Parameter induction in continuous univariate distributions : Well-established G families

The art of parameter(s) induction to the baseline distribution has received a great deal of attention in recent years. The induction of one or more additional shape parameter(s) to the baseline distribution makes the distribution more flexible especially for studying the tail properties. This parameter(s) induction also proved helpful in improving the goodness-of-fit of the proposed generalized family of distributions. There exist many generalized (or generated) G families of continuous univariate distributions since 1985. In this paper, the well-established and widely-accepted G families of distributions like the exponentiated family, Marshall-Olkin extended family, beta-generated family, McDonald-generalized family, Kumaraswamygeneralized family and exponentiated generalized family are discussed. We provide lists of contributed literature on these well-established G families of distributions. Some extended forms of the Marshall-Olkin extended family and Kumaraswamy-generalized family of distributions are proposed.


INTRODUCTION
There has been an increased interest in developing generalized (or generated) G families of distributions by introducing one or more additional shape parameter(s) to the base-line distribution.There is no doubt that the popularity and the use of Euler-beta and -gamma functions in some G families of distributions have attracted the attention of statis-ticians, mathematicians, scientists, engineers, economists, demographers and other applied researchers.One reason might be the computational and analytical facilities available in programming softwares like R (packages), ox5, Python, Matlab, Maple and Mathematica, through which researchers can easily tackle problems involved in computing incomplete-beta and -gamma functions in G families.The second reason is the tail properties of G distributions that can easily be explored by inducting one or more additional shape param-eter(s) to the baseline distribution.Thirdly, this parameter(s) induction has also proved to be helpful in improving the goodness-of-fit of the proposed G family of distributions.Fourthly, G families have the ability to fit skewed data better than existing The genesis of this family can be traced back to the first half of the nineteenth century when Gompertz (1825) and Verhulst (1838Verhulst ( , 1845Verhulst ( , 1847) ) used the cumulative distribution function (cdf) G(t) = (1 − ρ e −λt ) α for t > λ −1 log ρ, where ρ, α and λ are positive real numbers.Ahuja and Nash (1967) introduced the generalized Gompertz-Verhulst family of distributions to study growth curve mortality.Gompertz-Verhulst's cdf was the first member of the EF of distributions.The exponentiated exponential (EE) distribution is its particular case PARAMETER INDUCTION IN DISTRIBUTIONS for ρ = 1.The properties and estimation methods for parameters of the EF of distributions have been studied by many authors, see Mudholkar and Srivastava (1993), Mudholkar and Hutson (1996), Mudholkar et al. (1995), Gupta and Kundu (1999, 2001a, b, 2007), Pal et al. (2006), Nadarajah and Kotz (2006a), Nadarajah (2011) and Nadarajah et al. (2013b).The EF of distributions is also known as Lehmann alternatives (LAs) (Lehmann 1953) or proportional reversed hazard rate model (PHRM) (see Gupta et al. 1998, Gupta and Gupta 2007, Martínez-Florez et al. 2013), while other authors referred to the EF of distributions as max-stable family (Sarabia and Castillo 2005) and F α -distributions (Gupta et al. 1998, Al-Hussaini, 2010a, b, 2012, Shakil and Ahsanullah 2012, Hamedani 2013and Ghitany et al. 2013).
In literature there exist four different ways for obtaining the EF of distributions.
LEHMANN ALTERNATIVE 1 (LA1) The method of Lehmann alternative 1 (LA1) (due to Lehmann (1953)) has received a great deal of attention in developing the EF of distributions.
If G(z) is the cdf of the baseline distribution, then an EF of distributions is defined by taking the αth-power of G(z) as where α > 0 is a positive real parameter.The variable z can take any of the form z = x or z = x − µ or z = δ .The probability density function (pdf) corresponding to (2.1) is where g(z) = dG(z)/dz denotes the pdf of G.For any lifetime random variable t, the survival (reliability) function (sf), F (t), the hazard (failure) rate function (hrf), h(t), the reversed hazard rate function (rhrf), r(t), and the cumulative hazard rate function (chrf), H(t), associated with (2.1) and (2.2) are and The method of Lehmann alternative 2 (LA2) (due to Lehmann (1953)) has received less attention.
is the sf of the baseline distribution, then an EF of distributions is defined by taking one minus the αth-power of G(z) as where α is a positive real parameter.The LA2 cdf may also be written as (2. 3) The pdf corresponding to (2.3) is (2.4) MUHAMMAD H. TAHIR and SARALEES NADARAJAH For any lifetime random variable t, the sf, hrf, rhrf and chrf associated with (2.3) and (2.4) are Nadarajah and Kotz (2003Kotz ( , 2006a)), Nadarajah (2006) and Rao et al. (2013) used the LA2 approach for introducing exponentiated Fréchet, exponentiated Gumbel and exponen-tiated log-logistic distributions.For more applications of the LA2 approach, the reader is referred to Abd-Elfattah and Omima (2009), Abd-Elfattah et al. (2010), Rao et al. (2012, 2013), and Al-Nasser and Al-Omari (2013).
USING TRANSFORMATION z = log(x), x > 0 Nadarajah (2005a) developed exponentiated distributions by applying the transformation z = log(x) to (2.3).The cdf, pdf and the hrf of the exponentiated distribution are Nadarajah (2005b) developed exponentiated distributions by applying the transformation z = − log(x) to (2.3).The cdf, pdf and the hrf of the exponentiated distribution are A list of papers on the EF of distributions is presented in Table I.

