A new method for generating families of continuous distributions
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Abstract
In this paper, a new method is proposed for generating families of continuous distributions. A random variable \(X\), “the transformer”, is used to transform another random variable \(T\), “the transformed”. The resulting family, the \(T\)\(X\) family of distributions, has a connection with the hazard functions and each generated distribution is considered as a weighted hazard function of the random variable \(X\). Many new distributions, which are members of the family, are presented. Several known continuous distributions are found to be special cases of the new distributions.
Keywords
Hazard function Pearson system Betafamily Generalized distribution Entropy Shannon entropy1 Introduction
Statistical distributions are commonly applied to describe real world phenomena. Due to the usefulness of statistical distributions, their theory is widely studied and new distributions are developed. The interest in developing more flexible statistical distributions remains strong in statistics profession. Many generalized classes of distributions have been developed and applied to describe various phenomena. A common feature of these generalized distributions is that they have more parameters. Johnson et al. [19] stated that the use of fourparameter distributions should be sufficient for most practical purposes. According to these authors, at least three parameters are needed but they doubted any noticeable improvement arising from including a fifth or sixth parameter.
This article presents yet another technique to generate families of continuous probability distributions. The article is organized as follows: Sect. 2 presents a new technique for generating families of continuous distributions. Section 3 gives examples of classes of generalized families developed using the technique in Sect. 2. The paper ends with a summary and conclusion in Sect. 4.
2 Method for generating families of continuous probability distributions
The betagenerated family of distributions in (1.10) and the \(KW\)\(G\) family of distributions in (1.11) are generated by using distributions with support between \(0\) and \(1\) as the generator. The beta random variable and the \(KW\) random variable lie between \(0\) and \(1\), so is the c.d.f. \(F(x)\) of any other random variable. The limitation of using a generator with support lying between \(0\) and \(1\) raises an interesting question: ‘Can we use other distributions with different support as the generator to derive different classes of distributions?’ This section will address this question and introduce a new technique to derive families of distributions by using any p.d.f. as a generator.
Definition:

The c.d.f. in (2.2) is a composite function of \((R\mathbf \huge . W\mathbf \huge . F)(x)\).

The p.d.f. \(r(t)\) in (2.2) is “transformed” into a new c.d.f. \(G(x)\) through the function, \(W(F(x))\), which acts as a “transformer”. Hence, we shall refer to the distribution \(g(x)\) in (2.3) as transformed from random variable \(T\) through the transformer random variable \(X\) and call it “TransformedTransformer” or “\(T\)\(X\)” distribution.

The random variable \(X\) may be discrete and in such a case, \(G(x)\) is the c.d.f. of a family of discrete distributions.

The distribution (1.12) introduced by Ferreira and Steel [13] is a special case of (2.3) by defining \(W(F(x))=F(x)\) and \(r(.)\) plays the same role as the weight function.
 1.
When the support of \(T\) is bounded: Without loss of generality, we assume the support of \(T\) is \([0,\, 1]\). Distributions for such \(T\) include uniform \((0,\, 1)\), beta, Kumaraswamy and other types of generalized beta distributions. \(W( F(x))\) can be defined as \(F(x)\) or \({{F}^{\alpha }}(x)\). This is the betagenerated family of distributions which have been well studied during the recent decade.
 2.
When the support of \(T\) is \([a,\, \infty ), a \ge 0\): Without loss of generality, we assume \(a = 0. W(F(x))\) can be defined as \(\log (1F(x)), F(x)/(1F(x)), \log (1{{F}^{\alpha }}(x))\), and \({{F}^{\alpha }}(x)/(1{{F}^{\alpha }}(x))\), where \(\alpha >0\).
 3.
When the support of \(T\) is \((\infty ,\, \infty )\): \(W( F(x))\) can be defined as \(\log [\log (1F(x))], \log [F(x)/(1F(x))], \log [\log (1{{F}^{\alpha }}(x))]\), and \(\log [{{F}^{\alpha }}(x)/(1{{F}^{\alpha }}(x))]\).
