Some Cubic Transmuted Exponentiated Pareto-1 Distribution

Email: heledum@yahoo.com Abstract: In this study, we introduce two new generalized versions of the Exponentiated Pareto-I distribution called (CTEP-I-G) and (CTEP-I-R). Statistical properties of the two distributions such as reliability function, hazard function, moments and moment generating function are studied. Models parameters are estimated by the maximum likelihood method. Finally, an application of CTEP-I-G and CTEP-I-R distributions to two real datasets and compared with some distributions based on exponentiated Pareto-I distribution is illustrated. The applications suggest that the CTEPI-G performs better than CTEP-I-R.


Introduction
In order to obtain more flexible statistical models, several methods for generalizing of the well-known distributions have been widely used. For examples, (Gupta et al., 1998) proposed the exponentiated family of distributions to model the failure time data. Several authors use (Gupta et al., 1998) method to propose new distributions for examples see (Gupta and Kundu, 1999;2001;2007). Eugene et al. (2002) introduced a general class of distributions generated from the logit of the beta random variable. Another family of distributions, known as Kw-G distributions, proposed by (Cordeiro and de Castro, 2011) by using cdf of the Kumaraswamy distribution. Shaw and Buckley (2009) suggested a new generalization method called transmution map. According to (Shaw and Buckley, 2009) the cumulative distribution function of the Quadratic Ranking Transformation Map (QRTM) is: where, G(x) is the Cumulative Distribution Function (cdf) of the base distribution. Observe that, when λ= 0, the new distribution turns out to the base one. This method has been further studied and extended by many authors see for examples (Aryal and Tsokos, 2009;Nofal et al., 2017;Alizadeh et al., 2016;Merovci et al., 2016). Al-Kadim (2018) introduced generalized formula for transmuted distribution proposed by Shaw and Buckley (2009), the cdf and pdf of the Cubic Ranking Transformation Map (CRTM) respectively are: and: This method used by (Al-Kadim and Mohammed, 2017) to develop cubic transmuted Weibull distribution and utilized by (Ansari and Eledum, 2018;2020) to introduce the Cubic Transmuted Pareto distribution and cubic transmuted exponentiated Pareto-I respectively. Another two classes of Cubic Transmuted distributions with two transmuted parameters have been developed, one by (Granzotto et al., 2017) the other by (Rahman et al., 2018b). The cdf and pdf of the Granzottoa formula are as follows: and: with, l, [0, 1] and 2  [-1, 1]. This method used by many authors to generalized a various of well-known distributions for examples (Granzotto et al., 2017;Celik, 2018). Concerning to (Rahman et al., 2018b) method, the cdf and pdf are given respectively by:

Some Special Cases
It can be easily see that the cubic transmuted family of distributions proposed by (Al-Kadim, 2018) in Equation (2) turned out to be a special case of the cubic transmuted family distribution proposed by: Moreover, the quadratic transmuted family of distributions of (Shaw and Buckley, 2009)  Further, the cubic transmuted family distribution introduced by (Rahman et al., 2018b) in Equation (6) is a special case of the family proposed by (Granzotto et al., 2017) of Equation (4) for 1 = 1-1 and 2 = 2-1.

Exponentiated Pareto Distribution-1
The Exponential Pareto-1 distribution (EP-1) proposed by (Nadarajah, 2005) taking the logarithm of the classical Pareto random variable. Let X be a random variable with the exponentiated Pareto-1 distribution. The Probability Density Function (pdf) and the Cumulative Distribution Function (cdf) are defined, respectively, as: and: where, z = k a e -ax , k is the (necessarily positive) minimum possible value of X and a is a shape parameter:

Cubic Transmuted Exponentiated Pareto-1-G Distribution (CTEP-1-G)
This section pertains to the first version of the Cubic Transmuted Exponentiated Pareto-1-G (CTEP-1-G) using the cubic transmuted distributions proposed by Granzotto et al. (2017). Discussing the Probability Density Function (PDF), Cumulative Distribution Function (CDF), survival and hazard functions, some statistical properties and parameters estimates.

Density and Cumulative Function for CTEP-1-G Distribution Theorem 1
Let X be a random variable with the CTEP-1-G distribution. The Cumulative Distribution Function (CDF) and the Probability Density Function (PDF) are defined, respectively, as:

Lemma 1
The limit of CTEP-1-G density as x  ln k is a1 and as x   is 0.

