Exponential power distribution - Probability How To

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The exponential power distribution (also called the symmetric generalized normal distribution or generalized error distribution) is a generalization of the normal distribution. It allows for more flexibility with kurtosis and has become popular in many fields of study, including economics, engineering, and biology. In this article, we will explore what exactly the exponential power distribution is and why it is so useful.

Exponential power distribution properties

Graph of the PDF for λ= 1, κ = 2 (blue) and lambda = 2, k = 1 (green).

The exponential power distribution can be thought of as a generalized normal distribution that adds an extra parameter, κ.

The probability density function (PDF) is

Where

λ = a positive scale parameter,
κ = a positive shape parameter.

The cumulative distribution function (CDF) on the support of X (x > 0) is

F(x) = P(X ≤ x) = 1 − e^{1−e^{λx^κ.}}

General features include:

Heavy tails,
Symmetry around the mean,
Unimodality,
Can take on a bathtub shaped hazard function.

Advantages of the Exponential Power Distribution

The exponential power distribution (EPD) allows for greater flexibility with kurtosis by adding an extra parameter to the equation. This allows more variety in the shapes that can be represented by a single model. Not only does this make it easier to represent data with asymmetric values, but it also allows for greater accuracy when modeling real-world scenarios such as modeling the lifetimes of electronic and mechanical components [1]. This additional parameter makes EPD particularly useful in fields such as economics, engineering, and biology where data may have varying degrees of skewness or kurtosis.

In addition to its versatility, EPD has several other notable advantages over standard distributions. For example, it’s more robust when dealing with outliers because it can better account for extreme values in a dataset without having to transform or discard them altogether [2]. Additionally, EPD can be extended into higher dimensions by using multiple parameters instead of just one. This makes it especially attractive in fields such as machine learning where complex datasets are common.

The primary advantage of EPD is its flexibility with respect to representing different types of data shapes while still maintaining accuracy. However, there are some drawbacks associated with using this type of model as well. One major drawback is that EPD does not allow for asymmetric data. Fortunately though, there are methods available to derive a slightly different model which includes skew if needed [3]. Additionally, because EPD requires an extra parameter compared to standard distributions like the normal distribution or Poisson distribution — it can be computationally expensive and may require additional resources depending on your application.

References

[1] Viroy C. Koh, Lawrence M. Leemis, Statistical procedures for the exponential power distribution, Microelectronics Reliability, Volume 29, Issue 2, 1989,
Pages 227-236, ISSN 0026-2714, https://doi.org/10.1016/0026-2714(89)90570-2.

[2] Konunjer, I. Asymmetric Power Distribution: Theory and Applications to Risk Measurement.

[3] Ayebo, A. and Kozubowski, T. An asymmetric generalization of Gaussian and Laplace laws. Retrieved Auhust 12, 2019 from: https://wolfweb.unr.edu/homepage/tkozubow/0skeexp1.pdf