A Characterization of the Members of a Subfamily of Power Series Distributions

doi:10.4236/am.2011.26099

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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

E-mail: [email protected]

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

kk. Suppose that the generation size



;0, 1

Xn X



p







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





k is observed and

the offspring distribution is known, the likelihood func-

tion is give n b y



!1 0

Ln PXkX









)

where



s is the nth functional iteration of the off-

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







s1 with 0

 and



is the kth de-

rivative of



with respect to s. T he maxi mum l ike -

lihood estimator of n can be obtained by the method of

calculus if

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,

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,

PXkp k  and pk > 0

at least for0, 1k



. If the p.g.f.





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





0fp

0 ,0a





. But then (1.1)

turns out to be









0, 0fs

 ,s1 which implies



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

 

log ,

scask

where k1 is an arbitrary constant. Since





and

kca



 , the above solution reduces to



exp1 ,[0,1].

cs s









Note that ca cannot be negative because if 0ca



then which is impossible. Thus



0f10ca and



s is the p.g.f. of a Poisson distribution with mean

ca.

4) Let and 0, c0a0b



. Then



'fsc

sab

s

. Solving this differential equation, we get

 

skabs, where k is a constant. Since



11f,



kab



 . Hence



abs

fs ab











(2.1)

Note that if , then

0ab



s in (2.1) does not

define a p.g.f.

Also, (2.1) can be expressed as



**,

sabs (2.2)

where *a

aab

, *b

bab

, and .

1ab

Since and hence



001, 0fp a

1,ab

1

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



s in (2.2) is the

p.g.f. of a binomial





,Nb -distribution.

4b) Let 0c



. Then 0cbb. Suppose that cN



.

Then,





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



bb.

sa s











(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







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