Cubic Rank Transmuted Modified Burr III Pareto Distribution: Development, Properties, Characterizations and Applications

In this paper, a flexible lifetime distribution called Cubic rank transmuted modified Burr III-Pareto (CRTMBIII-P) is developed on the basis of the cubic ranking transmutation map. The density function of CRTMBIII-P is arc, exponential, left-skewed, right-skewed and symmetrical shaped. Descriptive measures such as moments, incomplete moments, inequality measures, residual life function and reliability measures are theoretically established. The CRTMBIII-P distribution is characterized via ratio of truncated moments. Parameters of the CRTMBIII-P distribution are estimated using maximum likelihood method. The simulation study for the performance of the maximum likelihood estimates (MLEs) of the parameters of the CRTMBIII-P distribution is carried out. The potentiality of CRTMBIII-P distribution is demonstrated via its application to the real data sets: tensile strength of carbon fibers and strengths of glass fibers. Goodness of fit of this distribution through different methods is studied.


Introduction
In recent decades, many continuous univariate distributions have been developed but various data sets from reliability, insurance, finance, climatology, biomedical sciences and other areas do not follow these distributions.Therefore, modified, extended and generalized distributions and their applications to the problems in these areas is a clear need of day.
The modified, extended and generalized distributions are obtained by the introduction of some transformation or addition of one or more parameters to the well-known baseline distributions.These new developed distributions provide better fit to the data than the sub and competing models.Shaw and Buckley (2009) proposed ranking quadratic transmutation map to solve financial problems.

Quadratic Ranking Transmutation Map
Theorem 1.1: Let 1 Z and 2 Z be independent and identically distributed (i.i.d.) random variables with the common cumulative distribution function   Gz .Then, the ranking quadratic transmutation map is and (2) If we take 2 1,     the distribution in equation ( 2) is known as ranking quadratic transmutation map or transmuted distribution.

Cubic Ranking Transmutation Map
Theorem 2.1: Let 1 Z , 2 Z and 3 Z be i.i.d.random variables with the common cumulative distribution function   Gz.Then, the cubic ranking transmutation map is (3)

Proof
Consider the following order: , with probability 3 where and Pr min , , Pr 1 Pr max , , .Arnold et al. (1992) showed that Pr min , , 1 1 If we take 1  the distribution in equation ( 4) is known as Cubic ranking transmutation map or transmuted distribution of order 2.

Definition 2.1
The cumulative distribution function (cdf) and probability density function (pdf) for the cubic rank transmuted distribution are given, respectively, by and Afify et al. (2017) proposed the beta transmuted-H family of distributions.Al-Kadim and Mohammed (2017) presented the cubic transmuted Weibull distribution in terms of basic mathematical properties.Nofal et al. (2017) studied a generalized transmuted-G family of distributions.Alizadeh et al. (2017) developed generalized transmuted family of distributions.Bakouch et al. (2017) introduced a new family of transmuted distributions.Granzotto et al. (2017) proposed a cubic ranking transmutation map and studied different properties.They studied properties of Cubic rank transmuted and Here, the CRTBMIII-P distribution is introduced with the help of ( 6) and ( 7).The cdf and pdf of CRTMBIII-P distribution are given, respectively, by and   are parameters.

Structural Properties of CRTMBIII-P Distribution
The survival, hazard, cumulative hazard and reverse hazard functions and the Mills ratio of a random variable X with CRTMBIII-P distribution are given, respectively, by and The elasticity     lnF( ) ln The elasticity of CRTMBIII-P distribution shows the behavior of the accumulation of probability in the domain of the random variable.

Shapes of the CRTMBIII-P Density
The following graphs show that shapes of CRTMBIII-P density are arc, exponential, positively skewed, negatively skewed and symmetrical (Fig. 1).The plots of hrf (Fig. 2) are also given.

Sub-Models
The CRTMBIII-P distribution has the following sub models.

Descriptive Measures Based Quantiles
The quantile function of CRTMBIII-P distribution is the solution of the following equation Median of CRTMBIII-P distribution is the solution of the following The random number generator of CRTMBIII-P distribution is the solution of the following , where and the random variable Z has uniform distribution on   0,1 .Some measures based on quartiles for location, dispersion, skewness and kurtosis for the CRTMBIII-P distribution respectively are: Median M=Q (0.5); Quartile deviation and Moors kurtosis measure based on Octiles . The quantile based measures exist even for distributions that have no moments.The quantile based measures are less sensitive to the outliers.

