 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 fs 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 0kk. Suppose that the generation size 0;0, 1nXn XpnXk 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 ). When the generation size nXk is observed and the offspring distribution is known, the likelihood func-tion is give n b y (00 ,!1 0nnknnLn PXkXfkf) where nfs is the nth functional iteration of the off-spring probability generating function (p.g.f.) 0kkkfsps1 with 0s and knf is the kth de- rivative of nfsn with respect to s. T he maxi mum l ike -lihood estimator of n can be obtained by the method of calculus if fsn has a closed form expression. When the offspring distribution is binomial, Poisson or nega-tive binomial, fs does not have a closed form ex-pression. Ades et al.  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,nPXk 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,kPXkp k  and pk > 0 at least for0, 1k. If the p.g.f. 0kkkfsps1, 0s, 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 10fp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  1log ,fscask1 where k1 is an arbitrary constant. Since 1f and 1kca , the above solution reduces to exp1 ,[0,1].cs safs Note that ca cannot be negative because if 0ca, then which is impossible. Thus 0f10ca and fs is the p.g.f. of a Poisson distribution with mean ca. 4) Let and 0, c0a0b. Then 'fscfsabs. Solving this differential equation, we get  cbfskabs, where k is a constant. Since 11f, cbkab . Hence .cbabsfs ab (2.1) Note that if , then 0abfs in (2.1) does not define a p.g.f. Also, (2.1) can be expressed as **,cbfsabs (2.2) where *aaab, *bbab, and . **1abSince and hence *0001, 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 fs in (2.2) is the p.g.f. of a binomial *,Nb -distribution. 4b) Let 0c. Then 0cbb. Suppose that cN. Then, fs 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 ***1ccbbb.fsa sa (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 0kkkfsp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  M. Stigler, “Estimating the Age of a Galton-Watson Br an- ching Process,” Biometrika, Vol. 57, No. 3, 1972, pp. 505-512.  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 Copyright © 2011 SciRes. AM