Proof
The proof is straightforward.

Lemma 2
The limit of CTEP-1-G distribution function as x  ln k is 0 and the limit as x   is 1.

Proof
The proof is straightforward.

Lemma 3
f(x) of Equation (11) is a probability density function.

Proof
To prove f(x) is a pdf, we need to prove f(x) 0 and Therefore, Lemma is Proved. Some of possible shapes of the pdf and cdf of CTEP-1-G for selected values of 1 and 2 setting a = 0.5 and k = 1 are illustrated in Fig. 1a and 1b respectively, note that the pdf and cdf of the base distribution EP-1 are shown as a solid black curves.

Survival and Hazard Function
The survival function is defined as s(x) = 1-F(x) and for the CTEP-1-G distribution is given as: The hazard function is defined as for the CTEP-1-G distribution is given as:

Statistical Properties
In this subsection, some statistical properties for CTEP-1-G distribution are explained. Including moments, moment generating function and order statistics.

Theorem 2
If X is a random variable having the CTEP-1-G then the rth moment of X about the origin is:

Moments Generating Function
Theorem 3 If X is a random variable having the CTEP-1-G then the moments generating function of X is: where, a-t > 0,2a-t > 0 and 3a-t > 0. Further, the mean and variance of the distribution are given respectively as: performing some algebra calculations we get: The mean and variance have been obtained by differentiating Equation (14) i times (i = 1,2) with respect to t and setting t = 0. Table 1 and 2 show the mean and variance of CTEP-1-G for various combinations of model parameters respectively.
From Table 1 it is observed that, holding the transmuted parameters 1 and 2 constants, as the shape parameter a increases the mean decreases. On the other hand, holding a constant, as the transmuted parameters 1 and 2 increase the mean also decreases. From Table 2 it is clear that, keeping the shape parameter a and transmuted parameters 1 constants, as the transmuted parameter 2 increases the variance decreases. Further, keeping a constant, as 1 increases the variance increases this for negative values of 2, while, for 2 > 0 the variance decreases.

Order Statistics
Let X1, X2,…, Xn ~i id CTEP-1-G(a, k, 1, 2). Then the pdf of the X(n) is given by: where, z = k a e -ax , x  ln k, a, k > 0. Using j = 1 and j = n we get the pdf of smallest X1:n and largest Xn:n order statistic respectively:

Parameters Estimation
This subsection discusses the Maximum Likelihood Estimation (MLE) for parameters of CTEP-1-G distribution.
Let X1, X2,…, Xn be a random sample of size n from CTEP-1-G distribution. Then the likelihood function is given by: Since x  (ln k, ), the MLE of ln k is the firstorder statistic x(1). The log-likelihood can be maximized either directly or by solving the non-linear likelihood equations obtained by differentiating Equation (15)

Cubic Transmuted Exponentiated Pareto-1-R Distribution (CTEP-1-R)
This section pertains to the second version of the Cubic Transmuted Exponentiated Pareto-1-R (CTEP-1-R) using the cubic transmuted distributions proposed by (Rahman et al., 2018b), studding the Probability Density Function (pdf), Cumulative Distribution Function (CDF), survival and hazard functions, some statistical properties and parameters estimates.

Density and Cumulative Function for CTEP-1-R Distribution Theorem 4
Let X be a random variable with the CTEP-1-R distribution. The Cumulative Distribution Function (CDF) and the Probability Density Function (PDF) are defined, respectively, as: where, a, k are a shape and location parameters respectively and 1, 2 are transmuted parameters.

Proof
The proof is straightforward. Equation (16) is obtained by substituting Equation (8) into Equation (6) and performing some algebra calculations while Equation (17) is gotten from differentiating Equation (16) with respect to x.

Lemma 4
The limit of CTEP-1-R density as x  ln k is a(1+1) and as x   is 0.

Lemma 5 f(x) of Equation
This completes the proof of Lemma. Some of possible shapes of the pdf and cdf of CTEP-1-R for different choices of parameters 1 and 2 setting a = 0.5 and k = 1 are illustrated in Fig. 3a and 3b respectively. Note that the pdf and cdf of the base distribution EP-1 are shown as a solid black curves.

