Some Generalized Multivariate G-Function Distributions of Matrix Argument
and Statistical Properties and Integral Transform
Dr A. K. Vyas Department of Mathematics
JodhpurNationalUniversity Jodhpur
Abstract:- In the present paper we have discussed various generalized multivariate G-Function distributions of matrix argument and statistical properties and integral transform few known or new properties such as characteristic function, moment generating function, moment, mean and variance and Mellin and Laplace transforms of the distribution function and some new table are also established for distribution function.
Introduction
A unified approach to generate probability distributions with the help of special functions technique was initiated by Mathai and Saxena(1966), Saxena and Mishra (1991) have investigated a more comprehensive from of
univariate distributions by taking a generalized Laplace integral incorporating 2
F
2 function. From this result wide classes of distributions have been obtained as particular cases. Sethi and Poonam (1992) have investigated a morecomprehensive form of univariate distribution by taking a generalized Laplace integral involving 2
F
2 of matrix argument. From this result, wide classes of distributions have been obtained as particular cases.In this paper, a generalized multivariate G-function distribution are introduced from which almost all well-known multivariate and univariate G-function distributions can be obtained as special cases. It depends on the multivariate Laplace integral, our distribution in analogues to the generalized probability distribution given by Poonam and Sethi P.L. (1992C) and has applications in multivariate analysis. It may provide some interesting special random variables obtained in various fields of statistical studies particularly in the field of biosciences, agricultural, social and behavioral sciences along with the resent techniques utilized in the geo-statistical and atmospheric sciences.
In this paper expression for the characteristic function. The Moment Generating Function and the
r
th Moment about the origin, mean, variance, Mellin and Laplace Transform have been worked out. In what follows, the argument and the parameter and restricted to take only those values for which the density functions are non-negative and have a meaning.THE DISTRIBUTION
In the multivariate Laplace type integral
0
2 1
X
m
dX
X
X
BX
etr
I
Taking
s r q
p s
r
b
a
b
a
RX
X
G
...
1 ... 1 ,
,
)
1
.
2
(
...
,
2
1
.... 1
... ... ... 1 1 1 ,
,
1
r
s q
p s r
a
a
b
b
m
RB
G
B
I
For
(
1
,...,
)
2
1
min
Re
b
j
m
j
p
and B is positive definite symmetric matrix and R is an arbitrary complex symmetricm
m
matrix.SThe result (2.1) is a direct consequence of the result [Mathai and Saxena 1978,P.114] Thus the function
X
a
a
b
b
B
R
f
X
f
(
)
;
1,
1...,
r,
1,..
s;
,
=
s q
p s r
s r q
p s r m
b
b
a
a
m
RB
G
B
b
b
a
a
RX
G
X
BX
etr
r
... ... ,... 1
,... 1 1
1 ,
, 1
... 1 1 ,
, 2
) 1 (
,
2
1
...
Where
2
1
min
Re
b
j
m
= 0, elsewhere ……… (2.2)
Provides a probability density function(p.d.f.)
Special cases
(i) Replacing R=I letting B tends to null matrix and using the result due to Mathai(1976)
s p j
r q j
j j
q j
j p
j
j
X s
r q
p s r m
a
m
b
m
m
a
m
m
b
m
dX
b
b
a
a
X
G
X
1 1
1 1
0 1,...
1 , , 2
) 1 (
2
1
2
1
)
(
....
Where
)
,...
1
(
;
2
1
)
Re(
)
,..,
1
(
;
2
1
)
Re(
n
j
m
a
m
j
m
b
j j
…………..(2.3)
*
2
1
2
1
)
m(b
)
(
2 ,,1 1
1 1
1 1
j
q p
s r m s
p j
r q j
j j
p j
q j
j
G
X
a
m
b
m
m
a
m
m
X
f
Where
s r q
p s r q
p s r
b
b
a
a
X
G
G
*
,....,
...,
1 , 1 , , ,
,
Where
elsewhere
X
X
n
j
m
a
m
j
m
b
j j
,
0
0
),
,...
