Applied Mathematics E-Notes, 15(2015), 105-120 c
Available free at mirror sites of http://www.math.nthu.edu.tw/ amen/
ISSN 1607-2510
The Exponentiated Gamma Distribution Based On
Ordered Random Variable
Devendra Kumary
Received 15 October 2014
Abstract
In this paper, we gave some new explicit expressions and recurrence relations
for marginal and joint moment generating functions of dual generalized order statistics from exponentiated gamma distribution. The results for order statistics
and lower record values are deduced from the relation derived. Further, characterizing result of this distribution on using a recurrence relation for marginal
moment generating functions dual generalized order statistics is discussed.
1
Introduction
The concept of generalized order statistics (gos) was introduced by Kamps [1] as a
general framework for models of ordered random variables. Moreover, many other
models of ordered random variables, such as, order statistics, k-th upper record values,
upper record values, progressively Type II censoring order statistics, Pfeifer records
and sequential order statistics are seen to be particular cases of gos. These models
can be e¤ectively applied, e.g., in reliability theory. However, random variables that
are decreasingly ordered cannot be integrated into this framework. Consequently, this
model is inappropriate to study, e.g. reversed ordered order statistic and lower record
values models. Burkschat et al. [2] introduced the concept of dual generalized order
statistics (dgos). The dgos models enable us to study decreasingly ordered random
variables like reversed order statistics, lower k record values and lower Pfeirfer records,
through a common approach below: Suppose Xd (1; n; m; k); : : : ; Xd (n; n; m; k), (k 1,
m is a real number), are n dgos from an absolutely continuous cumulative distribution
function cdf F (x) with probability density function pdf f (x), if their joint pdf is of the
form
0
1
!
n
n
Y1
Y1
m
A
k@
[F (xi )] f (xi ) [F (xn )]k 1 f (xn );
(1)
j
j=1
i=1
for F 1 (1) > x1 x2 : : : xn > F 1 (0), where j = k + (n j)(m + 1) > 0 for all
j, 1 j n, k is a positive integer and m
1. If m = 0 and k = 1, then this model
reduces to the (n r + 1)-th order statistic, from the sample X1 ; X2 ; : : : ; Xn and (1)
Mathematics Subject Classi…cations: 62G30, 62E10.
of Statistics, Amity Institute of Applied Sciences Amity University, Noida-201 303,
India.
y Department
105
106
The Exponentiated Gamma Distribution
will be the joint pdf of n order statistics. If k = 1 and m = 1, then (1) will be the
joint pdf of the …rst n record values of the identically and independently distributed
(iid) random variables with cdf F (x) and corresponding pdf f (x).
In view of (1), the marginal pdf of the r-th dgos, is given by
fXd (r;n;m;k) (x) =
Cr 1
[F (x)]
(r 1)!
r
1
r 1
f (x)gm
(F (x)):
(2)
The joint pdf of r-th and s-th dgos, is
=
fXd (r;n;m;k);Xd (s;n;m;k) (x; y)
Cs 1
r 1
[F (x)]m f (x)gm
(F (x))
(r 1)!(s r 1)!
[hm (F (y))
where
Cr
1
=
r
Y
i=1
i,
s r 1
hm (F (x))]
hm (x) =
[F (y)]
1
m+1
m+1 x
lnx
s
1
f (y);
for m 6=
for m =
(3)
1;
1;
and
gm (x) = hm (x)
hm (1);
for x 2 [0; 1).
Ahsanullah and Raqab [3], Raqab and Ahsanullah [4, 5] have established recurrence relations for moment generating functions (mgf ) of record values from Pareto
and Gumble, power function and extreme value distributions. Recurrence relations
for marginal and joint mgf of gos from power function distribution, Erlang-truncated
exponential distribution and extended type II generalized logistic distribution are derived by Saran and Singh [6], Kulshrestha et al. [7] and Kumar [8] respectively. Kumar
[9, 10, 11] have established recurrence relations for marginal and joint mgf of dgos
from generalized logistic, Marshall-Olkin extended logistic and type I generalized logistic distribution respectively. Al-Hussaini et al. [12, 13] have established recurrence
relations for conditional and joint mgf of gos based on mixed population. Kumar [14]
have established explicit expressions and some recurrence relations for mgf of record
values from generalized logistic distribution. Recurrence relations for single and product moments of dgos from the inverse Weibull distribution are derived by Pawlas and
Szynal [15]. Ahsanullah [16] and Mbah and Ahsanullah [17] characterized the uniform
and power function distributions based on distributional properties of dgos respectively. Characterizations based on gos have been studied by some authors, Keseling
[18] characterized some continuous distributions based on conditional distributions of
gos. Bieniek and Szynal [19] characterized some distributions via linearity of regression of gos. Cramer et al. [20] gave a unifying approach on characterization via linear
regression of ordered random variables.
