Probability Distributions > Power Series Distributions
Contents:
- What are power series distributions?
- Explicit PMF (Normalization Constant)
- Special properties
- Variations and compounds of power series distributions
- 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:
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]:
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: 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 |
References
- 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.
- 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.
- 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
- Chahkandi, M. & Ganjali, M. (2009). On some lifetime distributions with decreasing failure rate. Computational Statistics & Data Analysis 53(12): 4433-4440.