, n; n + k, n + h; cγ2 ν2 Σ−1
2 F2
22 ν2 ,
k!
2
2 2
2
2
(2.12)
where 2 F2 is the generalized hypergeometric function of scalar argument (see [6]).
3. Distribution of R 2 under mixture normal model. The density function f (u) of
U = 1 − R 2 is obtained by taking the inverse Mellin transform of E(U h ) as
N
N γ
(3.1)
f (u) =
(1 − )N−γ fγ (u)
γ
γ=0
with
fγ (u) = (2π ι)−1
C
Eγ U h u−h−1 dh,
0 < u < 1,
(3.2)
√
where ι = −1 and C is a suitable contour. Substituting (2.12) in (3.2), we obtain
∞ 2 t
cγ
1
Γ (1/2)n
fγ (u) = exp − cγ2 ν Σ−1
ν
0
2
Γ (1/2)(p − 1) Γ (1/2)(n − p + 1) t=0
2
k+=t
−1 k
(1/2)n k (1/2)n ν12 /σ11
ν2 Σ22 ν2
u(1/2)n+k+−1 (1 − u)(1/2)(p−3)
×
k! !
(1/2)n t
1
1
1
(p − 1) + k, (p − 1) + ; (p − 1); 1 − u , 0 < u < 1,
× 2 F1
2
2
2
(3.3)
where 2 F1 is the Gauss hypergeometric function (see [6]). To obtain (3.3) we have used
the result
1
1
1
1
(p − 1) + k, (p − 1) + ; (p − 1); 1 − u du
u(1/2)n+h+k+−1 (1 − u)(1/2)(p−3) 2 F1
2
2
2
0
Γ (1/2)(p − 1) Γ (1/2)(n − p + 1) + h Γ (1/2)n + t + h
=
.
Γ (1/2)n + k + h Γ (1/2)n + + h
(3.4)
254
H. ALI AND D. K. NAGAR
The density of R 2 = 1 − U is now derived from the density of U as
N γ
(1 − )N−γ gγ R 2 ,
γ
γ=0
N
g R2 =
(3.5)
where
∞
cγ2 t
1
Γ (1/2)n
ν
gγ R 2 = exp − cγ2 ν Σ−1
0
2
Γ (1/2)(p − 1) Γ (1/2)(n − p + 1) t=0
2
k+=t
−1 k
(1/2)n k (1/2)n ν12 /σ11
ν2 Σ22 ν2
×
k!!
(1/2)n t
(1/2)n+k+−1 2 (1/2)(p−3)
× 1 − R2
R
× 2 F1
1
1
1
(p − 1) + k, (p − 1) + ; (p − 1); R 2 ,
2
2
2
0 < R 2 < 1.
(3.6)
By using the result 2 F1 (a, b; c; z) = (1 − z)c−a−b 2 F1 (c − a, c − b; c; z), the above density
can be rewritten as
2
Γ (1/2)n
1 2 −1
gγ R = exp − cγ ν Σ0 ν
2
Γ (1/2)(p − 1) Γ (1/2)(n − p + 1)
∞
cγ2 t (1/2)n (1/2)n
(1/2)(p−3) (1/2)(n−p−1) k
× R2
1 − R2
2
(1/2)n
t
t=0
k+=t
−1 k
2
ν2 Σ22 ν2
ν1 /σ11
1
2
×
,
2 F1 − k, −; (p − 1); R
k!!
2
0 < R 2 < 1.
(3.7)
It is interesting to note that if ν = 0, then the density g(R 2 ) reduces to
Γ (1/2)n
g R2 = Γ (1/2)(p − 1) Γ (1/2)(n − p + 1)
× R
2 (1/2)(p−3)
1−R
2 (1/2)(n−p−1)
,
(3.8)
2
0 < R < 1.
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[2]
[3]
[4]
[5]
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Hydar Ali: Department of Mathematics and Computer Science, the University of
the West Indies, St. Augustine, Trinidad and Tobago
Daya K. Nagar: Departamento de Matemáticas, Universidad de Antioquia, Medellín,
A. A. 1226, Colombia