1
(
;
2
1
)
Re(
)
,..,
1
(
;
2
1
)
Re(
…………..(2.4)
(ii) Putting
p
1
,
q
0
,
r
0
,
s
1
,
B
I
,
then (2.1) takes the form
0
0 , 1
1 , 0 2
1
)
(
X
m
dX
a
RX
G
X
X
etr
I
……….(2.5)We know that
G
10,,01
RX
a
R
aX
ae
trRXWhere
X
X
0
……….(2.6) Use of (2.5) and (2.6) we have
0
2 1 )
(
X
m a X I R tr a
dX
X
e
R
I
The integral reduces to
( a)a
R
I
a
m
R
……….(2.6)(2.2) takes the form
( ) 21 ) ( ) (
)
(
a m a X
R I tr
R
I
a
m
X
e
X
f
Where
)
7
.
2
(
...
...
...
,
0
0
,
0
Re
,
2
1
Re
elsewhere
X
X
R
I
m
a
Which is a gama distribution.
Taking
,
2
1
)
(
a
m
(2.7) take the form2 ) 1 ( ) (
2
1
)
(
m X R I tr
R
I
m
m
e
X
Where
0
,
0
Re
I
R
X
X
=0 elsewhere ………..(2.8)
Taking
,
2
1
)
(
,
2
)
(
a
n
I
R
T
1 (2.7) yields Wishart distribution with scale matrixT
andn
offreedom
2 1
2 1 2 2
2
1
2
1
n m n X T tr
T
n
m
X
e
X
f
2 1 2
) 1 ( 2
2
2
1
2
n m
n n
T
n
m
X
T
etr
X
For
X
X
0
,
T
0
,
m
n
=
0
,
elsewhere
……….(2.9)THE CHARACTERISTIC FUNCTION
The characteristic function
X
T
ofX
is given by
T
E
itr
XT
X
exp
=
exp
...
...
.(
2
.
10
)
0
X
dX
X
f
XT
itr
Where
i
1
12,
T
ism
m
symmetric matrix of characteristic function variables andE
denotes mathematical expectation.By Virtue of (2.10), we get characteristic function as
s r q
p s r
s r q
p s r X
b
b
a
a
m
RB
B
b
b
a
a
m
iT
B
R
iT
B
T
G
G
,...,
,...,
,
2
1
,...,
,....,
,
2
1
1 1 1
1 ,
, 1
1 1 1
1 ,
, 1
Where
,
Re(
)
0
,
Re(
)
Re(
)
2
1
)
min
Re(
,
2
1
)
Re(
m
b
j
m
B
B
R
…(2.11)
0
X
dX
X
f
XT
etr
XT
etr
E
……….(2.12)By virtue of (2.2),we get moment generating function as
s r q
p s r
s r q
p s r
b
b
a
a
m
RB
B
b
b
a
a
m
T
B
R
T
B
XT
etr
E
G
G
,...,
,...,
,
2
1
,...,
,....,
,
2
1
)
(
1 1 1
1 ,
, 1
1 1 1
1 ,
, 1
Where
,
Re(
)
0
,
Re(
)
Re(
)
2
1
)
min
Re(
,
2
1
)
Re(
m
b
jm
B
B
R
…(2.13)THE MOMENT OF THE DISTRIBUTION
The
r
thorder moment about origin of the matrix random variableX
with p.d.f. (2.2) is given by
0
X r r
dX
X
f
X
X
E
……….(2.14)
s r q
p s r
s r q
p s r r
b
b
a
a
m
RB
b
b
a
a
r
m
RB
B
G
G
,...,
,...,
,
2
1
,...,
,....,
,
2
1
1 1 1
1 ,
, 1
1
1 1
1 ,
, 1
Where the various parameter are governed by same restrictions as mentioned in (2.15)
MEAN AND VARIANCE OF THE DISTRIBUTION BY DEFINITIONS
0
X
dX
X
f
X
X
E
s r q
p s r
s r q
p s r
b
b
a
a
m
RB
b
b
a
a
m
RB
B
G
G
,...,
,...,
,
2
1
,...,
,....,
,
1
2
1
1 1 1
1 ,
, 1
1
1 1
1 ,
, 1 1
……..(2.16)
s r q p s r s r q p s rb
b
a
a
m
RB
b
b
a
a
m
RB
B
G
G
,...,
,...,
,
2
1
,...,
,....,
,
2
2
1
1 1 1 1 , , 1 1 1 1 1 , , 1 2
…….(2.17) Variance
, 1
2 , 1 2 1 , , 1 , , 1 1 , , 1 2]
[
]
[
]
[
]
[
G
G
G
G
q p s r q p s r q p s r q p s rB
Where
s r q p s r q p s rb
b
a
a
m
RB
G
G
,
,...