MARSHALL-OLKIN EXTENDED (MOE) FAMILY OF DISTRIBUTIONS
Marshall and Olkin (1997) proposed a flexible semi-parametric family of distributions and defined a new sf F MO (x) by introducing an additional parameter α > 0. Marshall and Olkin (1997) called α a tilt parameter and interpreted α in terms of the behavior of the hrfs of F MO and G. Their ratio is increasing in t for α ≥ 1 and decreasing in t for 0 < α < 1. Nanda and Das (2012) reinterpreted α as a tilt parameter since the hrf of the new family is shifted below (α ≥ 1) or above (0 < α ≤ 1) the hrf of the underlying distribution.
Specifically, for all t ≥ 0, h MO (t) ≤ h(t) when α ≥ 1, and h MO (t) ≥ h(t) when 0 < α ≤ 1, where h MO (t) and h(t) are the hrfs of the MOE and baseline distributions.
For any baseline pdf g(t), cdf G(t) = P (T ≤ t) and sf G(t) = P (T > t) of the baseline distribution, the sf F MO (t) of the MOE family of distributions is defined by where −∞ < t < ∞, α > 0 and a = 1 − α.The cdf and pdf associated with (3.1) are and where −∞ < t < ∞, α > 0 and a = 1 − α.If α = 1, then we have F MO (t) = G(t).Other reliability measures like the hrf, rhrf and chrf associated with (3.1) are or ah(t) a + aG (t) , and where h(t) is the hrf of the baseline distribution.
Note that if we define For more general results on the MOE family of distributions, the reader is referred to Barakat et al. (2009), Jose (2011), Krishna (2011), Barreto-Souza et al. (2013) and Cordeiro et al. (2014c).

EXISTING GENERALIZED MOE FAMILY OF DISTRIBUTIONS
In this section, we describe existing generalized Marshall-Olkin families of distributions.
The cdf and the pdf associated with (3.2) are Other reliability measures like the hrf, rhrf and chrf associated with (3.2) are where h(t) is the hrf of the baseline distribution.MUHAMMAD H. TAHIR and SARALEES NADARAJAH

