Exponentiated Cubic Transmuted Weibull Distribution: Properties and Application

This work is focused on the four parameters Exponentiated Cubic Transmuted Weibull distribution which mostly found its application in reliability analysis most especially for data that are non-monotone and Bi-modal. Structural properties such as moment, moment generating function, Quantile function, Renyi entropy, and order statistics were investigated. The maximum likelihood estimation technique was used to estimate the parameters of the distribution. Application to two real-life data sets shows the applicability of the distribution in modeling real data.


Introduction
Statistical distributions are very useful tools in analyzing and predicting real-life phenomena. In recent times, several distributions have been suggested and studied. There is always new ground for development in statistical distributions to blend with the current situations which allows for wider applications that can be achieved by inducing flexibility into the standard probability distribution to allow for fitting specific real-world scenarios. This has served as a motivating factor for many researchers to work towards developing new and more flexible distributions. There are several ways to extend standard probability distributions, and one of the most popular methods is the use of distribution generators such as the exponentiated method by Lehmann [1]; the Marshall-Olkin method developed by Marshall and Olkin [2]; the beta distribution method proposed by Alexander, et al. [3] and Eugene, et al. [4]; the gamma distribution method by Cordeiro, et al. [5], Ristic and Balakrishnan [6], and Zografos and Balakrishnan [7]; the McDonald method proposed and studied by MacDonald [8]; and the exponentiated generalized method developed by De Andrade, et al. [9]. The cubic rank transmutation map was proposed and studied by Granzotto, et al. [10]. The Weibull distribution mostly used for modeling lifetime data and phenomenon with monotone failure rates. However, in real life problems one may encounter real life data which is bimodal and also exhibits non-monotone failure rate which the Weibull distribution does not provide a reasonable parametric fit. Then the need to extend the Weibull distribution is of interest to allow for a wider class of applications. Several variant of Weibull distribution have been proposed and discuss in literature and these includes: Cubic Transmuted Weibull (CTW) distribution was studied by Abed Al-Kadim [11] the Additive Weibull (AW) distribution was developed and studied by Xie and Lai [12], Afify, et al. [13] studied the Transmuted Complementary Weibull Geometric (TWG) distribution. Transmuted Modified Weibull (TMW) distribution was studied by Khan and King [14], developed and studied the Modified Weibull (MW) distribution, the Transmuted Generalised Inverse Weibull (TGIW) distribution was studied by Merovci, et al. [15], the Beta Transmuted Weibull (BTW) distributionwas proposed and studied by Pal and Tiensuwan [16], the beta modified Weibull (BMW) distribution by Silva, et al. [17], Elbatal and Aryal [18] proposed and studied the transmuted additive Weibull distribution etc. The main focus of this study is to redefine the Weibull distribution to obtain Exponentiated Cubic Transmuted Weibull distribution which a more flexible distribution and that it can be used to model real life data even those that possesses bi-modal property.
A random variable is said to follow a Weibull distribution if its cumulative distribution function is given by [ And the probability density function given as [ With shape parameter and the scale parameter

Cubic Transmuted Family of Distributions
Rahman, et al. [19], proposed an extension of the quadratic transmuted distributions called the cubic transmuted family of distribution meant to address the problem of bi-modality of the data which the quadratic family of distribution is not capable of handling. This was done by adding one or more parameter in (refqrtm) to the transmuted family of distribution. ] and Al-kadim and Mohammed (2017), proposed and studied a case of (3) by letting in (3). The is given by | | The density function corresponding to (4), is given by [ ] | | In this article, we redefine the Cubic transmuted family of distributions proposed and studied by Abed Al-Kadim [11] using the cdf of Exponentiated generalised-G (EXG-G) family proposed by Cordeiro, et al. [5]. The cdf of EXG-G class is given by [ ] Where a and are positive shape parameters. Taking in (6), we have another class of EXP-G family called the Lehmann type 1 which cdf is given by [ ] And the corresponding pdf is given by [ ]

Generalized Exponentiated Cubic Transmuted Family of Distributions
Another family of distribution can be derived called the Exponentiated Cubic transmuted family of distribution by Putting (4) in (7

Expansion for the PDF and the CDF
To simplify the expression given in (10), we use the Binomial series expansion given by ] Applying the binomial series expansion given (12) to (13)

Exponentiated Cubic Transmuted Weibull Distribution
By putting (1) in (9), we obtain the of the Exponentiated Cubic Transmuted Weibull ) distribution given by And its associated is given as ( ) Corresponding, using the property given in (14), we can re-write (16) as Andthe expressions for the Survival and Hazard function are respectively given by The graph of the distribution, density, survival and the hazard functions are given below in figure (1, 2, 3 and 4).

Statistical Properties of ECTW Distribution
In this section, emphases are on some statistical properties of Exponentiated Cubic Transmuted Weibull distribution given in (16). These properties include moment, moment generating function, quantile function, Random number generation and Renyl entropy.

