A General Transmuted Family of Distributions

In this article we have proposed a general transmuted family of distributions with emphasis on the cubic transmuted (CT ) family of distributions. This new class of distributions provide additional flexibility in modeling the bi-modal data. The proposed cubic transmuted family of distributions has been linked with the T −X family of distributions proposed by Alzaatreh et al. (2013). Some members of the proposed family of distributions have been discussed. The cubic transmuted exponential distribution has been discussed in detail and various statistical properties of the distribution have been explored. The maximum likelihood estimation for parameters of cubic transmuted exponential distribution has also been discussed alongside Monte Carlo simulation study to assess the performance of the estimation procedure. Finally, the cubic transmuted exponential distribution has been fitted to real datasets to investigate it’s applicability.


Introduction
Probability distributions has been popularly used in many areas of life.The standard probability distributions has been used in practice from long.In recent years the extension and generalization of probability distributions have attracted several statisticians.A simple extension of the probability distributions is proposed by Gupta et al. (1998) and is known as the exponentiated family of distributions.The exponentiated family of distributions provides flexibility by adding one more parameter to the distribution and has attracted several authors, see for example Gupta andKundu (1999, 2007), among others.Eugene et al. (2002) have proposed the Beta-G distributions using logit of the Beta distributions.The Beta-G distributions extends the distribution of order statistics.Another family of distributions, known as Kum-G distributions, has been proposed by Cordeiro and Castro (2010) by using cdf of the Kumaraswamy distribution.The Beta-G and Kum-G families of distributions uses cdf of bounded random variables and thus has limited extension.Alzaatreh et al. (2013), have proposed a general method of extending probability distributions known as T-X family of distributions which used cdf of any random variable.
The Beta-G, Kum-G and T-X families of distributions uses some baseline distribution.Another method of generating families of distributions has been proposed by Shaw and Buckley (2007), which used quadratic transmutation map to generate new probability distribution using any baseline distribution.The quadratic transmuted family of distributions of Shaw and Buckley (2007) , where G(x) is the cdf of the baseline distribution.Observe that at λ = 0, we get the baseline cdf .
The quadratic transmuted family of distributions by Shaw and Buckley (2007), has opened doors to extend the existing probability models to capture the quadratic behavior of the data.The quadratic transmuted class has been further studied and extended by Aryal andTsokos (2009, 2011) In this paper, we have proposed a family of general transmuted distributions and have developed the cubic transmuted family of distributions as well.This family of distributions captures the complexity of the data arising in engineering, environmental, financial and other areas of life.The layout of the paper is given below.

Organization of the Article
The paper is structured as follows.The new family of general transmuted distributions is provided in Section 2. Section 3 provides a new class of cubic transmuted distributions.Some examples related to the cubic transmuted distributions are presented in Section 4. The cubic transmuted exponential distribution has been explored in detail in Section 5. Section 6 describes the expressions for moments, quantiles, reliability function and random number generation for the proposed cubic transmuted exponential distribution.In Section 7, we have presented estimation of the parameters and two real-life applications along with simulation study is given in Section 8. Finally, in Section 9, some concluding remarks are given.

General Transmuted Family of Distributions
In this section, we have proposed a general transmuted family of distributions that can be used to generate new families.The family is defined below.
Let X be a random variable with cdf G(x), then a general transmuted family; called k− transmuted family; is defined as with The general transmuted family reduces to the base distribution for We now give the cubic transmuted family of distributions in the following.

Cubic Transmuted Family of Distributions
In this section we have discussed the cubic transmuted family of distributions.The cubic transmuted family of distributions is obtained by setting k = 2 in (1) and is given as which can also be written as where It can be easily seen that the cubic transmuted family of distributions proposed by AL-Kadim and Mohammed (2017), turned out to be a special case of (3) for λ 2 = −λ 1 .The cubic family of transmuted distributions reduces to the quadratic transmuted family of distributions of Shaw and Buckley (2007), for λ 2 = 0. Also, the family (3) is different as compared with the family proposed by Granzotto et al. (2017).
We now present the genesis of cubic transmuted family of distributions given in (3) in context with distribution of order statistics and with T − X family of distributions in following two theorems.
Theorem 3.1.Let X 1 , X 2 and X 3 be iid random variables each with distribution function G(x), then the cubic transmuted family of distributions (3) can be obtained as a weighted sum of three order statistics.
where p(t) indicates the pdf with support on [0, 1].Using , we obtain the cubic transmuted family (3).Now, suppose that the density p(t) is written as weighted sum of three densities as Hence, using p On simplification, the cdf (7) reduces to the cdf given in (3).
We now give the density function of cubic transmuted family in the following.

Definition 3.1 (Cubic Transmuted Distribution).
A random variable X is said to have a cubic transmuted distribution, with parameters where g(x) is the pdf associated with baseline cdf G(x).
Observe that for λ 2 = 0, the pdf of cubic transmuted distribution reduces to the pdf of quadratic transmuted distribution.Also, observe that, at λ 1 = λ 2 = 0, we have the pdf of the baseline random variable, as it should be.The density function of cubic transmuted family of distributions given in ( 8) can be obtained by using k = 2 in the density function of general transmuted family of distributions given in (2).
Proof.The proof is simple.

