Properties of new derived power series distribution

Introduction

Suppose that a = (a0, a1, a2,…) is a sequence of nonnegative real numbers. The partial sum of order n ∈ N is gn(θ) =  ak k , k=0, 1 … n for all θ ∈ R. The power series is then defined by g(θ) = limn→∞ gn(θ) for θ ∈ R for which the limit exists, and is denoted gn(θ) = ak k , k = 0,1 …  θ ∈ R. A random variable n with values in N has the power series distribution associated with the function g (or equivalently with the sequence a), with parameter θ ∈ [ 0, r ) , if N has discrete probability density function fθ(n) = anθ n / g(θ), n ∈ N. Let P1(s) and P2(s); | s | 1 are the probability generating functions of two power series distributions. Jayasree and Swamy (2006) [2] defined a family of new power series distributions with the convolution of P1(s) and [P2(s)] -1 called Derived Power Series Distributions (DPSD). Consider a DPSD for which the probability mass function is given by

The mean and variance of derived power series distributions are μ = μ1 -μ2 and σ 2 = σ1 2 -σ2 2 where μ1, σ1 2 are the mean and variance of P1(s) and μ2, σ2 2 are the mean and variance of P2(s) power series distributions. A power series distribution derived by Kulasekera and Tonkyn (1992) [5] is presented in definition 1.1.

Definition 1.1:

A random variable X is said be Kulasekera and Tonkyn probability distribution if it satisfies the probability law P [X = x] = x  q x / ( x  q x ) ;  x 1, 2, … ;  <  < ; 0 < q < 1; p+q =1. For fixed value of α, the distribution belongs to the family of power series distribution. The mean and variance of the distribution are (1+q) p -1 and 2qp -2 .

A power series distribution derived by Jayasree and Bhatra Charyulu (2017) [2] were presented is like as follows. Let X1 be the random variable follows a power series probability distributions of Kulasekera and Tonkyn (1992) [5] , and X2 be the random variable follows Geometric distribution. Let P1(s) and P2(s) are the probability generating functions of the two power series distributions in s and convergent for | s | 1,

Where P [X1 = x1] = x  q1 x / (

The convolution of P1(s) and [P2(s)] -1 provides a power series distribution with random variable X satisfies the probability law

Under appropriate conditions for the coefficients of the power series P(s) to be non-negative, one can consider this as the probability generating function of a random variable, where from, the p.m.f. can be obtained by identifying the coefficients in P(s). The P(s) is a proper Power series probability distribution for specified vector of probabilities with The mean (1+q1) p1 -1 -p2 -1 and variance

Properties of the Power Series Distribution

Some properties of new derived power series distribution of Jayasree and Bhatra Charyulu (2017) [2] are studied and presented below. (2.1)

where u1= μ+1, u2 = 2(u1 2 -2u1 -2σ 2 ), u3= (13u1 2 -8u1 -8σ 2 ) 1/2 , and c = σμ -1 which is the coefficient of variation of the distribution .

Proof:

The moment equations for the estimators are given by μ

The solution of these equations yields 2 p = u2 -1 {-3u1+u3}. Substituting 2 p in (2.3), one can obtain (2.4).

Further 2 p is real if 13u1 2 -8u1 -8σ 2 > 0. Equivalently, if μ + μ 2 > σ 2 .In terms of u1 and c, the moment estimators are given by (2.3) and (2.4) if u1(u1-1) -1 > c 2 .

Theorem 2.2:

The derived power series distribution belongs to the two-parameter exponential family of distributions, if the k th power of   2 1 q , q x; ω is negligible for k ≥ 3, where ω( x; q1, q2) = |x -1 -q1 -1 q2| .

Proof:

The probability mass function of the new Derived Power series distribution specified in (1.1) can be expressed as Px

When the k th power of ω( x; q1, q2) = |x -1 -q1 -1 q2| is negligible for k ≥ 3, considering loge{1+(x -1 -q2q1 -1 )}, one has Px = exp[x logeq1+ logex + q2 (xq1) -1 +x -1 -(2x 2 ) -1 +λ(q1,q2)] (2.6) where λ(q1,q2)= loge { (1-q1) 2 (1-q2) -1 }-q2 q1 -1 -q2 2 (2q1 2 ) -1 Let 1 = logeq1; 2= q2q1 -1 , so that q1 = e 1 and q2 = 2 e 1 then by the re-parameterization, one has,

(2.7) From (2.7), we can obtain maximum likelihood estimators for the parameters 1 and 2.

where H(x) = logx + x -1 -(2x 2 ) -1 and λ(1, 2) = 2log(1-e ϴ1 ) -log(1-2e 1 ) -2 -(2 2 ) /2 Thus, the maximum likelihood equations are given by A

where, A is the arithmetic mean and H is the Harmonic mean of the sample. Since the solution of these equations seem to be difficult, one can adopt an iterative procedure like the Newton-Raphson, to obtain the same.

Example 2.1: Consider distribution of spiders among patches of habitat or prey data presented in table 2.1. The fitting of derived distribution is presented below. When a distribution is fit, using the method of moments for estimating the parameters one can obtain the estimates of 1 p ˆ= 0.6277, 2 p ˆ= 0.9792, 1 q = 0.3723 and 2 q = 0.0208 and the expected frequencies are obtained as 0.1975, 0.4310, 0.125, 0, 0.1428 and 0.6666, so that the observed value of χ 2 is given by χ 2 = 1.5629. Note: It can be noted that the derived power series distribution of Swamy and Jayasree (2006) SJS-4 is also an equally good fit to the same data.

Acknowledgement

AcknowledgementsThe authors are grateful to Prof. R.J.R. Swamy for his valuable suggestions in pursuing this work.

References

1. Jayasree G. Derived Power Series Distributions and Applications, unpublished Ph.D. thesis submitted to Osmania University, Hyderabad, India, 2003.
2. Jayasree G, Bhatra charyulu N.Ch. New derived power series distribution and its properties, International Journal of Mathematical Archive, ISSN: 2229-5046., 2017.
3. Jayasree G, Swamy RJR. Some New Discrete Probability Distributions Derived from Power Series Distributions, Communications in Statistics Theory and Methods, Taylor and Francis series, 2006; 35(9):1555-1567.
4. Jayasree G, Swamy RJR. Statistical Inference about Parameters of Derived Power Series Distributions, International Journal of Advances in Management, Technology & Engineering Sciences, 2013; 2(6v):100-106, ISSN: 2249-7455.
5. Kulasekera KB, Tonkyn DW. A new discrete distribution, with applications to survival, dispersal and dispersion, Commun. Statist. Simula, 1992; 21(2):499-518.
6. Patil GP. Certain properties of the generalized power series distribution. Ann. Inst.Statist. Math, 1962; 14:179-182.