Recent Developments in Distribution Theory: A Brief Survey and Some New Generalized Classes of distributions

The generalization of the classical distributions is an old practice and has been considered as precious as many other practical problems in statistics. These generalizations started with the introduction of the additional location, scale or shape parameters. In the last couple of years, this branch of statistics has received a great deal of attention and quite a few new generalized classes of distributions have been introduced. We present a brief survey of this branch and introduce several new families as well. Keyword: Generalized classes of distributions; Exponentiated family; Marshall Olkin family; Transmuted family; Kumaraswamy family; Alpha power transformation; ZubairG family; Construction of new families.


1.
Introduction The recent development in distribution theory stresses on problem solving faced by the researchers and proposes a variety of models so that lifetime data sets can be better assessed and investigated in different applied areas. In other words, there is a need to introduce useful models for the better exploration of the real phenomenon of nature. Nowadays, the trends and practices in proposing new probability models totally differ in comparison to the models suggested before 1997. One main objective for proposing, extending or generalizing (models or their classes) is to explain how the lifetime phenomenon arises in fields like physics, computer science, insurance, public health, medical, engineering, biology, industry, communications, life-testing and many others. The well-known and fundamental distributions such as exponential, Rayleigh, Weibull and gamma are very limited in their characteristics and are unable to show wide flexibility. For example, the exponential distribution is capable of modeling with constant hazard function, whereas, the Rayleigh distribution has increasing hazard function only. However, the Weibull is much flexible and capable of modeling with increasing, decreasing or constant hazard function.
Unfortunately, the Weibull model is not capable of modeling with non-monotonic (such as unimodal, modified unimodal or bathtub shaped) failure rate function. The gamma distribution does not have a closed form of cumulative distribution function (cdf) which causes difficulties in describing its mathematical properties. For complex phenomenon in human mortality studies, reliability studies, lifetime testing, engineering modeling, electronic sciences and biological surveys, the failure rate behavior can be bathtub, upsidedown bathtub and other shaped but not usually monotone increasing or decreasing. Thus, in order to cope with both monotonic and non-monotonic failure rate shapes, researchers have proposed several generalized classes of distributions which are very flexible to study needful properties of the model and its fitness. In the last two decades, several generalization approaches were adopted and practiced, which have received increased attention.
The objectives of the present study are three-fold: Firstly, we present an up-to-date account of the extended classes of distributions for the readers of modern distribution theory. Secondly, this survey will motivate the researchers to fill up the gap and to furnish their work in remaining applied areas. Thirdly, we propose some new classes of distributions which might be helpful as a tutorial to the beginners of the generalized modeling art.
The rest of the article is organized as follows. In Section 2, some extended classes of distributions are reviewed. In section 3, we present some new families. Section 4 presents certain characterizations of the distributions listed in Section 3. Finally, concluding remarks are provided in Section 5.

2.
Review of the existing family of distributions In this section, we present up-to-date review of the extended families of distributions. Mudholkar and Srivastava (1993) proposed another method of introducing an extra parameter to a two-parameter Weibull distribution. The cumulative distribution function of the Mudholkar and Srivastava (1993)'s proposed exponentiated family has the following form

The exponentiated family of distributions
where 0   is an extra shape parameter. Due to the presence of an extra shape parameter, the proposed exponentiated distributions are more flexible than the traditional models. Using (1), a number of modifications of the existing distributions have been proposed in the literature. A brief list of these modifications is presented in Table 1:  where,  = − . Clearly, for  =, we obtain the baseline distribution, i.e., ( ) ( ) Using (2), the extended versions of the existing distributions have been proposed. Based on the MO family, a detail review of the existing distributions is provided in Table 2:

Transmuted family of distributions
Shaw and Buckley (2009) pioneered another prominent method of adding a parameter into a family of distributions and several authors used their method to extend well-known distributions in the last couple of years. If ( ) ; Fx  denotes the cdf of a parent distribution depending on the vector parameter  , then the cdf of the transmuted family is given by From (3), for 0  = , we obtain the baseline distribution, i.e., Using (3), the extended versions of the existing distributions have been proposed, for detail we refer to Tahir and Cordeiro (2016).

