< Probability Distributions < Power function distribution
The power function distribution (PFD) is a flexible model used to analyze income distribution data, lifetime data, and failure processes. One of its key strengths is its mathematical simplicity, especially when compared to more complex distributions like the Weibull distribution.
The PFD is a special case of the beta distribution [1]—also called the Pearson type I distribution—and has an inverse relationship with the standard Pareto distribution [2], such that the moments of the PFD correspond to the negative moments of the Pareto distribution [3].
Sometimes, the graph of a power function of the form f(x) = axp is called a power function distribution, and various parameterizations of the probability density function (pdf) and cumulative distribution function (CDF) exist, reflecting the distribution’s versatility.
Power-Function Distribution PDF & CDF
A random variable has a (standard) power function distribution if its PDF is:
The CDF for the standard power function distribution is:
To allow a different support (0,θ) for some θ > 0, the pdf can be written as:
With CDF:
Note on “Power Functions”
A power function has the form f(x) = axp. There are an infinite number of possible power functions, depending on your choice of scaling factor a and exponent p. Each power function has an associated distribution (or graph). Therefore, the term power function distribution could refer to one of these distributions (rather than “the” power function distribution described above).
References
- Pandey, A. & Saran, J. (2004). ESTIMATION OF PARAMETERS OF A POWER FUNCTION DISTRIBUTION AND ITS CHARACTERIZATION BY K-TH RECORD VALUES. STATISTICA, anno LXIV, n.3. Retrieved December 6, 2021 from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.953.5181&rep=rep1&type=pdf
- Kleiber C, Kotz S. Statistical size distributions in economics and actuarial sciences: John Wiley & Sons; 2003.
- Johnson NL, Kotz S. Distributions in Statistics: Continuous Univariate Distributions: Vol 1. John Wiley & Sons; 1970.