,....,
,
2
2
1
]
[
1 1 1 1 , , 1 1 , , 1
s r q p s r q p s rb
b
a
a
m
RB
G
G
,
,...
,....,
,
2
1
]
[
1 1 1 1 , , 1 1 , , 1
s r q p s r q p s rb
b
a
a
m
RB
G
G
,
,...
,....,
,
1
2
1
]
[
1 1 1 1 , , 1 1 , , 1
…….(2.18)Where the various parameter are governed by same restriction as mentioned in (2.2)
MELLIN AND LAPLACE TRANSFORM OF THE DISTRIBUTION
By definition of Mellin transform is
0 2 ) 1 ()
(
)
(
X mdX
X
f
X
X
f
M
s r q p s r s r q p s r mb
b
a
a
m
RB
b
b
a
a
m
m
RB
B
G
G
,...,
,...,
,
2
1
,...,
,....,
,
2
1
2
1
1 1 1 1 , , 1 1 1 1 1 , , 1 2 ) 1 (
…….(2.19)By definition of Laplace transform is
s r q
p s r
s r q
p s r
b
b
a
a
m
RB
B
b
b
a
a
m
T
B
R
T
B
X
f
L
G
G
,...,
,...,
,
2
1
,...,
,....,
,
2
1
1 1 1
1 ,
, 1
1 1 1
1 ,
, 1
Where
2
1
)
min
Re(
,
2
1
)
Re(
m
b
j
m
0
Re
)
Re(
)
Re(
,
0
)
Re(
B
B
R
B
T
=0, elsewhere ………….(2.20)
Special function of statistical distributions.
T
X
E
etr
XT
,E
X
r.
VarianceM
f
X
,L
f
X
are defined already through equation (2.10),(2.12),(2.14),(2.17),(2.19),(2.20) respectively.TABLE 1
( )2 ) 1 ( ) ( ) 1 (
1
am a X
R tr
R
a
m
X
e
X
f
Where
,
Re(
1
)
0
,
0
2
1
)
Re(
a
m
R
X
X
T
X
E
etr
XT
rX
E
a
a
R
I
T
R
I
a
a
R
I
T
R
I
a
m
R
I
r
a
m
r
Variance=
E
X
2
E
X
2M
f
X
L
f
X
2 1 2
)
(
)
1
(
2
a
m
R
I
a
m
a
m
R
I
a
m
a
m
R
I
m
s
a
m
m s
21
2
1
) (
) (
)
(
a a
R
I
T
R
I
2 ) 1 ( ) 1 (2
1
m X R trR
I
m
m
e
X
f
0
,
0
)
Re(
:
I
R
X
X
Where
T
X
E
etr
XT
rX
E
2 ) 1 ( 2 ) 1 (
m mR
I
T
R
I
2 ) 1 ( 2 ) 1 (
m mR
I
T
R
I
2
1
2
1
m
m
R
I
r
m
m
rVariance=
E
X
2
E
X
2M
f
X
L
f
X
2 1 2
2
1
1
2
1
2
1
2
2
1
m
m
R
I
m
m
m
m
R
I
m
m
2
1
2 1m
m
R
I
s
m
m s
2 1 2 1)
(
m mR
I
T
R
I
TABLE 3
2 1 2 ) 1 ( 22
2
1
2
n m n nT
n
m
X
T
etr
X
X
f
For
X
X
0
,
T
0
,
m
n
T
X
E
etr
XT
rX
E
2 2 12
n ns
iT
s
2 2 12
n ns
T
s
2
2
2
n
m
s
r
n
m
r r Variance=
2
2X
E
X
E
M
f
X
L
f
X
2 1 1 2 2
2
1
2
2
2
2
2
2
n
m
s
n
m
n
m
s
n
m
2
2
1
2
2
2 1 2 1n
m
s
m
s
n
m
m s m s 2 2 12
n ns
T
References
1. Mathai A.M. and Saxena R.K.(1966): on a generalized hypergeometric distribution,Metrika,11,127-132 2. Saxena R.K. and Mishra K.N.(1991):A generalized Probability
Distribution,Mat.Aacd.Math;India(Gorakhpur) Vol:9
3. Sethi P.L. and Poonam(1992):use of unsymmetric fractional integral operations in Weber-Orr transforms,Proc.Nat.Acad,Sci.,India,Part-A, Vol.62
4. Mathai A.M. and Saxena R.K.(1978): The H-function with applications in statistics and other disciplines, Wiley Eastern limited, New Delhi, India
