Applied Mathematics, 2011, 2, 750-751

doi:10.4236/am.2011.26099 Published Online June 2011 (http://www.SciRP.org/journal/am)

Copyright © 2011 SciRes. AM

A Characterization of the Members of a Subfamily of

Power Series Distributions

G. Nanjundan

Department of Statistics, Bangalore University, Bangalore, India

E-mail: nanzundan@gmail.com

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

s satisfy the functional equation

absfs cfs

where a, b

and c are constants and

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

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

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

s is the nth functional iteration of the off-

spring probability generating function (p.g.f.)

0

k

k

k

sp

s1 with 0

and

k

n

is the kth de-

rivative of

n

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

s

n

has a closed form expression. When

the offspring distribution is binomial, Poisson or nega-

tive binomial,

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

sp

s

1

,

0

, 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

cNp1qp

.

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

,s1 which implies

0b

and then (1.1) has no meaning. Thus 0c

.

2) Let 0c

. If 0a

, (1.1) reduces to

cf sbsf s

,

0,s 1. Then for , we get 0s

(1.1)