Page 1
173
Pak. J. Statist.
2008 Vol. 24(3), 173178
A FAMILY OF ABEL SERIES DISTRIBUTIONS OF ORDER k
Rupak Gupta1, Kishore K. Das2 and Dulumoni Das3
1 Faculty of Engineering and Technology, The ICFAI University,
Agartala799210, Tripura, India. Email: rupak.icfai@gmail.com
2 Department of Statistics, Gauhati University, Guwahati781014,
Assam, India
3 Department of Statistics, Cotton College, Guwahati781001, Assam, India
ABSTRACT
the Abel Series Distributions of order k (ASD(k)) generated by suitable functions of real
valued parameters in the Abel polynomials. A new distribution called the Quasi
Logarithmic Series Distribution of order k (QLSD(k)) is derived from ASD(k) and many
other distributions, viz. Quasi Binomial distributions of order k (QBD(k)), Generalized
Poison distribution of order k (GPD(k)) and Quasi negative binomial distribution of order
k (QNBD (k)) have been derived as particular cases of ASDs of order k. Some properties
have also been discussed.
KEY WORDS
Abel polynomials, Abel series distribution of order k, quasi binomial distribution of
order k, Quasi negative binomial distribution of order k, quasi logarithmic series
distribution of order k, Generalized Poisson distribution of order k.
AMS 2000 Subject Classification: 62E15; 62E17
1. INTRODUCTION
Feller (1968) introduced the discrete distributions of order k, when he extended the
notion of success to a success run of length k. A Run is usually defined as an
uninterrupted sequence of symbols like (‘S’ or ‘F’). Thus the distributions associated
with runs of ‘k’ like outcomes are distributions of order k.
The only explicit relationship between distributions of order ‘k’ and that of order ‘1’
is that “Every distribution of order 1 is the usual corresponding discrete distribution”.
In this paper, we propose a simple series distribution called the Abel Series
distribution of order k (ASD (k)). The ASD (k) may be obtained by the expansion of
suitable functions in power series. Using suitable functions in ASD (k), we derive some
new distributions of order k such as the QLSD (k), QBDI (k), GPD (k) and QNBD (k).
In this paper we have considered a class of univariate discrete distributions of order k,
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A Family of ABEL Series Distributions of Order k
174
2. THE ABEL SERIES DISTRIBUTION OF ORDER k
Comtet (1974, p.128) introduced the Abel polynomials as,
1
1
x
( , )
A a z
()
!
x
x
a axz
−
=−
;
0,1,2,.....
x =
Extending this polynomial to order k, we may get,
k
A a za a
=
Consider a finite and positive function
(1(1) )
k
=
is a nonnegative integer. For any real z, we have the Abel series expansion
of order k as,
( )
{}
12
0,,...,
k
xx xx
=
⎡
δ
⎢
δδδ
⎢
⎣
with
12
( ,,...,)
k
aa aa
=
,
12
( , ,....,)
k
x x xx
=
, where in the inner sum, the summation is
over all nonnegative integers
12
, ,...,
k
x xx such that
xx
++
δ
=
δ
()
1
,
1
( , )()/!
i
x
x kiiii
i
x zx
−
=−
∏
; (2.1)
( )
f a of
()
123
,,,...,
k
a a a aa
=
, where each
ia i
,
( , )
a z D f kz
( )
k
x k
f aA
∞
= ∑∑
=
{}
()
12
12
12
1
...
0, ,...,
1
12
( )
f a
a
!
,,...,
i
k
k
k
x
xxx
k
iii
x
k
xx
xx xx
i
i
a xz
=
a ax z
x
aa
−
+ + +
∞
∑
=
=
⎤
⎥
⎥
⎦
−
∑
∏
, (2.2)
12
2...
k
xxkxx
+++= and
( )
f kz
( )
f a
( )
k
( )
k

a xz
=
DD
=
( )
x
k
a
δ
12
1
....