Moment
In statistical analysis, moments are very useful tool in describing certain properties of the distributions. An expression for raw moment of exponentiated cubic transmuted Weibull distribution is given in the following Lemma.
Lemma 1 Suppose that the random variable follows the Exponentiated cubic transmuted Weibull distribution, then the moment of is given as  (17), in (21) and by simplifying the expression we obtain the moment of exponentiated cubic transmuted Weibull distribution given in (20). The mean and the variance can easily be obtained by taking in (20).

Moment Generating Function
Moment generating function is a very useful function that can be used to describe certain properties of the distribution. It can be used to obtain moments of a distribution. The moment generating function of exponentiated cubic transmuted Weibull distribution is given in the following lemma.
Lemma 2. Let follows the exponentiated cubic transmuted Weibull distribution, then the moment generating function, is Where Proof the moment generating function of a random variable is given by ∫ Where is given in (17). Using series expansion for given by ∑ Using (24), we can re-write equation (23) as follows
It can be observed from the series expansion of (25) that moments are the coefficients of .  From table 1 drawn above we can conclude that, for any parameter values of the ECTW distribution, the first four moments is the same, the distribution is negatively skewed and can be used to model data with varying degree of kurtosis.

Quantile Function
The quantile function of the exponentiated Cubic transmuted Weibull distribution is obtained by solving (9) for and is given as,

( )
Where, ( ) The three quartiles, lower quartile , middle quartile and the upper quartile can be obtained by taking q to be 0.25, 0.5 and 0.75 in (26) respectively.

Random Number Generation
Random number can be generated from the Exponentiated Cubic transmuted Weibull distribution by equating the cdf of the ECTW distribution with a uniform random number and inverting the expression. More so, the random number from ECTW distribution is obtained by solving for . the random sample for ECTW distribution can also be expressed as

( )
Where y is given in equation (27); with and

Renyi Entropy
The Renyi entropy of a random variable represents a measure of uncertainty. A large value of entropy indicates the greater uncertainty in the data. The Renyi [20], introduced the Renyi entropy defined as With simple mathematical operation, we have

Order Statistics
Order statistics are among the most fundamental tools in non-parametric statistics and inference.

Estimation of the Parameters
The likelihood function of ECTW distribution is given by The log-likelihood function of the ECTW distribution is given by Equation (34) can be maximized by solving the nonlinear likelihood equations obtained by differentiating this with respect to and . The components o the score vector are given by By setting the above partial equations above to zero, the equations obtained are not in closed form and values of the parameters must be found by iterative methods. The maximum likelihood estimates of the parameters, denoted by ̂ is obtained by solving nonlinear equation ( ) using a numerical method.
The fisher information matrix is given by * + ( ) , can be numerically obtained by R or MATLAB software. The Fisher information matrix can be approximated by To obtain numerical solution for the values of the estimates of ECTW distribution we may employ software such as R, Maple, OX Program etc.

Applications
In this section, we illustrate the usefulness and applicability of the ECTW distribution by fitting it to tworeal life data sets. We fit the density function of Exponentiated Cubic Transmuted Weibull (ECTW), Cubic Transmuted Weibull (CTW), distribution and the Weibull (W) distributions. To demonstrate the tractability/ flexibility of the new model proposed model, we consider the following measures of fits: Anderson-Darling , Probability Value (P-Value), Akaike Information Criteria (AIC), Bayesian Information Criteria (BIC), Consistent Akaike Information Criteria (CAIC) and Komogorov Smirnoff (KS) Statistic. The best model among the competing model will be the one having the smallest AIC, BIC, CAIC and and the highest P-Value. The first data set consists of data of cancer patients. The data represents the remission times (in months) of a random sample of 128 bladder cancer patients from Lee and Wang [21]. This data have been used by Rahman, et al. [19] to fit Kumaraswamy exponentiated Burr XII and generalized transmuted log-logistic distribution. The starting point of the iterative processes for the cancer patient's data set is (1:0; 0:009; 10:0; 0:1; 0:1). The data point is given as:0.08, 2.09, 3 Exploratory Data Analysis of the data is given in Table 2, Table 3 gives the estimates of the parameters of ECTW, CTW, W distribution (standard error in parentheses) and Table 4 contains the measures of goodness of fit for the cancer data. Figure 5 provides the Total Time on Test Plot and the Box plot for the cancer data.The graphs show the data is unimodal and skewed to the right.     Table 5. Table 6 gives the estimates of the parameters of ECTW, CTW, W distribution (standard error in parentheses) and Table 7 contains the measures of goodness of fit for the glass fibre data. Figure 6 provides the Total Time on Test Plot and the Box plot for the data. The graph shows the data exhibits increasing failure rate and skewed to the left.

Conclusion
In this study, a four-parameter model called Exponentiated Cubic Transmuted Weibull distribution is proposed and its statistical properties are derived. The estimators for the parameters of the distribution are developed using maximum-likelihood estimation. The applications of the Exponentiated Cubic Transmuted Weibull distribution are demonstrated by using two real life datasets. The performance of the distribution was compared with other distribution regards to providing good fit to the data sets is assessed. The results show that the Exponentiated Cubic Transmuted Weibull distribution provides a more reasonable parametric fits to the data sets.