Rest of the paper is organized as follows: In Section 2 exact expressions and recurrence relations for marginal and joint mgf of dgos from exponentiated Gamma distribution (EGD) are presented, while in Section 3, the exact expressions and recurrence
D. Kumar
107
relations for joint mgf for dgos from EGD are discussed In Section 4, a characterization of EGD is obtained by using the recurrence relation for marginal mgf of dgos.
Some …nal comments in Section 5 conclude the paper.
Gupta et al. [21] introduced the EGD. This model is ‡exible enough to accommodate both monotonic as well as nonmonotonic failure rates. The cdf and pdf of EGD
are given, respectively by
F (x) = [1
f (x) = xe
x
x
e
[1
(x + 1)] ; x > 0;
x
e
(x + 1)]
1
> 0,
; x > 0;
(4)
> 0:
(5)
Note that for F GD;
xF (x) = [e
x
(x + 1)]f (x):
(6)
For
= 1, the above distribution corresponds to the gamma distribution G(1; 2).
2
Relations for Marginal Moments Generating Functions
In this Section the exact expressions and recurrence relations for marginal mgf of dgos
from EGD are considered. For the EGD when m 6= 1,
Z 1
Z 1
Cr 1
r 1
tx
etx [F (x)] r 1 f (x)gm
(F (x))dx:
e f (x)dx =
MXd (r;n;m;k) (t) =
(r 1)! 1
1
(7)
On using (4) and (5) in (6) and simpli…cation of the resulting equation we get
MXd (r;n;m;k) (t)
=
Cr 1
1)!(m + 1)r
(r
r u
1
1
p
r 1X
1 X
X
( 1)u+p
u=0 p=0 q=0
and for m =
1;k) (t)
=
1 1 r 1+v+p
( k)r X X X
( 1)v
(r 1)! p=0 v=0 w=0
r
p (r
(8)
1
MXd (r;n;
where
1
u
(q + 2)
;
(p + 1 t)q+2
p
q
p
r
1
X
e
p=1
see Balakrishnan and Cohan [22].
(r 1+p)x
px
1)
k
1
v
(w + 2)
;
(r + v + p t)w+2
1+v+p
w
1) is the coe¢ cient of e
p (r
(x + 1)r
(x + 1)p
p
!r
1+p
1
;
in the expansion of
(9)
108
The Exponentiated Gamma Distribution
Di¤erentiating both sides of (8) and (9) with respect to t, j times we get
(j)
MXd (r;n;m;k) (t)
=
Cr 1
1)!(m + 1)r
(r
1
( 1)u+p
r
u=0 p=0 q=0
1
r u
p
r 1X
1 X
X
p
q
p
1
u
(j + q + 2)
(p + 1 t)j+q+2
(10)
and
(j)
MXd (r;n;
1;k) (t)
=
1 1 r 1+v+p
( k)r X X X
( 1)v
(r 1)! p=0 v=0 w=0
r
If
p (r
1)
k
1
v
(j + w + 2)
:
(r + v + p t)j+w+2
1+v+p
w
(11)
is a positive integer, the relations (10) and (11) then give
(j)
MXd (r;n;m;k) (t)
=
(r
Cr 1
1)!(m + 1)r
r u
1
r 1
X
r u
X
u=0
1
p=0
( 1)u+p
r
1
u
q=0
(j + q + 2)
(p + 1 t)j+q+2
p
q
p
1 p
X
(12)
and
(j)
MXd (r;n;
1;k) (t)
=
1 k 1 r 1+v+p
( k)r X X X
( 1)v
(r 1)! p=0 v=0 w=0
r
1+v+P
w
p (r
1)
k
1
v
(j + w + 2)
:
(r + v + p t)j+w+2
(13)
By di¤erentiating both sides of equation (12) and (13) with respect to t and then setting
t = 0, we obtain the explicit expression for single moments of dgos and k record values
from EGD in the form
E[Xdj (r; n; m; k)]
=
(r
Cr 1
1)!(m + 1)r
r u
1
1
p
r 1
X
u=0
p
q
r u
X
p=0
1 p
X
( 1)u+p
r
1
u
q=0
(j + q + 2)
(p + 1)j+q+2
(14)
and
E[Xdj (r; n; 1; k)]
=
1 k 1 r 1+v+p
( k)r X X X
( 1)v
(r 1)! p=0 v=0 w=0
r
Special Cases:
1+v+p
w
p (r
1)
(j + w + 2)
:
(r + v + p)j+w+2
k
1
v
(15)
D. Kumar
109
(i) Putting m = 0, k = 1 in (12) and (14), relations for order statistics can be
obtained as
(j)
MXr:n (t)
=
Cr:n
(r+u) 1 p
X X
n
Xr
u=0
p=0
(r + u)
p
( 1)u+p
q=0
1
n
r
u
(j + q + 2)
(p + 1 t)j+q+2
p
q
and
j
E[Xr;n
]
=
Cr:n
(r+u) 1 p
X X
n
Xr
u=0
p=0
(r + u)
p
Cr:n =
n!