Some Examples of Cubic Transmuted Distributions
In this section, we have discussed some members of the cubic transmuted family of distributions given in (3) for different choices of baseline G(x).These are discussed in the following subsections.

Cubic Transmuted Normal Distribution
Suppose that the random variable X has normal distribution with cdf Φ(x), then the cubic transmuted normal distribution has cdf The density function of cubic transmuted normal distribution can be similarly written.

Cubic Transmuted Gamma Distribution
Let X is a gamma random variable with shape parameter α and scale parameter 1/β , then the cdf of cubic transmuted gamma distribution is given as where α, β ∈ R + , Γ(α) is complete gamma function and Γ(α, βx) is the upper incomplete gamma function.Using (3), the cdf of the cubic transmuted gamma distribution is given by where η(x) = Γ(α, βx).

Cubic Transmuted Log-logistic Distribution
Let X has Log-logistic distribution with cdf where α, β ∈ R + are the scale and shape parameters respectively.Using (3), the cdf of the cubic transmuted log-logistic distribution is given by

Cubic Transmuted Pareto Distribution
Suppose X has Pareto distribution with cdf where k, θ ∈ R + are the scale and shape parameters.Using (3), the cdf of cubic transmuted Pareto distribution is given by

Cubic Transmuted Rayleigh Distribution
Let X has Rayleigh distribution with cdf where σ ∈ R + is the scale parameter of the distribution.Using (3), the cdf of cubic transmuted Rayleigh distribution is given by

Cubic Transmuted Gumbel Distribution
Let random variable X has Gumbel distribution with cdf where µ ∈ R, β ∈ R + are the location and scale parameters respectively.Using (3), the cdf of the cubic transmuted Gumbel distribution is The plot of density function for above mentioned distributions is given in Figure 1.
We now discuss the cubic transmuted exponential distribution in detail in the following section.

Cubic Transmuted Exponential Distribution
The exponential distribution is a widely used lifetime distribution.Several researchers have attempted to generalize exponential distribution for the limited applicability and they have proposed beta exponential (Nadarajah and Kotz, 2006), generalized exponential (Gupta andKundu, 1999, 2007), exponentiated exponential (Gupta and Kundu, 2001) and quadratic transmuted exponential (Owoloko et al., 2015) distribu- where θ ∈ [0, ∞) is the scale parameter.Owoloko et al. (2015), have developed the quadratic transmuted exponential distribution with cdf given as where λ ∈ [−1, 1].Using the pdf and cdf of exponential distribution in (8), the pdf of cubic transmuted exponential distribution is given in the following.
Proposition 5.1.Let X has exponential distribution with parameter θ ∈ [0, ∞), then the pdf of cubic transmuted exponential distribution with parameters θ ∈ [0, ∞), Proof.Using the cdf of exponential distribution in (3), the cdf of cubic transmuted exponential distribution is The pdf of cubic transmuted exponential distribution is easily obtained by differentiating (10) and is given in (9).
Figure 2 shows some of the possible shapes for density and distribution functions of cubic transmuted exponential distribution for various values of λ 1 and λ 2 at θ = 1.5.
Table 1: Mean of the CT exponential distribution

Statistical Properties
In this section, we have discussed some distributional properties of the cubic transmuted exponential distribution given in (9).These properties include expressions for moment, quantile, reliability and hazard functions.We have also discussed the random number generation from cubic transmuted exponential distribution.

Moments
The rth moment of the cubic transmuted exponential distribution is given as Using f (x) as given in ( 9) and simplifying, the rth moment for cubic transmuted exponential distribution is given as The mean and variance can be obtained by using (11), and are given as The higher moments can also be obtained from (11).Table 1 provides the mean for various combinations of the parameters.

Quantile Function
The quantile function is obtained by solving (10) for x and is given as where The three quartiles can be obtained by using q = 0.25, 0.50 and 0.75 in ( 12), respectively.

Reliability Analysis
The reliability function describes the probability of an element not failing prior to some time t, and is defined by R(t) = 1 − F (t).The reliability function of cubic transmuted exponential distribution can be obtained as The hazard function, h(t), is defined by which, for cubic transmuted exponential distribution, is given as where . The hazard function indicates the instantaneous rate of failure at time t, given that the component has survived up to time t.Some of the possible shapes of the reliability and hazard functions for the selected values of model parameters λ 1 and λ 2 keeping θ = 1.5 are presented in Figure 3.We have observed that the reliability and distribution functions behaved complement to each other.In order to failure time t, we observed increasing, increasing then decreasing and decreasing then increasing and constant hazard rates from the shapes.

