Generalization methods of probability distributions

in engineering applications

Gokarna Aryal

Purdue University Calumet

Hammond, IN 46323, USA

Chris P.Tsokos

University of South Florida

Tampa, FL, 33620, USA

In the last few decades there has been a growing interest on generalizing a probability distribution and investigation of possible areas of applications. In this note we present some generalization methods applied to both symmetric and asymmetric distributions. Most of these generalizations have wide applications in Engineering.

A frequently occurring problem in statistics is model selection and related issues. This includes the identification of the underlying probability distribution. In the last few decades there has been a growing interest on generalizing a probability distribution and investigation of possible areas of application. In particular univariate skew-symmetric models have been considered by several authors. A classical example is the skew normal distribution with its probability density function (pdf) given by f(x)=2g(x)G(lx) where, g(ћ) and G(ћ) respectively, denote the pdf and cumulative distribution function (cdf) of the standard normal distribution. This generalization of probability distribution was introduced by O'Hagan and extensively studied by Azzalini. This method of generalization has been extended to the other univariate symmetric models. Several authors study these extensions and find their area of applications. An extensive bibliography has been made available.

Azzalini's approach to generate a flexible family of probability distributions is restricted to symmetric distributions. In this study we summarize recent generalization methods applicable to symmetric and asymmetric probability distributions. This is by no means an exhaustive summary of the generalization methods. The notations:

probability density function (pdf); cumulative distribution function (cdf).

In the present study, we have briefly reviewed the generalizations methods of probability distributions. This is by no means an exhaustive review of the generalization methods. We expect that this study will serve as a reference and help to advance and generate further fruitful applications in the subject area.