1)!(n
(r
n
(j + q + 2)
;
(p + 1)j+q+2
p
q
r)!
r
u
q=0
1
where
( 1)u+p
:
(ii) Putting k = 1 in (13) and (15), relations for record values can be obtained as
r
(j)
MXL(r) (t)
=
(r
1 X1 r 1+v+p
X
X
1)! p=0 v=0
r
( 1)v
p (r
1)
w=0
1
v
(j + w + 2)
(r + v + p t)j+w+2
1+v+p
w
and
j
E[XL(r)
]
1 X1 r 1+v+p
X
X
r
=
(r
1)! p=0 v=0
r
( 1)v
p (r
1)
w=0
1+v+P
w
1
v
(j + w + 2)
:
(r + v + p)j+w+2
A recurrence relation for mgf of dgos from cdf (4) can be obtained in the following
theorem.
THEOREM 1. For 2
r
t
1
MXd (r;n;m;k) (t)
(j)
MXd (r 1;n;m;k) (t)
1
2 and k = 1; 2; : : : ;
(j)
r
=
nn
+
j
(j 1)
r
MXd (r;n;m;k) (t)
tE[ (Xd (r; n; m; k))] + E[ (Xd (r; n; m; k))] ;
r
(16)
110
The Exponentiated Gamma Distribution
where
(x) = xj
e(t+1)x
1
etx
(x) = xj
and
e(t+1)x
2
etx :
PROOF. Integrating by parts of (7) and using (6), we get
MXd (r;n;m;k) (t)
= MXd (r
1;n;m;k) (t)
j
+
MXd (r;n;m;k) (t)
E[h(Xd (r; n; m; k))] ;
(17)
r
where
h(x) =
e(t+1)x
x
etx
x
:
Di¤erentiating both the sides of (17) j times with respect to t, we get the result given
in (16). By di¤erentiating both sides of equation (17) with respect to t and then setting
t = 0, we obtain the recurrence relations for moments of dgos from EGD in the form
E[Xdj (r; n; m; k)]
= E[Xdj (r
r
j n
E[Xdj
+
j
1; n; m; k)] +
2
E[Xdj
(r; n; m; k)]
o
E[ (Xd (r; n; m; k))] ; (18)
(r; n; m; k)]
r
where (x) = xj
1
2 x
e :
REMARK 2.1. Putting m = 0, k = 1 in (16) and (18), relations for order statistics
can be obtained as
(j)
MXr:n (t)
=
t
1
+
(j)
(r
1)
1
(r
1)
MXr
ftE[ (Xr
1:n
(t)
1:n )]
j
(r
(j 1)
1)
+ jE[ (Xr
MXr
1:n
(t)
1:n )]g
and
j
E(Xr:n
) = E(Xrj
1:n )
j
(r
1)
REMARK 2.2. Putting k =
be obtained as
1
t
k
(j)
MXL(r) (t)
n
E(Xrj
1
1:n )
+ E(Xrj
2
1:n )
E( (Xr
1;n ))
(j)
= MXL(r
and
1) ]
:
1 in (16) and (18), relations for k record values can
1)
(t) +
j
(j 1)
M
(t)
k XL(r)
1
tE[ (XL(r) )] + jE[ (XL(r) )]
k
j
j
E[XL(r)
] = E[XL(r
o
+
j n
j 1
j 2
E[XL(r)
] + E[XL(r)
]
k
o
E[ (XL(r) )] :
D. Kumar
3
111
Relations for Joint Moment Generating Functions
In this Section exact moments and recurrence relations for joint mgf of dgos from