Random Numbers Generation
The random sample from cubic transmuted exponential distribution can be obtained by setting the distribution function (10) equal to u, where u is a uniform random variable, that is the random number from cubic transmuted exponential distribution is obtained by solving where u ∼ U (0, 1).On simplification, this can be presented as where y is given in ( 13) with d = 1 − u.The random sample from cubic transmuted exponential distribution can be obtained by using (14) for various values of the parameters θ, λ 1 and λ 2 .

Parameter Estimation and Inference
In this section, we have obtained maximum likelihood estimators (M LEs) for parameters of the cubic transmuted exponential distribution.For this, let X 1 , X 2 , • • • , X n be a random sample of size n from cubic transmuted exponential distribution.The likelihood function is, then, given by The log-likelihood function l = ln(L) is given as The M LEs of θ, λ 1 and λ 2 are obtained by maximizing (15).The derivatives of (15) wrt the unknown parameters are given as .
The likelihood equations are given as where The asymptotic variance and covariance matrix of the estimates θ, λ1 and λ2 ; see Appendices 1, is obtained by inverting ( 16).An approximate 100(1−α)% asymptotic confidence intervals for θ, λ 1 and λ 2 are, respectively, given by where Z α is the αth upper percentile of the standard normal distribution.

Numerical Studies
In this section, an extensive Monte Carlo simulation study is carried out to assess the performance of estimation method.We have also considered two real-life datasets to investigate the applicability of the proposed model.A Monte Carlo simulation study is carried out for samples of sizes 50, 100, 200, 500 and 1000, drawn from cubic transmuted exponential distribution.The samples have been drawn for θ = 2, λ 1 = 0.5 and λ 2 = −0.5 and maximum likelihood estimators for the parameters θ, λ 1 and λ 2 are obtained.The procedure has been repeated for 10000 and the mean and mean square error for the estimates are computed.The results are summarized in Table 2.We have found that the simulated estimates are closed to the true values of parameters and hence the estimation method is adequate.We have also observed that estimated mean square errors (M SEs) consistently decreases with increasing sample size.

Life Test Data
This dataset represents the lifetimes of 50 devices and has been used by Aarset (1987) and Lai et al. (2003).The summary statistics of the data presented in Table 3.In order to assess the performance of the cubic transmuted exponential distribution we have computed various measures for quadratic transmuted exponential, beta exponential and exponentiated exponential distributions.The estimated values of   4. Estimated pdf and cdf of the lifetimes of 50 devices are plotted over empirical density and distribution functions respectively and presented in the upper panels of Figure 4. Table 5 provides the log-likelihood, Akaike's information criterion (AIC), corrected Akaike's information criterion (AICc) and Bayesian information criterion (BIC).From Table 5, we can see that the cubic transmuted exponential distribution

Electronics Data
The dataset provides lifetimes of 20 electronic components and has been used by Murthy et al. (2004).The summary statistics are presented in Table 3.We have fitted quadratic transmuted exponential, cubic transmuted exponential, beta exponential and exponentiated exponential distributions to the data.The M LEs with their corresponding SEs are given in Table 6.Estimated pdf and cdf of the lifetimes of 20 electronic components are plotted over empirical density and distribution functions respectively and presented in the lower panels of Figure 4.The computed Log-likelihood, AIC, AICc and BIC values are provided in Table 7.We have observed, on the basis of the criteria used, that the cubic transmuted exponential distribution is the most appropriate model for this data.

Concluding Remarks
In this article, we have introduced a general family of transmuted distributions with special reference to the cubic transmuted family of distributions.We have found the cubic transmuted distributions are flexible enough and are capable to capture the bi-modality of the data.In order to assess the performance of this new class of where η i = 1 − e − x i θ .
; Nofal et al. (2017); Alizadeh et al. (2016); Merovci et al. (2016) and Yousof et al. (2015).At this time, quadratic transmuted distributions are familiar in the literature, several quadratic transmuted distributions are provided by Tahir and Cordeiro (2016).Alizadeh et al. (2017), have noted that the quadratic transmuted family of distribution can be linked with the T-X family of distributions for suitable choice of a bounded density function between [0, 1].In order to capture the complexity of the data and increases the flexibility, new classes of cubic transmuted distributions have been developed by Granzotto et al. (2017); AL-Kadim and Mohammed (2017).

Figure 2 :
Figure 2: Density and distribution functions are plotted for the CT exponential distribution for different values of model parameters λ 1 and λ 2 , setting θ = 1.5.

Figure 3 :
Figure 3: Reliability and hazard functions are plotted for the CT exponential distribution for different values of model parameters λ 1 and λ 2 , setting θ = 1.5.

Figure 4 :
Figure 4: Upper panels: Estimated pdf and cdf for life test dataset; Lower panels: Estimated pdf and cdf for electronics dataset.

Table 2 :
Average estimates of model parameters and M SEs

Table 3 :
Summary statistics for selected datasets

Table 4 :
M LEs of the parameters and respective SEs for selected models

Table 5 :
Selection criteria estimated for selected models

Table 6 :
M LEs of the parameters and respective SEs for selected models