A NEW GENERALIZED MOE FAMILY OF DISTRIBUTIONS
Here, we propose another generalization of the Marshall and Olkin (1997) family of distributions.
Using the LA2 approach to the sf of the MOE family of distributions, we obtain where −∞ < t < ∞, α > 0, and θ > 0 is the additional shape parameter.When θ = 1, F G2MO (t) = F MO (t).
The cdf and the pdf associated with (3.3) are and ) After simplification, the above pdf can be rewritten as Other reliability measures like the hrf, rhrf and chrf associated with (3.3) are and where r(t) is the rhrf of the baseline distribution.
The construction in (3.3) is similar to that due to Jayakumar and Mathew (2008).But there is an important distinction.Suppose that a system consists of θ independent components.Suppose too that each component has a lifetime with the sf given by αG(t) / [ 1 − aG(t)].Then (3.2) is the sf of the minimum of the lifetimes and (3.3) is the sf of the maximum of the lifetimes.So, (3.2) can be used to model the minimum of the lifetimes and (3.3) can be used to model the maximum of the lifetimes.

SEMI-TYPE PROCESSES BASED ON CHARACTERISTIC FUNCTION
In this section, we briefly discuss semi-Pareto, semi-Burr, semi-Laplace, semi-logistic and semi-Weibull distributions based on the characteristic function (cf) ψ(t) of the baseline distribution.The concept of semitype distributions arose from the minification process.Tavares (1980) defined a minification process as observations in a process generated by where n ≥ 1, k > 1 is a constant and {² n } is an innovation process of independent and identically distributed random variables.Here, {X n } is called the first order autoregressive AR(1) minification process.There exists many modified minification processes.PARAMETER INDUCTION IN DISTRIBUTIONS Linnik (1963) introduced the α-Laplace distribution, a symmetric distribution defined on (−∞, ∞).For α = 2, the Linnik distribution reduces to the Laplace distribution.Pillai (1985) generalized the Linnik distribution and introduced the semi-α-Laplace distribution.Yeh et al. (1988) modified (3.4) and introduced the first auto-regressive Pareto minification process having Pareto marginals.Arnold and Robertson (1989) introduced minification processes with logistic marginals.Pillai (1991) and Pillai et al. (1995) introduced semi-Pareto minification processes.Balakrishna (1998) investigated some properties and estimated the unknown parameters of Pillai's semi-Pareto minification process.Pillai (1985) proposed the semi-α Laplace distribution.Its sf is , where ψ(t) satisfies the functional equation where α > 0 and 0 < p < 1.The solution of (3.5) is ψ(t) = |t| α η(t), where η(t) is periodic in log |t|.In the particular case η(t) = c, the semi-α-Laplace distribution reduces to the Linnik distribution.
A random variable T is said to have the semi-Pareto distribution if its sf is , where t > 0 and ψ(t) satisfies the functional equation where 0 < p < 1, t > 0 and ° > 0. The solution of (3.6) is ψ(t) = t ° η(t), where η(t) is periodic in log t with period ´.
A random variable T is said to have the semi-Burr distribution if its sf is where t > 0, β > 0 and ψ(t) satisfies the same functional as (3.6).Cifarelli et al. (2010) expressed the sf of the semi-Burr distribution as where ψ(t) satisfies the same functional as (3.6) and b > 0.
According to Arnold (1992) and Jayakumar and Mathew (2005), a random variable T is said to have the semi-logistic distribution if its sf is MUHAMMAD H. TAHIR and SARALEES NADARAJAH , where ψ(t) is a nondecreasing and right-continuous function satisfying where 0 < p < 1, t > 0, and σ > 0.
According to Jose (1994) and Thomas and Jose (2005), a random variable T is said to have the semi-Weibull distribution if its sf is where ψ(t) satisfies the functional equation ´, where ° > 0 and 0 < p < 1.Note that (3.8) yields the iterative solution ´. ´.
If 0 < α < 1 and φ(t) is a valid cf then is also a valid cf.Using this fact, Krishna and Jose (2011) defined the Marshall-Olkin generalized asymmetric Laplace distribution as that having the cf George and George (2013) defined the Marshall-Olkin Esscher transformed Laplace distribution as that having the cf Jose and Uma ( 2009) defined the Marshall-Olkin Linnik and Mittag-Leffler distributions as those having the cfs respectively, where ν > 0, 0 < α ≤ 2, and β > 0.
A list of papers on the MOE family is presented in Table II.Consider the cdf of a beta random variable of type 1 with two shape parameters a and b given by (4.1) is the beta function.