EGD are considered. For the EGD when m 6= 1;
MXd (r;n;m;k);Xd (s;n;m;k) (t1 ; t2 )
Z 1Z x
et1 x+t2 y fXd (r;n;m;k)Xd (s;n;m;k) (x; y)dxdy:
=
1
1
Z 1Z x
Cs 1
r 1
et1 x+t2 y [F (x)]m f (x)gm
(F (x))
=
(r 1)!(s r 1)! 1 1
hm (F (x))]s
[hm (F (y))
r 1
[F (y)]
s
1
f (y)dydx:
(19)
On using (4) and (5) in (19) and simpli…cation of the resulting equation we get
MXd (r;n;m;k);Xd (s;n;m;k) (t1 ; t2 )
a X
r 1
1 X
X
2
=
(r
1)!(s
a=0 b=0 u=0
l w+1
X
X
p!(l + 1
(w + 2) (p + a + 2)
t2 )w+2 p (a + l + 2 t1
s
( 1)a+l+u+v
1
(s
r
a
b
a
l=0 w=0 p=0
r
v
1
s v
v + u)(m + 1)
l
1
t2 )p+a+2
r
1
u
l
w
:
(20)
1;
MXd (r;n;
=
2
sX
r 1X
1
v=0
For m =
Cs 1
r 1)!(m + 1)s
(r
1;k);Xd (s;n; 1;k) (t1 ; t2 )
( k)s
1)!(s r
1)!
sX
r 1X
1 s+b+p
Xa 2
a=0 b=0
c+1 X
1 X
1 X
1 a+q+v
X
X
( 1)s
c=0
r 1+a+v
p (s
a
2)
q (a)
d=0 p=0 q=0 v=0 w=0
s
r
a
1
d!(s + b + p
s+b+p
c
1
a
2
k
1
v
(c + 2) (w + d + 2)
t2
+b+p+q+v
)c d+2 (s
t1
a+q+v
w
t2 )w+d+2
:
(21)
112
The Exponentiated Gamma Distribution
Di¤erentiating both side of (20) and (21) i times with respect to t1 and then j times
with respect to t2 , we get
(i;j)
MXd (r;n;m;k);Xd (s;n;m;k) (t1 ; t2 )
1 X
a X
r 1
X
2
=
(r
Cs 1
r 1)!(m + 1)s
1)!(s
2
a=0 b=0 u=0
sX
r 1X
1
l i+w+1
X
X
p!(l + 1
(i + w + 2) (j + p + a + 2)
t2 )i+w+2 p (a + l + 2 t1 t2 )j+p+a+2
v=0
( 1)a+l+u+v
l=0 w=0
s
r
v
(s
r
a
b
a
p=0
1
1
s v
v + u)(m + 1)
l
1
r
1
u
l
w
(22)
and
(i;j)
MXd (r;n;
=
1;k);Xd (s;n; 1;k) (t1 ; t2 )
( k)s
1)!(s r
(r
1)!
sX
r 1X
1 s+b+p
Xa 2
a=0 b=0
1 a+q+v
1 X
i+c+1
1 X
X
X X
c=0
( 1)s
r 1+a+v
p (s
a
2)
q (a)
d=0 p=0 q=0 v=0 w=0
s
r
a
1
d!(s + b + p
If
s+b+p
c
1
a
2
k
1
a+q+v
w
v
(i + c + 2) (j + w + d + 2)
t2 )i+c d+2 (s + b + p + q + v
t2 )j+w+d+2
t1
: (23)
is a positive integer, the relations (22) and (23) then give
(i;j)
MXd (r;n;m;k);Xd (s;n;m;k) (t1 ; t2 )
2
=
(r
1)!(s
Cs 1
r 1)!(m + 1)s
1
sX
r 1 (s r v+u)(m+1)
X
v=0
l=0
r
1
u
p!(l + 1
s
r
v
1
2
s v
X
a=0
1 a r 1
XX
b=0 u=0
l i+w+1
X
X
( 1)a+l+u+v
w=0
p=0
(s
r
v + u)(m + 1)
l
(i + w + 2) (j + p + a + 2)
t2 )i+w+2 p (a + l + 2 t1 t2 )j+p+a+2
s v
1
a
b
a
1
l
w
(24)
D. Kumar
113
and
(i;j)
MXd (r;n;
=
1;k);Xd (s;n; 1;k) (t1 ; t2 )
( k)s
1)!(s r
(r
1)!
