Methods for generating families of univariate continuous distributions in the recent decades

Abstract

There has been a renewed interest in developing more flexible statistical distributions in recent decades. A major milestone in the methods for generating statistical distributions is undoubtedly the system of differential equation approach. There is a recent renewed interest in generating skewed distributions. Generally speaking, the methods developed prior to 1980s may be summarized into three categories: (1) method of differential equation, (2) method of transformation, and (3) quantile method. Techniques developed since 1980s may be categorized as ‘methods of combination’ for the reason that these methods attempt to combine existing distributions into new distributions or adding new parameters to an existing distribution. This article discusses five general methods of combination and their variations. These five are (1) method of generating skew distributions, (2) method of adding parameters (e.g., exponentiation), (3) beta generated method, (4) transformed‐transformer method, and (5) composite method. WIREs Comput Stat 2013, 5:219–238. doi: 10.1002/wics.1255 This article is categorized under: Algorithms and Computational Methods > Random Number Generation Statistical and Graphical Methods of Data Analysis > Modeling Methods and Algorithms Statistical and Graphical Methods of Data Analysis > Nonparametric Methods

On generating T-X family of distributions using quantile functions

“…The T-X{Y} framework provides an easy way for generating distributions of the T-X(W) family introduced by Alzaatreh et al (2013b). Existing methods like the methods of combination reviewed in Lee et al (2013) for generating univariate continuous distributions can be derived using the T-X{Y} framework. T-X{exponential}, T-X{Weibull} and T-X{Rayleigh} can be viewed as families of distributions arising from hazard functions.…”

“…The interest in developing new methods for generating new or more flexible distributions continues to be active in the recent decades. Lee et al (2013) indicated that the majority of methods developed after 1980s are the methods of 'combination' for the reason that these new methods are based on the idea of combining two existing distributions or by adding extra parameters to an existing distribution to generate a new family of distributions. A brief summary of some methods in the literature that are related to the method proposed in this article is provided.…”

“…The T-geometric family generates the discrete analogue to the distribution of any continuous random variable T (Alzaatreh et al 2012b). For a review of methods for generating univariate continuous distributions, one may refer to Lee et al (2013).…”

T-normal family of distributions: a new approach to generalize the normal distribution

“…Various extensions of SN(λ) have been proposed and studied (e.g., Arellano-Valle et al 2004;Arnold and Beaver 2002;Arnold et al 2007;Choudhury and Abdul 2011;Balakrishnan 2002;Gupta and Gupta 2004;Sharafi and Behboodian 2008;Yadegari et al 2008). For reviews on skewed normal and its generalization, one may refer to Kotz and Vicari (2005) and Lee et al (2013). Pourahmadi (2007) showed that the skewed normal distribution http://www.jsdajournal.com/content/1/1/16 SN(λ) approaches half-normal as λ → ∞.…”

On The Gamma-Half Normal Distribution and Its Applications

“…in the literature (e.g., Alzaatreh, et al (2013a); Alzaatreh, et al (2013b); Alzaatreh, et al (2012a); Alzaatreh, et al (2012b); Lee, et al (2013)). …”