Cubic Transmuted family of distributions
Recently, Aslam et al. (2018) proposed Cubic transmuted-G family by using the T-X idea of Alzaatreh (2013).
The general transmuted family reduces to the base distribution for 0 i  = for i = 1;2; · · · ; k. Kumaraswamy (1980) (for short Ku) proposed a two-parameter distribution on (0,1), called Kumaraswamy distribution, is defined by

Kumaraswamy-G family of distributions
where and  are shape parameters. The density function corresponding to (6) is The Ku density has the same basic shape properties as to the beta distribution: Using (8), a number of modifications of the existing distributions have been proposed in the literature. A brief list of these modifications is presented in Table 3: Table 3: Contributed work on Ku-G distributions.  .
Using (10), some new extensions of the parent distributions have been introduced. A list of distributions based on alpha power transformation is provided in Table 5. Table 5: Contributed work on alpha power transformation.

The Zubair-G family
Using (11), some new modified versions of the parent distributions have been proposed. A list of distributions based on the Zubair-G method is provided in Table 6.  ; R x  is a baseline cdf. We can also take ( ) ; R x  as any function of cdf, which obey the properties of cdf, or we may combine two or more distribution functions to propose a new class of distributions. For the sake of simplicity we omit the dependency on the vector parameter and we simply write ( ) ( ) we can also define a new function as  (13), we arrive at the Zubair-G distribution.  Table 7, we define ( ) R x for the sub-models of the EZ-G class of distributions.

The Cosine-X family of distributions
in (9), we define the cosine-X family as The pdf corresponding to (16), is given by

The Cosine exponentiated-X family of distributions
A random variable X is said to follow the Cosine exponentiated-X distribution if its cdf is given by

The extended Cosine-X family of distributions
A random variable X is said to follow the extended Cosine-X (for short 'EC-X') if its cdf is given by

The extended Cosine exponentiated-X family of distributions
A random variable X is said to follow the extended cosine exponentiated-X (for short 'ECE-X') distribution, if its cdf is given by

3.6.
Another extended Cosine-X family of distributions (14), we define the cdf of the another extended cosine-X (for short 'AEC-X') family as The pdf of the AEC-X family can easily be obtained by simply differentiating (24).

3.8.
The The pdf of the APTC-X can easily be obtained by simply differentiating (29). 3.11. The alpha power transformed Cosine exponentiated-X family A random variable X is said to have the alpha power transformed cosine exponentiated-X (for short 'APTCE-X') family, if its cdf is given by  (28), we introduce the extended alpha power transformed-

The extended alpha power transformed-X family
By differentiating (33), we get the density function of the EAPT-X family.

Characterization Results
In designing a stochastic model for a particular modeling problem, an investigator will be vitally interested to know if their model fits the requirements of a specific underlying probability distribution. To this end, the investigator will rely on the characterizations of the selected distribution. Thus, the problem of characterizing a distribution is an important problem in various fields and has recently attracted the attention of many researchers. Consequently, various characterization results have been reported in the literature. These characterizations have been established in different directions. This section deals with various characterizations of 12 proposed distributions listed in Section 3. These characterizations are based on a simple relationship between two truncated moments. It should be mentioned that one important advantage of our characterization is that the cdf need not have a closed, and moreover, it depends on the solution of a first order differential equation, which provides a bridge between probability and differential equation. In the subsection 4.1 we provide the characterizations of the Extended Zubair-G (EZ-G) family of distributions. Similar characterizations can be stated for the other 11 distributions.

Characterizations based on two truncated moments
This subsection deals with the characterizations of the EZ-G distribution based on the ratio of two truncated moments. Our first characterization employs a theorem of Glänzel (1987) Now, in view of Theorem 1, X has density (12).  , , q x q x x  satisfying the conditions of Theorem 1.

Concluding Remarks
The need of compounding and generalizing distributions were first felt in the financial and actuarial science and later in many other fields which researchers adopted this approach for lifetime and reliability modeling. In this way, the possible available compound and generalized G-classes are surveyed and using these basic principles nearly 12 new classes are proposed. The goal of providing a variety of new class classes is to test the flexibility of the proposed models to cope with the data available in complex situations. The parameters inducted in this way might be helpful in describing the phenomenon generated from real-lifetime data sets. We expect that these distributions will be an addition to the art of constructing useful probability models. One can imagine its motivation and usefulness in the fields which are not touched so far. Lastly, we offer more choices to the learners and practitioners of modeling to compare different models and to illustrate usefulness of old and new classes of distributions.