sX
r 1X
1 s+b+p
Xa 2
a=0 b=0
i+c+1
1 X
1 X
k 1 a+q+v
X X
X
d=0 p=0 q=0 v=0
s
r
a
1
c=0
( 1)s
1
p (s
a
2)
q (a)
w=0
s+b+p
c
d!(s + b + p
r 1+a+v
a
2
k
1
a+q+v
w
v
(i + c + 2) (j + w + d + 2)
t2 )i+c d+2 (s + b + p + q + v
t2 )j+w+d+2
t1
: (25)
By di¤erentiating both sides of equation (24) and (25) with respect to t1 , t2 and then
setting t1 = t2 = 0, we obtain the explicit expression for product moments of dgos and
k record values from EGD in the form
E[Xdi (r; n; m; k); Xdj (s; n; m; k)]
2
=
(r
1)!(s
Cs 1
r 1)!(m + 1)s
1
sX
r 1 (s r v+u)(m+1)
X
v=0
l=0
r
1
s
u
r
v
1
2
s v
X
a=0
1 a r 1
XX
b=0 u=0
l i+w+1
X
X
( 1)a+l+u+v
w=0
s v
p=0
(s
r
1
a
b
a
v + u)(m + 1)
l
1
l
w
(i + w + 2) (j + p + a + 2)
p!(l + 1)i+w+2 p (a + l + 2)j+p+a+2
(26)
and
E[Xdi (r; n; 1; k); Xdj (s; n; 1; k)]
=
(r
( k)s
1)!(s r
1)!
sX
r 1X
1 s+b+p
Xa 2
a=0 b=0
i+c+1
1 X
1 X
k 1 a+q+v
X X
X
d=0 p=0 q=0 v=0
s
r
a
1
d!(s + b + p
Special Cases:
( 1)s
c=0
r 1+a+v
p (s
a
2)
q (a)
w=0
s+b+p
c
a
2
k
1
v
(i + c + 2) (j + w + d + 2)
:
1)i+c d+2 (s + b + p + q + v)j+w+d+2
a+q+v
w
(27)
114
The Exponentiated Gamma Distribution
(i) Putting m = 0, k = 1 in (24) and (26), relations for order statistics can be
obtained as
(i;j)
MXr:n ;Xs:n (t1 ; t2 )
2
=
(r+v) 1 a n s s r 1 (s r v+u) 1
X XX X
X
Cr;s:n
a=0
b=0 u=0 v=0
l i+w+1
X
X
(r + v)
a
( 1)a+l+u+v
w=0
p=0
n
s
s
r
v
u
l=0
1
(s
1
r
a
b
v + u)
l
1
l
w
(i + w + 2) (j + p + a + 2)
t2 )i+w+2 p (a + l + 2 t1 t2 )j+p+a+2
p!(l + 1
and
i
j
E[Xr:n
; Xs:n
]
2
=
Cr;s:n
(r+v) 1 a s r 1 (s r v+u) 1 i+w+1 n s
X X X
X
X X
a=0
l
X
b=0 v=0
(r + v)
a
( 1)a+l+u+v
w=0
s
r
v
1
(s
p=0
l=0
r
1
a
b
v + u)
l
u=0
n
s
u
1
l
w
(i + w + 2) (j + p + a + 2)
;
p!(l + 1)i+w+2 p (a + l + 2)j+p+a+2
where
Cr;s;n =
(r
n!
r 1)!(n
1)!(s
s)!