1

....
k
k
x
a xz
=
x
f a
a
+
.
definition of the Abel series distribution of order k.
Definition:
A family of discrete distributions {
k, if it has the following probability function (p.f.),
⎡
⎢
=
⎢
⎣
where, ( )
f a is stated in (2.2)
Since (2.2) is the Abel series expansion of order k, so we propose the following
}
( );
x x =
0,1,2,...
k p
is said to be ASDfamily of order
12
12
12
1
....
{ ,
x x
,....,}
1
12
()
( )
f a
∂
( )0,1,2,....
!
,,....,
i
k
k
k
x
xxx
k
iii
k
x
k
xx
x
i
i
a xz
=
a a x z
p xx
x
aaa
−
+++
=
⎤
⎥
⎥
⎦
−
∂
=
∂∂
∑
∏
, (2.3)
and then for z = 0, we get the common power series distribution (Noack, 1950).
For z = 0, (2.3) reduces to power series distribution of order k.
In particular, if k = 1, then (2.3) reduces to simple ASDfamily (Nandi and Das, 1994)
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Gupta, Das and Das
175
3. QUASI LOGARITHMIC SERIES DISTRIBUTION OF ORDER k
given by,
Consider the Abel series expansion of order k of the logarithmic series function
( )
f a
()
()
1
+
2
x
12
1
0{ ,
x
,..,}
1
2...
log(1)1
i
i
k
k
x
k
x
iii
i
xx xx
i
i
kxx
a ax z
x
ax z
−
∞
∑
−
=∋
=
+ +=
−
−−=−
∑
∏
. (3.1)
Then the probability function of the ASD is,
{
x
}
()
1
+
2
x
12
1
, ,...,
+ +
1
2...
( )
x
(1)
i
i
k
k
x
k
iii
x
ki
x xx
i
i
kxx
a ax z
x
px z
−
−
∋
=
=
−
= θ−
∑
∏
, (3.2)
where
1
log(1
⎡
⎣
)
ia
−
θ = −−
⎤
⎦.
(Johnson and Kotz, 1969, p. 166).
4. QUASINEGATIVE BINOMIAL DISTRIBUTION OF ORDER k
Suppose the function ( )()
f aba−
=−
The p.f. (3.2) is called the Quasi Logarithmic Series distribution (QLSD) of order k.
For z = 0, the p.f. (3.2) becomes the logarithmic series distribution of order k.
For k = 1 and z = 0, the p.f. (3.2) becomes the common logarithmic series distribution
n
. Then the Abel series expansion of
()()
1
+
2
x
12
1
0{ ,
x
,..,}
1
12
2 ...
1
x
()
, ,..,
ii
k
k
k
xn x
− −
i
n
iiiii
x x xx
i
k
kxx
xn
baa ax zbx z
x x
∞
∑
−
−
=∋
=
+ +=
+ −
⎛
⎜
⎝
⎞
⎟
⎠
−=−−
∑
∏
(4.1)
Suppose
()()()
,&
ii
iiiiii
a
−
b
−
z
−
PQ
bababa
== φ =
. (4.2)
Using (4.2), we get the p.f. of a particular case of ASD of order k as,
1
+
2
x
12
1
{ ,
x x
x
,..,}
1
12
2...
1
x
( )()()
,,..,
ii
k
k
k
i
xn x
− −
kii
x
i
k
kxx
xn
p xP PxQx
x x
−
∋
=
+ +=
+ −
⎛
⎜
⎝
⎞
⎟
⎠
=−φ−φ
∑
∏
, (4.3)
where QP = 1 and ()1
i
Px
− φ < .
4.1 Some Special Cases:
i) If k =1, then (4.3) reduces to QNBD.
The probability function (4.3) is termed as the QNBD of order k.