Probability density functions of some \(T\)\(X\) families based on different \(W(.)\) functions
Support of \(T\)  \(W(F(x))\)  \(g(x)\) 

\([0,\,\infty )\)  \(\frac{F(x)}{1F(x)}\)  \(\frac{f(x)}{{{(1F(x))}^{2}}}\,\,r\left\{ \frac{F(x)}{1F(x)} \right\} \) 
\([0,\,\infty )\)  \(\log (1{{F}^{\alpha }}(x))\)  \(\frac{\alpha f(x){{F}^{\alpha 1}}(x)}{1{{F}^{\alpha }}(x)}r\{\log (1 {{F}^{\alpha }}(x))\}\) 
\([0,\,\infty )\)  \(\frac{{{F}^{\alpha }}(x)}{1{{F}^{\alpha }}(x)}\)  \(\frac{\alpha f(x){{F}^{\alpha 1}}(x)}{{{(1{{F}^{\alpha }}(x))}^{2}}}\,\,r\left\{ \frac{{{F}^{\alpha }}(x)}{1{{F}^{\alpha }}(x)} \right\} \) 
\((\infty ,\,\infty )\)  \(\log (\log (1F(x)))\)  \(\frac{f(x)\,r\{\log (\log (1F(x)))\}}{(F(x)1)\log (1F(x))}\) 
\((\infty ,\,\infty )\)  \(\log \left( \frac{F(x)}{1F(x)} \right) \)  \(\frac{f(x)}{F(x)(1F(x))}\,\,r\left\{ \log \left( \frac{F(x)}{1F(x)} \right) \right\} \) 
\((\infty ,\,\infty )\)  \(\log (\log (1{{F}^{\alpha }}(x)))\)  \(\frac{\alpha f(x){{F}^{\alpha 1}}(x)}{({{F}^{\alpha }}(x)1)\log (1{{F}^{\alpha }}(x))}r\{\log (\log (1{{F}^{\alpha }}(x)))\}\) 
\((\infty ,\,\infty )\)  \(\log \left( \frac{{{F}^{\alpha }}(x)}{1{{F}^{\alpha }}(x)} \right) \)  \(\frac{\alpha f(x)}{F(x)(1{{F}^{\alpha }}(x))}\,\,r\left\{ \log \left( \frac{{{F}^{\alpha }}(x)}{1{{F}^{\alpha }}(x)} \right) \right\} \) 
In the remainder of this article, we will focus on the case when \(T\) has the support \([0,\, \infty )\) and \(W(F(x))=\log (1F(x))\). For simplicity, we will use the name \(T\)\(X\) family of distributions for the new family of distributions in (2.5).
 (a)
The p.d.f. in (2.5) can be written as \(g(x)=h(x)r(H(x))\) and the corresponding c.d.f. is \(G(x)=R( \log (1F(x)))=R(H(x))\), where \(h(x)\) and \(H(x)\) are hazard and cumulative hazard functions of the random variable \(X\) with c.d.f. \(F(x)\). Hence, this family of distributions can be considered as a family of distributions arising from a weighted hazard function.
 (b)
The fact that \(G(x)=R(\log (1F(x)))\) gives the relationship between random variables \(X\) and \(T\text{: }\; X={{F}^{1}}(1{{e}^{T}})\). This provides an easy way for simulating random variable \(X\) by first simulating random variable \(T\) from p.d.f. \(r(t)\) and computing \(X={{F}^{1}}( 1{{e}^{T}})\), which has the c.d.f. \(G(x)\). Thus, \(E(X)\) can be obtained using \(E(X)=E\{ {{F}^{1}}( 1{{e}^{T}})\}\).