PARAMETER INDUCTION IN DISTRIBUTIONS
Similarly, the cdf of a beta random variable of type 2 with parameters a and b is is the beta function.
The pdf corresponding to (4.2) is where a > 0, b > 0, and y > 0. The beta type 2 distribution is also known as inverted beta distribution as it can be obtained from (4.1) by the transformation Y = The cdf of a beta random variable of type 3 with parameters a and b is where a > 0, b > 0, and z 2 (0, 1).Eugene et al. (2002) and Jones (2004a) replaced the upper limit x of the integral in (4.1) with G(x).The resulting cdf of beta G family of distributions is (4.4) The pdf corresponding to (4.4) is where g(x) = dG(x)/dx denotes the pdf.The beta G family of distributions is also known as the beta logit family.For any lifetime random variable t, the sf, hrf, rhrf and chrf associated with (4.4) and (4.5) are and A list of papers on the beta G family of distributions is given in Table III.
where a > 0, b > 0 and c > 0 are the three shape parameters.The Mc distribution includes as special cases the beta type 1 distribution (c = 1) and the Kumaraswamy distribution (a = 1).The pdf corresponding to (5.1) is where 0 < x < 1.

EXISTING MCDONALD G FAMILY OF DISTRIBUTIONS
For any baseline cdf G(x), Alexander et al. (2012) replaced the upper limit x c of the integral in (5.1) with G(x) c .Lemonte and Cordeiro (2013) stated that this simple transformation facilitates the computation of several properties of the G family of distributions.
where I G(x) c (a, b) denotes the incomplete beta function ratio.The pdf corresponding to (5.2) is where a > 0, b > 0 and c > 0 are the three shape parameters.For a lifetime random variable t, the sf, hrf, rhrf and chrf associated with (5.2) and ( 5.3) are and The three shape parameters a, b and c introduce skewness, kurtosis, and vary tail weights.The parameters control skewness and kurtosis through altering the tail entropy (Alexander et al. 2012).They also control skewness and kurtosis through adding entropy to the center of the baseline distribution (Alexander et al. 2012).Cordeiro et al. (2014b) mentioned that a and b are skewness parameters that control relative tail weights but not the peak, but c provides the control over the peak.Alexander et al. (2012), Marciano et al. (2012), Cordeiro andLemonte (2012, 2014), Cordeiro et al. (2012aCordeiro et al. ( , b, 2013dCordeiro et al. ( , 2014b)), Lemonte and Cordeiro (2013)  A list of papers on the Mc G family of distributions is given in Table IV.

PARAMETER INDUCTION IN DISTRIBUTIONS
KUMARASWAMY DISTRIBUTIONS AND KUMARASWAMY G FAMILIES OF DISTRIBUTIONS Kumaraswamy (1980) argued that the beta distribution does not fairly fit hydrological random variables like rainfall, daily stream flow, etc. Jones (2009) commented that "beta distribution is fairly tractable, but in some ways not fabulously so.In particular its distribution function is an incomplete beta function ratio and its quantile function the inverse thereof".The Kumaraswamy (Kw) distribution is relatively much appreciated in comparison to the beta distribution, and has a simple form which can be unimodal, increasing, decreasing or constant, depending on the parameter values.
In this section, we give functional forms of Kw distributions.We also propose Kumaraswamy generalized families of distributions.

EXISTING KUMARASWAMY DISTRIBUTIONS
The Kw distribution has the cdf and the pdf specified by and respectively, where 0 < x < 1 and a > 0, b > 0 are both shape parameters.