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A Family of ABEL Series Distributions of Order k
176
ii) If k =1 and φ = 0, then (4.3) reduces to the common negative binomial distribution
(Johnson and Kotz, 1969, p. 122).
iii) If k =1 and n = 1, the p.f. (4.3) becomes,
11
( )
x
()()
xx
k p P PxQx
− − −
=− φ−φ
. (4.4)
and Kotz, 1969, p. 123).
5. QUASI BINOMIAL DISTRIBUTION OF TYPEI OF ORDER k (QBDI(k))
Consider the simple series function,
( )
f a
)n
ab
+
of order k is,
We call the p.f. (4.4), the quasigeometric distribution.
For φ = 0, the p.f. (4.4) transforms to the common geometric distribution (Johnson
()n
ab
=+
. Then the Abel Series expansion
of (
11
1
1
12
011
12
.....
,.....,
()
,,
kk
ii
ii
k
j
j
xnx
k
∑
kk
k
n
ii
mii
k
jxn m kx
= − −
xxxx
aba a
⎟ ⎜
⎝
⎠
zxbzx
x xx x
==
−−
−
===
∑∑
++++
⎛
⎜
⎝
⎞ ⎛
⎞
⎟
⎠
⎛
⎜
⎝
⎞
⎟
⎠
+=+−
∑
∑∑∑
.
Suppose
(5.1)
a
+
p
ab
=
,
b
+
q
a b
=
&
z
+
ab
φ =
(5.2)
Now using (5.2) in (5.1), we get the probability functions as,
11
1
12
11
12
.....
,.....,
( )
,,
kk
ii
ii
xnx
kk
k
kii
ii
k
xxxx
p xp p
⎟ ⎜
⎠
xqx
x xx x
==
−−
==
∑∑
++++
⎛
⎜
⎝
⎞ ⎛
⎞
⎟
⎠
⎛
⎜
⎝
⎞
⎟
⎠
= +φ−φ
⎝
∑∑
(5.3)
For
0,1,....,
n
k
x
⎡ ⎤
⎢ ⎥
⎣ ⎦
x , such that
=
; q = 1p, [a] means the largest integer not exceeding ‘a’ and the
integers
12
, ,....,
k
x x
123
23....
k
xxx kxn m
−
kx
++++=−
.
The p.f. (5.3) is termed as QBDI of order k (Gupta and Das , 2008).
6. GENERALIZED POISSON DISTRIBUTIONS OF ORDER k
Let us consider the exponential series function,
=
of order k is,
( )
f a
a
e
=
. Then the Abel series
expansion of ( )
a
f ae
( )
x
1
1
1
1
1
/!
k
k
i
i
i
i
j
j
x
zx
k
k
a
ii
jxxi
i
ea a
⎜
⎝
zxe
=
=
−
==
=
∑
∑
⎛⎞
⎟
⎠
=−
∑∑∑
∏
(6.1)
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Gupta, Das and Das
177
Then we get a special case of ASD of order k as,
( )
x
1
1
1
1
1
( )/!
k
k
i
i
i
i
j
j
x
a z
−
x
k
k
kii
jxxi
i
p x a a
⎜
⎝
zxe
=
=
⎛
⎜
⎜
⎝
⎞
⎟
⎟
⎠
−
−
==
=
∑
∑
⎛⎞
⎟
⎠
=−
∑∑∑
∏
(6.2)
6.1 Special Case:
If k = 1, then p.f. (6.2) reduces to the p.f. of GPD (Consul and Jain, 1973) given by,
()
( )/!
i
p ka axzex
=−
ACKNOWLEDGEMENTS
The p.f. (6.2) is known as GPD(k) (Gupta and Das, 2008)
()
1
x
a xz
−
−
−
(6.3)
One of the authors, Rupak Gupta is grateful to Dr. R.K. Patnaik, ProViceChancellor
of The ICFAI University, Tripura, India and Prof. J.J. Kawle, Director, INEUC for their
constant encouragements and motivations.
The authors acknowledges the financial support received from the ICFAI University,
Tripura and also, thanks the referees for their helpful suggestions and comments.
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