Theorem 1
Proof
Hence, \({{\eta }_{X}}=E\{ \log f( {{F}^{1}}( 1{{e}^{T}}))\}{{\mu }_{T}}+{{\eta }_{T}}\), which is the result in (2.7). \(\square \)
3 Some families of \(T\)\(X\) distributions with different \(T\) distributions
Families of generalized distributions derived from different \(T\) distributions
Name  The density \(r(t)\)  The density of the family \(g(x)\) 

Exponential  \(\theta {{e}^{\theta t}}\)  \(\theta f(x){{( 1F(x) )}^{\theta 1}}\) 
Betaexponential  \(\frac{\lambda {{e}^{\lambda \beta x}}{{(1{{e}^{\lambda x}})}^{\alpha 1}}}{B(\alpha ,\beta )}\)  \(\frac{\lambda f(x)}{B(\alpha ,\beta )}{{( 1F(x) )}^{\lambda \beta 1}}{{\{ 1{{( 1F(x) )}^{\lambda }}\}}^{\alpha 1}}\) 
Exponentiatedexponential  \(\frac{\alpha \lambda {{(1{{e}^{\lambda x}})}^{\alpha 1}}}{{{e}^{\lambda x}}}\)  \(\alpha \lambda f(x){{\{ 1{{(1F(x))}^{\lambda }} \}}^{\alpha 1}}{{(1F(x))}^{\lambda 1}}\) 
Gamma  \(\frac{1}{\Gamma (\alpha ){{\beta }^{\alpha }}}{{t}^{\alpha 1}}{{e}^{t/\beta }}\)  \(\frac{f(x)}{\Gamma (\alpha ){{\beta }^{\alpha }}}{{( \log ( 1F(x)))}^{\alpha 1}}{{(1F(x))}^{\frac{1}{\beta }1}}\) 
Half normal  \(\frac{1}{\sigma }{{(\frac{2}{\pi } )}^{1/2}}{{e}^{{{t}^{2}}/2{{\sigma }^{2}}}}\)  \(\frac{1}{\sigma }{{( \frac{2}{\pi } )}^{1/2}}\frac{f(x)}{1F(x)}\exp ({{\{ \log ( 1F(x))\}}^{2}}/2{{\sigma }^{2}})\) 
Levy  \({{( \frac{c}{2\pi } )}^{1/2}}\frac{{{e}^{c/2t}}}{{{t}^{3/2}}}\)  \({{( \frac{c}{2\pi })}^{1/2}}\frac{f(x)}{1F(x)}\frac{\exp ( c/2\log (1F(x)))}{{{\{ \log ( 1F(x) )\}}^{3/2}}}\) 
Log logistic  \(\frac{\beta {{( t/\alpha )}^{\beta 1}}}{\alpha {{\{ 1+{{( t/\alpha )}^{\beta }} \}}^{2}}}\)  \(\begin{array}{l} \frac{\beta }{{{\alpha }^{\beta }}}\frac{f(x)}{1F(x)}{{\{ \log (1F(x)) \}}^{\beta 1}} \\ \quad \times {{\{ 1+{{({\log ( 1F(x))}/{\alpha } )}^{\beta }}\}}^{2}} \end{array}\) 
Rayleigh  \(\frac{t}{{{\sigma }^{2}}}{{e}^{{{t}^{2}}/2{{\sigma }^{2}}}}\)  \(\frac{f(x)\log ( 1F(x))}{{{\sigma }^{2}}(1F(x))}\exp ({{\{ \log ( 1F(x))\}}^{2}}/2{{\sigma }^{2}})\) 
Type2 Gumbel  \(\alpha \beta {{t}^{\alpha 1}}{{e}^{\beta {{t}^{\alpha }}}}\)  \(\begin{array}{l}\frac{\alpha \beta f(x)}{1F(x)}{{\{ \log (1F(x)) \}}^{\alpha 1}} \\ \quad \times \exp ( \beta {{\{ \log (1F(x)) \}}^{\alpha }} ) \end{array}\) 
Lomax  \(\frac{\lambda k}{{{( 1+\lambda t )}^{k+1}}}\)  \(\frac{f(x)}{1F(x)}\frac{\lambda k}{{{\{ 1\lambda \log ( 1F(x) )\}}^{k+1}}}\) 
Inverted beta  \(\frac{{{t}^{\beta 1}}{{(1+t)}^{\beta \gamma }}}{B(\beta ,\;\gamma )}\)  \(\begin{array}{l} \frac{f(x)}{B(\beta ,\;\gamma )( 1F(x))}{{\{ \log ( 1F(x))\}}^{\beta 1}} \\ \quad \times {{\{ 1\log (1F(x))\}}^{\beta \gamma }} \end{array}\) 
Inverse Gaussian  \(\sqrt{\frac{\lambda }{2\pi {{t}^{3}}}}{{e}^{\frac{\lambda }{2{{\mu }^{2}}t}{{(t\mu )}^{2}}}}\)  \(\begin{array}{l} \frac{f(x)}{1F(x)}{{\left( \frac{\lambda }{2\pi {{\{ \log ( 1F(x))\}}^{3}}} \right) }^{1/2}} \\ \quad \times \exp \left( \frac{\lambda {{\{ \log ( 1F(x) )\mu \}}^{2}}}{2{{\mu }^{2}}\{ \log ( 1F(x) )\}} \right) \end{array}\) 
Weibull  \(\frac{c}{\gamma }{{( \frac{t}{\gamma } )}^{c1}}{{e}^{{{(t/\gamma )}^{c}}}}\)  \(\begin{array}{l} \frac{c}{\gamma }\frac{f(x)}{1F(x)}{{\{(1/\gamma )\log (1F(x)) \}}^{c1}} \\ \quad \times \exp ({{\{(1/\gamma )\log (1F(x)) \}}^{c}})\end{array}\) 
In each of the following subsections, we discuss the properties of the gamma\(X\) family, betaexponential\(X\) family, and Weibull\(X\) family.
