Applied Mathematics, 2011, 2, 750-751
doi:10.4236/am.2011.26099 Published Online June 2011 (http://www.SciRP.org/journal/am)
A Characterization of the Members of a Subfamily of
Power Series Distributions
G. Nanjundan
Department of Statistics, Bangalore University, Bangalore, India
Received April 7, 2010; revised April 22, 2011; accepted April 25, 2011
Abstract
This paper discusses a characterization of the members of a subfamily of power series distributions when
their probability generating functions

f
s satisfy the functional equation

 
absfs cfs

 where a, b
and c are constants and
f
 is the derivative of f.
Keywords: Galton-Watson Process, Probability Generating Function, Binomial, Poisson, Negative Binomial
Distributions
1. Introduction
Let a population behave like a Galton-Watson process
with a known offspring distribution
0
kk. Suppose that the generation size

0
;0, 1
n
Xn X

p


n
X
k

is
observed and n, the age in generations, is to be estimated.
Such a problem arises in many situations. For example,
one might be interested in the length of existence of a
certain species in its presen t form or how long ago a mu-
tation took place, etc. (See Stigler [1]).
When the generation size


n
X
k is observed and
the offspring distribution is known, the likelihood func-
tion is give n b y




(0
0
,
!1 0
nn
k
n
n
Ln PXkX
f
kf




)
where

n
f
s is the nth functional iteration of the off-
spring probability generating function (p.g.f.)

0
k
k
k
f
sp


s1 with 0
s
 and

k
n
f
is the kth de-
rivative of

n
f
s

n
with respect to s. T he maxi mum l ike -
lihood estimator of n can be obtained by the method of
calculus if
f
s
n
has a closed form expression. When
the offspring distribution is binomial, Poisson or nega-
tive binomial,

f
s does not have a closed form ex-
pression. Ades et al. [2] have obtained a recurrence for-
mula to compute when the
offspring p.g. f. sati sfi es the fu nct i onal eq uat ion

,1,2,3,
n
PXk k

absfs cfs


where a, b and c are constants and f' is the derivative of f.
We derive a characterization result using this differential
equation.
2. Characterization
We establish the following theorem.
Theorem: Let X be a non-negative integ er valued ran-
dom variable with


, 0,1,
k
PXkp k  and pk > 0
at least for0, 1k

. If the p.g.f.

0
k
k
k
f
sp

s
1
,
0
s

, satisfies (1.1), then the distribution of X is
Poisson, bin o mial, or negative binomial.
Proof: It is straight forward to verify that
1) when X has a Poisson distribution with mean

,
(1.1) hol ds with 1, 0 and ab c


.
2) when X has a binomial (N,p)-distribution, (1.1)
holds with , aqbp

 and with
cNp1qp

.
3) when X has a negative binomial (α,p)-distribution,
(1.1) holds with 1, abq

 and cq

 where
1qp

.
Now let us have a close look at the possible values of
the constants in (1.1).
1) If 0c

, then (1.1) reduces to



0absfs




0,s 1. In particular, for , this becomes 0s


af s
0

. Since


1
0fp
0 ,0a


. But then (1.1)
turns out to be

 

0, 0fs
 ,s1 which implies
0b

and then (1.1) has no meaning. Thus 0c

.
2) Let 0c

. If 0a

, (1.1) reduces to




cf sbsf s
,


0,s 1. Then for , we get 0s 
(1.1)

G. NANJUNDAN751

00cf  and hence which is a contradiction.
Therefore . 0c

0a0,c
3) Let . Suppose, if possible,
0a0b

.
Then (1.1) becomes ,

af s


cf s


0, 1s . Iden-
tifying this as a linear differential equation and solving,
we get
 
1
log ,
f
scask
1
where k1 is an arbitrary constant. Since

1f

and
1
kca

 , the above solution reduces to

exp1 ,[0,1].
cs s
a




fs
Note that ca cannot be negative because if 0ca

,
then which is impossible. Thus

0f10ca and

f
s is the p.g.f. of a Poisson distribution with mean
ca.
4) Let and 0, c0a0b

. Then


'fsc
f
sab
s
. Solving this differential equation, we get
 
c
b
f
skabs, where k is a constant. Since

11f,

c
b
kab

 . Hence

.
c
b
abs
fs ab





(2.1)
Note that if , then
0ab

f
s in (2.1) does not
define a p.g.f.
Also, (2.1) can be expressed as


**,
c
b
f
sabs (2.2)
where *a
aab , *b bab , and .
**
1ab
Since and hence

*
0
001, 0fp a
1,ab
1
0
*
0b. This also implies that . Thus, case (4)
reduces to 0, 0ca

 and . 0b
4a) Let . Then 0c0cb. Suppose that cNb

where N is a positive integer. Then

f
s in (2.2) is the
p.g.f. of a binomial


*
,Nb -distribution.
4b) Let 0c

. Then 0cbb. Suppose that cN

.
Then,


f
s in (2.2) is the p.g.f. of a negative binomial


*
,Nb -distribution.
Now it remains to verify whether cb can be a frac-
tion with 0c

. Note that (2.2) can be rewritten as


*
*
*
1
c
cb
bb.
f
sa s
a





(2.3)
The expansion of the RHS of (2.3) is a power series in
s with some coefficients being negative if cb is a frac-
tion, which is not permitted because the coefficients pk in

0
k
k
k
f
sp


s, being probabilities, are non-negative.
Now the proof of the theorem is complete.
3. Acknowledgements
The author is extremely grateful to Prof. M. Sreehari for
a very useful discussion.
4. References
[1] M. Stigler, “Estimating the Age of a Galton-Watson Br an-
ching Process,” Biometrika, Vol. 57, No. 3, 1972, pp.
505-512.
[2] M. Ades, J. P. Dion, G. Labelle and K. Nanthi, “Recur-
rence Formula and the Maximum Likelihood Estimation
of the Age in a Simple Branching Process,” Journal of
Applied Probability, Vol. 19, No. 4, 1982, pp. 776-784.
doi:10.2307/3213830