EXISTING KUMARASWAMY G FAMILY OF DISTRIBUTIONS
For a baseline cdf G(x) with pdf g(x), Cordeiro and de Castro (2011) defined the Kw G distribution specified by the cdf and the pdf   (6.4) where x > 0, g(x) = dG(x) = dx and a > 0, b > 0 are shape parameters in addition to those in the baseline distribution.They partly govern skewness and vary tail weights.For a lifetime random variable t, the sf, hrf, rhrf and chrf associated with (6.3) and (6.4) are and NOTES ON KUMARASWAMY G FAMILIES OF DISTRIBUTIONS Equations ( 6.3) and (6.4) do not involve any special function like the beta function, incomplete beta function, incomplete beta ratio, gamma function, incomplete gamma func-tion or the incomplete gamma ratio.Therefore, the generalization in (6.3) and (6.4) is computationally more efficient compared to beta G and Mc G families of distributions.
The Kw G families of distributions are more flexible than the baseline distribution in the sense that the families allow for greater flexibility of tail properties.Their second benefit is their ability to fit skew data that cannot be properly fitted by existing distributions.
NOTES ON KUMARASWAMY G FAMILIES OF DISTRIBUTIONS Equations (6.3) and (6.4) do not involve any special function like the beta function, incomplete beta function, incomplete beta ratio, gamma function, incomplete gamma func-tion or the incomplete gamma ratio.Therefore, the generalization in (6.3) and (6.4) is computationally more efficient compared to beta G and Mc G families of distributions.
The Kw G families of distributions are more flexible than the baseline distribution in the sense that the families allow for greater flexibility of tail properties.Their second benefit is their ability to fit skew data that cannot be properly fitted by existing distributions.
A list of papers on the Kw G family of distributions is given in Table V.  Setting X = − Y in (6.1) and ( 6.2), we obtain a distribution specified by the cdf and the pdf where 0 < x < 1 and a > 0, b > 0 are the shape parameters.
OTHER KW G FAMILIES OF DISTRIBUTIONS Replacing x with G(x) in (6.5), we obtain a Kw G distribution specified by the cdf where a > and b > 0 are both shape parameters.The pdf corresponding to (6.7) is Equations (6.6) and (6.7) are the cdf and the pdf of the Exp G family of distributions recently proposed by Cordeiro et al. (2013e).For a lifetime random variable t, the sf, hrf, rhrf and chrf associated with (6.6) and (6.7) are

