Power Series Distributions

Probability Distributions > Power Series Distributions

Contents:

  1. What are power series distributions?
  2. Explicit PMF (Normalization Constant)
  3. Special properties
  4. Variations and compounds of power series distributions
  5. Power series vs power law distributions

What are power series distributions?

Power series distributions are discrete probability distributions on a subset of natural numbers. The distributions are named because they are constructed from the power series. A discrete random variable X has a power series distribution if it follows this formula: power series distributions
Where f(θ) is a moment generating function which is positive, finite, and differentiable [1]. Many discrete distributions fall under this general category of distributions including the binomial distribution, negative binomial distribution, and Poisson distribution.

Explicit PMF (Normalization Constant)

The distribution is a member of the exponential family of distributions and can be expressed as follows [2]: general form of the power series distribution
Where a and g are functions of θ, an unknown parameter and c is a function of x. However, while the power series distribution can be expressed in a number of different ways, a normalizing constant is crucial to actually use the distribution in calculations or to derive mean/variance: power series distribution with normalizing constant Where:

  • s⊂{0,1,2,…} is the support of the distribution (often infinite but can be finite).
  • a(x) is a known function of (sometimes () ≡ 1, or it could be ()=(), etc.)
  • g(θ) is another function of .
  • f(θ) is the normalizing constant, which ensures that ∑∈ (=) = 1.

Special properties

The power series distribution has some special properties:

Condition Power Series Tends to…
If θ = p / (1 – p); f(θ) = (1 + θ)n; s = {1, 2, 3, … n} Binomial distribution.
If f(θ) = eθ and s = {0, 1, 2, 3, … ∞} Poisson Distribution.
If θ = p / (1 – p); f(θ) = (1 + θ)-n; s = {0, 1, 2, 3, … ∞} Negative Binomial Distribution
If f(θ) = -log (1 – θ) and s = {1, 2, …}, Logarithmic distribution

 

Variations and compounds of power series distributions

Several compounded distributions exist in the literature including the Weibull- power series distribution and the generalized Gompertz-power series distributions, obtained by compounding the generalized Gompertz distribution (a generalization of the exponential distribution) and the power series distributions [3].

Other variations include the exponential power series distribution, a composition of the exponential distribution with the power series distribution which gives a distribution with a decreasing failure rate [4].

Power series vs power law distributions

A power series distribution is a probability distribution defined by an infinite series of terms, often of the form xn, where n is a non-negative integer, and commonly used for phenomena that exhibit a long tail—a few large values and many small ones.

On the other hand, a power law distribution is defined by a power law function, f(x) = x − α, where α is a positive constant. It is often used to model scale-free phenomena whose values do not change when scaled by a constant factor.

One significant difference between these types of distributions is that the power series distribution has a finite mean and variance, while the power law distribution does not. The following table summarizes the differences between the two distributions:

Property Power Series Distribution Power Law Distribution
Number of terms Infinite Finite
Function Power series Power law
Mean Finite Undefined
Variance Finite Undefined
Long tail Yes Yes
Scale-free No Yes
Table showing the major differences between power law and power series distributions.

References

  1. Gupta, R. (1974). Modified Power Series Distribution and Some of Its Applications. The Indian Journal of Statistics, Volume 36, Series B, Pt. 3, pp. 288-298.
  2. Sakia, R. (2018). Application of the Power Series Probability Distributions for the Analysis of Zero-Inflated Insect Count Data. Open Access Library Journal, Vol.5 No.10.
  3. Tahmasebi, S. & Jafari, A. (2000). Generalized Gompertz-Power Series Distributions. Retrieved December 14, 2021 from: https://www.academia.edu/25858978/Generalized_Gompertz-Power_Series_Distributions
  4. Chahkandi, M. & Ganjali, M. (2009). On some lifetime distributions with decreasing failure rate. Computational Statistics & Data Analysis 53(12): 4433-4440.

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