3.1 Gamma\(X\) family
Lemma 1
The Shannon entropy of the gamma\(X\) family of distributions is given by \({{\eta }_{X}}=E\{ \log f({{F}^{1}}( 1{{e}^{T}}))\}+\alpha (1\beta )+\log \beta +\log \Gamma (\alpha )+(1\alpha )\psi (\alpha )\), where \(\psi \) is the digamma function.
Proof
It follows from Theorem 1 by using \({{\mu }_{T}}=\alpha \beta \) and the Shannon entropy for the gamma distribution, which is given by Song [36] as \({{\eta }_{T}}\!=\!\alpha \!+\!\log \beta \!+\!\log \Gamma (\alpha )\!+\!(1\alpha )\psi (\alpha )\). \(\square \)
From Fig. 2 and the corresponding data values (not included to save space), the Galton’s skewness is always positive which indicates that the gammaPareto distribution is right skewed. For fixed \(c\ge 1\), the Galton’s skewness is an increasing function of \(\alpha \). For fixed \(c < 1\), the Galton’s skewness is a decreasing function of \(\alpha \) and for fixed \(\alpha \), the Galton’s skewness is an increasing function of \(c\). The Moors’ kurtosis is an increasing function of \(\alpha \) and \(c\).
3.2 Betaexponential\(X\) family
Lemma 2
Proof
It follows from Theorem 1 by using the mean \({{\mu }_{T}}=[\psi (\alpha +\beta )\psi (\beta )]/\lambda \) and the Shannon entropy \({{\eta }_{T}}=\log ({{\lambda }^{1}}B(\alpha ,\beta ) )+(\alpha +\beta 1)\psi (\alpha +\beta )(\alpha 1)\psi (\alpha )\beta \psi (\beta )\) for the betaexponential distribution, which are given by Nadarajah and Kotz [30]. \(\square \)
 (1)
The betagenerated family in (1.10) is a special case of (3.4) when \(\lambda = 1\). Hence, the family of distributions in (3.4) can be used to generate all the distributions belonging to the betagenerated family.
 (2)
When \(\alpha =1\), the betaexponential\(X\) family reduces to the \(\text{ Exp }(1F)\) distributions. When \(\beta =1\) and \(\lambda =1\), the betaexponential\(X\) reduces to the \(\text{ Exp }(F)\) distributions.
 (3)
When \(\beta = 1\), (3.4) reduces to the exponentiatedexponential\(X\) family with p.d.f.
By using \(D(x)=1F(x)\) in (3.5), the exponentiatedexponential\(X\) family reduces to the \(KW\)\(G\) family.
From Fig. 4 and the corresponding data values (not included in order to save space), the exponentiatedexponentiallogistic distribution can be left skewed, right skewed, and symmetric. For fixed \(\lambda >1\), the Galton’s skewness is an increasing function of \(\alpha \), and for fixed \(\alpha \), the Galton’s skewness is a decreasing function of \(\lambda \). For fixed \(\alpha \), the Moors’ kurtosis is a decreasing function of \(\lambda \) when \(\lambda >1\), and for fixed \(\lambda \), the Moors’ kurtosis is a decreasing function of \(\alpha \) when \(\alpha >1\).
3.3 Weibull\(X\) family
Lemma 3
Proof
It follows from Theorem 1 by using the mean \({{\mu }_{T}}=\beta \,\Gamma (1+1/c)\) and the Shannon entropy \({{\eta }_{T}}=\gamma (11/c)\log (c/\beta )+1\) for the Weibull distribution, which is given by Song [36]. \(\square \)
Figure 6 and the corresponding data values (not included to save space) indicate that the Weibulllogistic distribution can be left skewed, right skewed, and symmetric. For fixed \(\beta \), the Galton’s skewness is a decreasing function of \(c\), and for fixed \(c\), the Galton’s skewness is an increasing function of \(\beta \). For fixed \(c\), the Moors’ kurtosis is an increasing function of \(\beta \) when \(c\le 1\) and a decreasing function of \(\beta \) when \(c>1\).