:
(ii) Putting k = 1 in (25) and (27), relations for record values in the form
(i;j)
MXL(r) ;XL(s) (t1 ; t2 )
s
=
(r
1)!(s
r
i+c+1
1 X
1
X X
1)!
sX
r 1X
1 s+b+p
Xa 2
a=0 b=0
X1 a+q+v
X
( 1)s
c=0
r 1+a+v
p (s
a
2)
q (a)
d=0 p=0 q=0 v=0 w=0
s
r
a
1
d!(s + b + p
s+b+p
c
1
a
2
1
a+q+v
w
v
(i + c + 2) (j + w + d + 2)
t2 )i+c d+2 (s + b + p + q + v
t1
t2 )j+w+d+2
D. Kumar
115
and
j
i
]
; XL(s)
E[XL(r)
s
=
(r
1)!(s
r
1)!
i+c+1
1 X
1
X X
sX
r 1X
1 s+b+p
Xa 2
a=0 b=0
X1 a+q+v
X
c=0
( 1)s
r 1+a+v
p (s
a
2)
q (a)
d=0 p=0 q=0 v=0 w=0
s
r
a
1
d!(s + b + p
s+b+p
c
a
2
1
v
a+q+v
w
(i + c + 2) (j + w + d + 2)
:
1)i+c d+2 (s + b + p + q + v)j+w+d+2
Making use of (6), we can derive recurrence relations for joint mgf of dgos.
THEOREM 2. For 1
t2
1
nn
2 and k = 1; 2; : : : ;
(i;j)
s
=
r<s
MXd (r;n;m;k)Xd (s;n;m;k) (t1 ; t2 )
(i;j)
MXd (r;n;m;k)Xd (s 1;n;m;k) (t1 ; t2 )
1
j
+
(i;j 1)
s
MXd (r;n;m;k)Xd (s;n;m;k) (t1 ; t2 )
t2 E[ (Xd (r; n; m; k)Xd (s; n; m; k))]
s
+jE[ (Xd (r; n; m; k)Xd (s; n; m; k))] ;
(28)
where
(x; y) = xi y j
1
et1 x+(t2 +1)y
et1 x+t2 y
and
(x) = xi y j
2
et1 x+(t2 +1)y
et1 x+t2 y :
PROOF. Integrating by parts of (19) and using (6), we get
MXd (r;n;m;k)Xd (s;n;m;k) (t1 ; t2 )
= MXd (r;n;m;k)Xd (s
1;n;m;k) (t1 ; t2 )
+
t2
MXd (r;n;m;k)Xd (s;n;m;k) (t1 ; t2 )
s
E[h(Xd (r; n; m; k)Xd (s; n; m; k))]
(29)
and
h(x; y) =
et1 x+(t2 +1)y
y
et1 x+t2 y
y
:
116
The Exponentiated Gamma Distribution
Di¤erentiating both the sides of above equation i times with respect to t1 and then
j times with respect to t2 and simplifying the resulting expression, we get the result
given in (28). By di¤erentiating both sides of equation (28) with respect to t1 , t2 and
then setting t1 = t2 = 0 , we obtain the recurrence relations for product moments of
dgos from EGD in the form
E[Xdi (r; n; m; k)Xdj (s; n; m; k)]
=
E[Xdi (r; n; m; k)Xdj (s 1; n; m; k)]
j n
E[Xdi (r; n; m; k)Xdj 1 (s; n; m; k)] + E[Xdi (r; n; m; k)Xdj
+
s
j
2
o
(s; n; m; k)]
E[ (Xd (r; n; m; k)(Xd (s; n; m; k)];
(30)
s
where
(x; y) = xi y j
2 y
e :
REMARK 3.1. Putting m = 0, k = 1 in (28) and (30), we obtain relations for order
statistics
(i;j)
MXr:n Xs:n (t1 ; t2 )
=
t t1
(r 1)
1
1
+
(r
1)
(i;j)
MXr
t1 E[ (Xr
1:n Xs:n
i
(t1 ; t2 )
1:n Xs:n )]
(r
+ jE[ (Xr
(i;j)
1)
MXr
1:n Xs:n
(t1 ; t2 )
1:n Xs:n )]
and
i
j
E(Xr:n
Xs:n
)
E(Xri
=
j
1:n Xs:n )
E( (Xr
REMARK 3.2. Putting m =
k record values in the form
t2
1
(i;j)
s
(i;j)
= MXd (r;n;
1
MXd (r;n;
i
(r
1;n Xs;n ))
1)
E(Xri
1
j
1:n Xs:n )
+ E(Xri
2
j
1:n Xs:n )
:
1 and k
1in (28) and (30), we obtain relations for
1;k)Xd (s;n; 1;k) (t1 ; t2 )
1;k)Xd (s 1;n; 1;k) (t1 ; t2 )
+
j
(i;j 1)
s
MXd (r;n;
t2 E[ (Xd (r; n; 1; k)Xd (s; n; 1; k))]
s
+jE[ (Xd (r; n; 1; k)Xd (s; n; 1; k))]
1;k)Xd (s;n; 1;k) (t1 ; t2 )
D. Kumar
117
and
E[Xdi (r; n; 1; k)Xdj (s; n; 1; k)]
=
E[Xdi (r; n; 1; k)Xdj (s 1; n; 1; k)]
j n
E[Xdi (r; n; 1; k)Xdj 1 (s; n; 1; k)] + E[Xdi (r; n; 1; k)Xdj