Survival and Hazard Function
The survival and hazard functions for CTEP-1-R distribution are given below:

Statistical Properties
In this subsection, some statistical properties for CTEP-1-R distribution are explained. These properties include moments and moment generating function.

Moments Theorem 5
If X is the random variable having the CTEP-1-R then the rth moment of X about the origin is:

Theorem 6
If X is a random variable having the CTEP-1-R then the moments generating function of X is: where, a-t > 0,2a-t > 0 and 3a-t > 0. Further, the mean and variance of the distribution are given respectively as:  (19) i times (i = 1,2) with respect to t and setting t = 0. Table 3 and 4 explain the mean and variance of CTEP-1-R for various combinations of model parameters respectively.
From Table 3 and 4 it is observed that, holding the transmuted parameters 1 and 2 constants, as the shape parameter a increases the mean and variance decrease. On the other hand, holding a and 1 constants, as 2 increases the mean and variance are also decrease.

Parameters Estimation
This subsection discusses the Maximum Likelihood Estimation (MLE) for Parameters of CTEP-1-R distribution.
Let X1, X2,…, Xn be a random sample of size n from CTEP-1-R distribution. Then the likelihood and log likelihood functions are given respectively by:

Application of CTEP-1-G and CTEP-1-R distributions
In this section the proposed distributions CTEP-1-G and CTEP-1-R are fitted to two real lifetime datasets to test their goodness of fit. Moreover, CTEP-1-G and CTEP-1-R are compared with some related distributions included the exponentiated Pareto-I (EP-I) and cubic transmuted exponentiated Pareto-I (CTEP-I) distributions. With a view to compare CTEP-1-G and CTEP-1-R some different comparison measures includes -log-likelihood and Kolmogorov-Smirnov (k-s) test are used. For the aim of the analysis we set ln k = exp[min(x)]. The computations are done by R.
The first data in Table 5 is leptokurtic, bimodal, highly right skewed reported by (Barlow et al., 1984) represents the failure times of Kevlar 49/epoxy strands when the pressure is at 90% stress level (FTK) and used by (Andrews and Herzberg, 2012;Cooray and Ananda, 2008;Al-Aqtash et al., 2014;Ansari and Eledum, 2020).
The second data in Table 6 is leptokurtic bimodal, moderate right skewed adduced by (Murthy et al., 2004) shows the Failure Times (in hours) of 50 Components (FTC). Table 7 shows the summary statistics of the two datasets. Tables 8 and 9 respectively demonstrate the results obtained from the two datasets including maximum likelihood estimates and comparison measures for the fitted distributions. Moreover, the estimated pdf's and cdf's of (FTK) and (FTC) are displayed in Fig. 5a and 5b respectively.
Based on the values of the k-s and its corresponding p-values in Table 8 we observe that all distributions fit the Kevlar 49/epoxy strands failure times data adequacy while the CTEP-1-G is the best one Fig. 5a. Moreover, CTEP-1-G has the minimum value of -log(L) (101.231). We conclude that the CTEP-1-G is the most appropriate model for FTK dataset.
From the results of FTC dataset in Table 9 it is clear that all distributions fit the Failure Times of components data adequacy while the CTEP-1-G is the best one Fig. 5b and it has the minimum value of -log (L) (74.017).
Hence we infer that the CTEP-1-G is the most appropriate model for FTC dataset.

Conclusion
In this study two new generalized models of the Exponentiated Pareto-I distribution Called (CTEP-I-G) and (CTEP-I-R) are introduced. Some statistical properties of the two distributions including survival and hazard functions, moments and moment generating function are studied. The models parameters are estimated by using the maximum likelihood method. Finally, an application of CTEP-I-G and CTEP-I-R to two real datasets and compared with the exponentiated Pareto-I and cubic transmuted exponentiated Pareto-1 distributions is illustrated. Based on applications results we conclude that the proposed distributions CTEP-I-G and CTEP-I-R have fitted the two datasets better than other distributions, further among the proposed distributions the CTEP-I-G is the best. We recommend the proposed distributions for modelling the failure times datasets adequacy and hope that they would receive significant applications in the future.

Acknowledgement
The author is grateful to the editor and the anonymous reviewers for their valuable comments and suggestions, which have substantially improved this paper.

Ethics
This article is original and contains unpublished material and no ethical issues involved.