4 Summary and conclusion
A method to generate new families of distributions is introduced. This technique defines new family of distributions using the composite function \((R\mathbf \huge . W\mathbf \huge . F)(x)\) with \(R\) and \(F\) being the c.d.f.s of the random variables \(T\) and \(X\), respectively. The \(W(.)\) function is defined to link the support of \(T\) to the range of \(X\). This technique generates a large number of new distributions as well as existing distributions as special cases. Table 1 contains several different variants of \(T\)\(X\) families using different \(W(.)\) functions.
This article focuses on \(W(F(x))=\log ( 1F(x) )\), where the support of \(T\) is \([0,\,\infty )\). Some properties of this \(T\)\(X\) family are studied. Besides using functions of moments for measuring skewness and kurtosis, we suggest Galton’s measure of skewness and Moors’ measure of kurtosis. Three subfamilies of \(T\)\(X\) family, namely gamma\(X\) family, betaexponential\(X\) family and Weibull\(X\) family are discussed. These subfamilies demonstrate that the \(T\)\(X\) family consists of many subfamilies of distributions. Within each subfamily, one can define many new distributions as well as relate its members to many existing distributions.
Table 2 summarizes various subfamilies based on different \(T\) distributions with the same \(X\) distribution. New distributions discussed include gammaPareto, exponentiatedexponentiallogistic and Weibulllogistic distributions. In general, it is difficult to see how the shapes of the \(T\) and \(X\) distributions will affect the \(T\)\(X\) distribution. We believe that a relationship may exist for some specific \(T\) and \(X\) distributions. For the gamma distribution, \(\alpha \) is a shape parameter while \(\beta \) is a scale parameter. For the Pareto distribution, \(\theta \) is a scale parameter and \(k\) is a shape parameter. After forming the gammaPareto distribution, \(\theta \) remains a scale parameter, \(\beta /k = c\) becomes a shape parameter. The study of the properties, parameter estimation and applications of these new distributions are currently under investigation. For example, Alzaatreh et al. [2] defined and studied the gammaPareto distribution, a member of the gamma\(X\) family. Three real data sets were used to illustrate the applications of the gammaPareto distribution. The illustration showed that the gammaPareto distribution is a good model to fit data sets with various kinds of shapes.
The variants of \(T\)\(X\) families in Table 1 will define many potential new distributions that deserve further study. Some of these variants are currently under investigation. Future research for the \(T\)\(X\) family may include (i) the investigation of general properties of distributions generated using different \(W(.)\) functions, (ii) defining and investigating the properties of specific new distributions, (iii) studying new methods for estimating the parameters in addition to the wellknown moments and maximum likelihood (ML) methods, and (iv) applying these new distributions to fit different types of data sets. Based on our experience, the ML method may be challenging for more than three parameters. A better estimation method will be needed for distributions with four or more parameters.
During the recent decade, many new distributions developed in the literature seem to focus on more general and flexible distributions. Using the technique that generates the \(T\)\(X\) family, one can develop new distributions that may be very general and flexible or for fitting specific types of data distributions such as highly lefttailed (righttailed, thintailed, or heavytailed) distribution as well as bimodal distributions. There are only a few existing distributions that are known to be capable of fitting bimodal shapes. One of such distributions that have been successfully applied to fit real world data sets is the betanormal distribution by Eugene et al. [10] and Famoye et al. [11]. Our limited investigation in the \(T\)\(X\) family suggests that there are new distributions that can fit not only unimodal and bimodal, but also multimodal distributions.
This article focuses on the case when both \(T\) and \(X\) are continuous random variables. This technique can be extended to develop discrete \(T\)\(X\) family of distributions where \(T\) is continuous and \(X\) is discrete. Different considerations for the \(W(.)\) functions will be needed.
Notes
Acknowledgments
The authors are grateful for the comments and suggestions by the referees and the EditorinChief. Their comments and suggestions have greatly improved the paper.
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