Transmuted Probability Distributions: A Review

Transmutation is the functional composition of the cumulative distribution function (cdf ) of one distribution with the inverse cumulative distribution function (quantile function) of another. Shaw and Buckley(2007), first apply this concept and introduced quadratic transmuted family of distributions. In this article, we have presented a review about the transmuted families of distributions. We have also listed the transmuted distributions, available in the literature along with some concluding remarks.


Background
The probability distributions are being used in many areas of life. The application of a statistical tools depends upon the underlying probability model of the data. As a result huge numbers of probability distributions are being developed by the researchers. However, there still remains large numbers of practical problems that does not follow any of the standard probability distributions. So, developing the new form of the probability distributions is very much common in statistical theory.
Generalizing probability distributions are very common practice in the theory of statistics. In order to generalize probability distributions various methods are proposed in literature which add extra parameter(s) to an existing baseline probability models so that the new model will increase the flexibility of the models to capture the complexity of the data. Several generalized (or G) classes are available in the literature, but our main focus in this paper is to present a review of transmuted-G class of distributions that increase the flexibility of the distribution along with the ability to explore its tail properties and to improve goodness-of-fits.
We first given describe some classical and recent families of distributions in the following.
Transmuted Probability Distributions: A Review

Classical Pearson and Burr Families
The normal distribution is not suitable for handling all practical problems. So generalization of distributions is very much essential for solving the practical problems that are not well fitted to normal distribution. The first approach for generalization of distributions was due to Karl Pearson and the second approach was due to Burr which are discussed in the following two subsections.

The Pearson Family of Distributions
Karl Pearson (1857Pearson ( -1936) developed a family of continuous probability distributions and named it as Pearson distribution. He has listed his four types of distributions; Type-I to Type-IV; in (Pearson, 1895) and has listed normal distribution as Type-V distribution. In an another article, (Pearson, 1901), he has redefined Type-V distribution and has proposed Type-VI distribution. In yet another article, (Pearson, 1916), he has further developed special cases and subtypes VII through XII. More details of the Pearson family of distributions, see Johnson et al.(1994Johnson et al.( , 1995. The Pearson probability density function (pdf ) y = f (x) is defined to be any valid solution to the differential equation (see Pearson,1895), as follows with: , and The Criterion k: The Pearson family of distributions is obtained by solving the differential Equation (1), which can also be written as where c is the constant of integration and I = (x−a) c0+c1x+c2x 2 dx.

Criterion k
Corresponding Frequency Curve Hence, y = f (x) depends on the roots of the equation c 0 + c 1 x + c 2 x 2 , which can be further expressed as 4c2c0 and is used as a criterion for obtaining the form of pdf . Basically the Pearson family of distributions is made up of seven distributions (Type I to VII), which are given in Table 1 on the basis of criterion k.

The Burr Family of Distributions
Burr(1942), introduced twelve cumulative frequency functions that could fit the real-life datasets. Most of these distributions are unimodal. The Burr distribution is very similar (and is, in some cases, the same) as many other probability distributions. The cdf , F (y), of various Burr distributions are given in Table 2 for k, r, c ∈ R + . More details on Burr distributions can be seen in Johnson et al.(1994Johnson et al.( , 1995; Tadikamalla and Pandu(1980) among others.

Cumulative Distribution Function
Corresponding Type The location and scale parameters µ (location) and σ (scale) can be easily introduced in all members of Burr family of distributions, given in Table 2, by using the transformation y = x−µ σ . Burr(1942), explored the Type XII distribution in details and the others were studied later.

Some Recent Families
Several families of distributions are available in the literature which extend a probability distributions by adding new parameter(s). Some of these families of distributions are discussed in the following.
Transmuted Probability Distributions: A Review

The Marshall-Olkin Family
Marshall and Olkin(1997) introduced an interesting method to add a new parameter to an existing model. They applied their method to extend exponential and Weibull distributions. The survival function of this family is is the survival function of the baseline probability distribution.

The Beta-G Family
Eugene et al. (2002) introduced a generalized family of distributions by using logit of the beta random variable. The proposed family is denoted by Beta − G and has the cdf where α ∈ R + , β ∈ R + and B(α, β) is the Beta function.

The Kumaraswamy-G Family
Kumaraswamy-G (K w − G) family of distributions is proposed by Cordeiro and de Castro(2011), by using Kumaraswamy (1980) distribution, and has the cdf where α, β ∈ R + are the shape parameters.

The T − X Family
An interesting method for obtaining families of continuous distributions has been introduced by Alzaatreh et al.(2013). In this technique "the transformer" random variable X is used to transform another "the transformed" random variable T . The resulting family is known as T − X family of distributions and is defined below.
Let X be a random variable with the pdf g(x) and cdf G(x). Also let T be a continuous random variable with the pdf r(t), defined on [a, b]. The cdf of a new family of distributions is defined by where W (G(x)) satisfies following conditions (2) is a composite function of (R · W · G)(x) and can be written as The layout plan of this paper follows: The transmuted family of distributions is described in Section 2, along with the expansion of transmuted distributions in Section 3. Section 4 provides the evolution related to cubic and quartic transmuted distributions. In Section 5, we have presented the recent development of general transmuted distributions. Finally, some concluding remarks are given in Section 6.

Transmuted Family of Distributions
Shaw and Buckley (2007), introduced an interesting method of adding new parameter to an existing distribution for solving the problems related to financial mathematics and named the family as quadratic transmuted family (QT-G, for short) of distributions. The cdf of the family has following simple quadratic form where λ ∈ [−1, 1] and G(x) is the cdf of the baseline distribution.  Table 3 below. The literature on the transmuted family of distributions is enrich enough and is also rapidly improving. We now describe cubic and quartic transmuted families of distributions in the following.

Developments in Quadratic Transmuted Distributions
Transmuted Probability Distributions: A Review

Developments in Cubic and Quartic Transmuted Distributions
The quadratic transmuted distributions, discussed above, capture the complexity of unimodal data but the real-life data can be more complex (multi-modal) and, sometimes, can not be fitted by using above those models. Several researchers have developed different types of cubic transmuted families (CT-G, for short) of distributions. Granzotto et al.(2017) have developed a cubic transmuted family which has following simple form where λ 1 ∈ [−1, 1] and λ 2 ∈ [−1, 1] and −2 ≤ λ 1 + λ 2 ≤ 1. It can be easily observed that the cubic transmuted family of distributions proposed by AL-Kadim and Mohammed(2017) turned out to be a special case of (4) for λ 2 = −λ 1 .
The third family of distributions is given as  Table 4. The cubic transmuted distributions shows better flexibility to handle more complex (bi-modal) data over quadratic transmuted distributions.
Using the family (6) and (7) there is huge scope to develop quartic (fourth rank) transmuted distributions. AL-Kadim(2018) has proposed a fourth rank transmuted Weibull distribution which is a special case of the family (6) for λ 2 = 0 and λ 3 = λ 1 .

Concluding Remarks
Generalization of probability distributions through transmutation was first applied in the area of financial mathematics. After that time, several researchers have successfully applied this technique to model lifetime and survival data. At present, this approach is being applied in the areas of biology, engineering, environmental, medical among others to handle more complex (bi-modal) data.