CONCLUSIONS
We first refer to some important surveys on the developments of continuous univariate distributions: Kotz and Vicari (2005) surveyed the developments in the theory of skewed continuous distributions; Gupta and Kundu (2009) described six different methods for the induction of shape and/or skewness parameter(s) in univariate probability distributions; Chakraborty and Hazarika (2011) surveyed the theoretical developments of the univariate skew-normal distribution, its extensions and generalizations; Lee et al. (2013) surveyed recent methods for generating families of univariate continuous distributions.They discussed five general methods for gen erating G families of distributions: (1) method for generating skewed distributions, (2) method for adding parameters (e.g., exponentiation), (3) beta G, (4) transformed-transformer (T-X) family, and (5) composite method.Recently, Nadarajah (2015a, 2015b) introduced the R package Newdistns which computes the pdf, cdf, quantiles and random numbers for nineteen general families of distributions.
In this paper, we have discussed the well-established and widely used G families of distributions: the EF of distributions, the MOE distributions, the beta G distributions, the Mc G distributions, the Kw G distributions and the Exp G distributions.We have provided exhaustive lists of papers on these families of distributions.We have cited 28 papers on the EF of distributions, 28 papers on the MOE distributions, 45 papers on the beta G distributions, 16 papers on the Mc G distributions, 21 papers on the Kw G distributions and 2 papers on the Exp G distributions.The literature review in Lee et al. (2013) appears less detailed.
We have introduced several new families of distributions relating to the MOE distribu-tions and the Kw G distributions.Of course, this is not an attempt to increase the frequency of articles on new families of distributions but rather to effectively explore real life phenom-ena through data sets available from different fields.We have noted that contributors (practitioners) have used different model selection criteria: the maximized log-likelihood ℓ (θ b ), the Akaike Information Criterion (AIC), the Bayesian Information Criterion (BIC), the Consistent Akaike Information Criterion (CAIC), the Hannan-Quinn Information Cri-terion (HQIC), the Cramer-von-Mises (W*), the Anderson-Darling (A*), the Wald (W ) statistic, the Kolmogorov-Smirnov (K-S) test and graphical inspection of the proximity of histograms to the fitted pdfs.
Tractability and effectiveness for modeling censored data require, among other things, closed form expressions for the cdf.So, the Kw G distributions can be tractable and effective models for censored data.The EF and MOE distributions can also be tractable and effective models for censored data, provided G is in closed form.However, beta G and Mc G distributions may not be tractable or effective models for censored data since their cdfs involve the incomplete beta function.
It is very appreciating that the contributors have expanded the horizon of applications with efficient statistical modeling.In this regard, the acknowledgements and appreciation go to Professors M. C.Jones, Narayanaswamy Balakrishnan, Kostas Zografos, Felix Famoye, Carl M. -S.Lee, Ramesh C. Gupta, Arjun Kumar Gupta, Rameshwar D. Gupta, Debasis Kundu, Mohamad E. Ghitany, and K. K. Jose.Special acknowledgements and apprecia-tion go to the Brazilian Statisticians Group headed by Professor Gauss M. Cordeiro for introducing the Mc G, Kw G, Exp G, beta extended G, Weibull G families and exploring their properties.We note that 58 of the listed papers in the References section belong to Professor Cordeiro.PARAMETER INDUCTION IN DISTRIBUTIONS sending PDF's of their published work.Both authors would also like to thank the Editor, the Associate Editor and the three referees for carefully reading and for providing comments which greatly improved the paper.

F
SEMI-TYPE MARSHALL-OLKIN DISTRIBUTIONS BASED ON CHARACTERISTIC FUNCTION Using (3.1), various authors have proposed Marshall-Olkin semi-type distributions from the baseline cf ψ(t).Alice and Jose (2003) introduced the Marshall-Olkin semi-Pareto (MOSP) distribution with sf extreme stability.Thomas and Jose (2005) and Alice and Jose (2005b) introduced the Marshall-Olkin semi-Weibull distribution with sf
and Gomes et al. (2013a) used Mc G distributions for developing McDonald normal, McDonald (extended) exponential, McDonald gamma, McDonald inverted beta, McDonald arcsine, McDonald Weibull, McDonald Birnbaum-Sanders (fatigue life), McDonald Lomax, McDonald Burr XII and McDonald Burr III distributions.These authors believe that the Mc G family of distributions can fit skew data better than existing distributions.The Mc G family of distributions is most applicable when G(x) and g(x) take simple analytical forms.The Mc G family of distributions reduces to the beta G family of distribution for c = 1 and to the Kw G family of distribution for a = c.Further, the Mc G family of distributions for G(x) = x contains as particular cases the beta type 1 distribution (c = 1) and the Kumaraswamy distribution (a = c).Zografos (2011) studied a family of distributions based on McDonald and Xu (1995)'s generalized beta distribution.This family was called the family of generalized beta generated (GBG) distributions.

TABLE III Contributed work on the beta G family of distributions.
MCDONALD DISTRIBUTIONS AND MCDONALD G FAMILIES OF DISTRIBUTIONSMCDONALD TYPE DISTRIBUTIONS McDonald (1984) replaced the upper limit x of the integral in (4.1) with x c , where c is an additional (third) shape parameter.The resulting cdf of the McDonald type (Mc) distribution is (5.1)

TABLE IV Contributed work on the Mc G family of distributions
.MUHAMMAD H. TAHIR and SARALEES NADARAJAH