+
s
j
2
o
(s; n; 1; k)]
E[ (Xd (r; n; 1; k)Xd (s; n; 1; k))]:
s
4
Characterization
This Section contains characterization of EGD by using the recurrence relation for
mgf of dgos. Let L(a; b) stand for the space of all integrable functions on (a; b) . A
sequence (fn ) L(a; b) is called complete on L(a; b) if for all functions g 2 L(a; b)
the condition
Z
b
g(x)fn (x)dx = 0;
a
n 2 N;
implies g(x) = 0 a.e. on (a; b). We start with the following result of Lin [23].
PROPOSITION 1. Let n0 be any …xed non-negative integer, 1
a<b
1
and g(x) 0 an absolutely continuous function with g 0 (x) 6= 0 a.e. on (a; b). Then
the sequence of functions f(g(x))n e g(x) ; n
n0 g is complete in L(a; b) i¤ g(x) is
strictly monotone on (a; b).
Using the above Proposition we get a stronger version of Theorem 1.
THEOREM 3. A necessary and su¢ cient conditions for a random variable X to be
distributed with pdf given by (5) is that
(j)
MXd (r;n;m;k) (t)
(j)
= MXd (r
1;n;m;k) (t) +
j n (j 1)
MXd (r;n;m;k) (t)
r
o
E[h(Xd (r; n; m; k))] :
(31)
PROOF. The necessary part follows immediately from equation (17). On the other
hand if the recurrence relation in equation (31) is satis…ed, then on using equation (2),
we have
Z 1
Cr 1
r 1
etx [F (x)] r 1 f (x)gm
(F (x))dx
(r 1)! 0
Z 1
Cr 1
r 2
=
etx [F (x)] r +m f (x)gm
(F (x))dx
2)! 0
r (r
Z 1 (t+1)x
tCr 1
e
r 1
[F (x)] r 1 f (x)gm
(F (x))dx
1)! 0
x
r (r
Z 1
tCr 1
r 1
+
xj 1 [F (x)] r 1 f (x)gm
(F (x))dx
1)! 0
r (r
118
The Exponentiated Gamma Distribution
and
tCr
r (r
1
1)!
Z
0
11
etx
[F (x)]
x
r
1
r 1
f (x)gm
(F (x))dx:
(32)
Integrating the …rst integral on the right-hand side of the above equation by parts and
simplifying the resulting expression, we get
Z 1
etx (x + 1)
tCr 1
r 1
etx [F (x)] r 1 gm
(F (x)) F (x)
f (x) dx = 0:
(33)
(r 1)! 0
x
It now follows from Proposition 1, that
xF (x) = [etx
(x + 1)]f (x)
which proves that f (x) has the form (4).
5
Concluding Remarks
(i) In this paper, we proposed new explicit expressions and recurrence relations for
marginal and joint moment generating functions of dgos from EGD. Further,
characterization of this distribution has also been obtained on using recurrence
relation for marginal moment generating functions of dgos. Special cases are also
deduced.
(ii) The recurrence relations for moments of ordered random variables are important
because they reduce the amount of direct computations for moments, evaluate
the higher moments in terms of the lower moments and they can be used to
characterize distributions.
(iii) The recurrence relations of higher joint moments enable us to derive single, product, triple and quadruple moments which can be used in Edgeworth approximate
inference.
Acknowledgments. The author is grateful to anonymous referees and the Editor
for very useful comments and suggestions.
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