Moments
Moments, incomplete moments, inequality measures, residual and reverse residual life function and some other properties are theoretically derived in this section.

Moments About the Origin
The  th ordinary moment of CRTMBIII-P distribution is, .
Mean and Variance of CRTMBIII-P distribution are The factorial moments for CRTMBIII-P distribution are given by  is Stirling number of the first kind.

Incomplete Moments
Incomplete moments are used to study mean inactivity life, mean residual life function and other inequality measures.The lower incomplete moments of a random variable X with CRTMBIII-P distribution are where is the incomplete beta function.
The upper incomplete moments for the random variable X with CRTMBIII-P distribution are The mean deviation about mean is 2 2 and the mean deviation about median is , where . Bonferroni and Lorenz curves for a specified probability p are computed from  

Residual Life Functions
The residual life, say   n mz , of X with CRTMBIII-P distribution has the n th moment The average remaining lifetime of a component at time z, say   1 mz , or life expectancy known as mean residual life (MRL) function is given by The reverse residual life, say,

 
n Mz of X with CRTMBIII-P distribution having n th moment is The waiting time z for failure of a component has passed with condition that this failure had happened in the interval [0, z] is called mean waiting time (MWT) or mean inactivity time.The waiting time z for failure of a component of X having CRTMBIII-P distribution is defined by

Reliability Measures
In this section, reliability measures are studied.

Stress-Strength Reliability for CRTMBIII-P Distribution
, , , , , , X , , , , , is the characteristic of the distribution of 1 2 and X

X
. Then reliability of the component for CRTMBIII-P distribution is computed as ,, and .5, it means that 1 X and 2 X are i.i.d. and there is equal chance that 1 X is bigger than 2 X .

Characterizations
In order to develop a stochastic function in a certain problem, it is necessary to know whether the selected function fulfills the requirements of the specific underlying probability distribution.To this end, it is required to study characterizations of the specific probability distribution.Certain characterizations of CRTMBIII-P distribution are presented in this section.

Characterization Through Ratio of Truncated Moments
The CRTMBIII-P distribution is characterized using Theorem 1 (Glä nzel; 1987) on the basis of a simple relationship between two truncated moments of functions of X. Theorem 1 is given in Appendix A.
The pdf of X is (11), if and only if   qx(in Theorem 1) has the form   .
After simplification, we have .
Therefore according to theorem 1, X has pdf (11).
Corollary 6.1.1.Let   : 0, X    be a continuous random variable and let The pdf of X is (11) if and only if functions   qx and   .
Remark 6.1.1.The general solution of the above differential equation is where D is a constant.

Characterization via Doubly Truncated Moment
Here CRTMBIII-P distribution is characterized via doubly truncated moment.

 
X: 0,    be a continuous random variable.Then, X has pdf (11) if and only if For random variable X with pdf (11), we have .
Conversely, if (30) holds, then Differentiating with respect to y, we have which is pdf of CRTMBIII-P distribution.

Maximum Likelihood Estimation
In this section, parameter estimates are derived using maximum likelihood method.The log-likelihood function for CRTMBIII-P distribution with the vector of parameters In order to estimate the parameters of CRTMBIII-P distribution, the following nonlinear equations must be solved simultaneously:

Simulation Study
In this section, we perform the simulation study to illustrate the performance of MLE.We consider the CRTMBIII-P distribution with  = 2.95,  = 0.7,  = 2.20,  = 3.95,  1 = 0.4,  2 = 0.1,  = 1.We generate 1000 samples of sizes 20, 50, 200.The simulation results are reported in Table 3.In the table, it reports the average estimated , , , ,  1 ,  2 and the standard deviation of the estimates within the parenthesis.From this Table , we observe that the MLE estimates approach true values as the sample size increases whereas the standard deviations of the estimates decrease, as expected.

Applications
In this section, the CRTMBIII-P distribution is compared with TMBIII-P, MBIII-P, BIII-P, IL-P, LL-P distributions.Different goodness fit measures like Cramer-von Mises (W), Anderson Darling (A), Kolmogorov-Smirnov (K-S) statistics with p-values, and likelihood ratio statistics are computed using R-package for tensile strength of carbon fibers and strengths of glass fibers.
The better fit corresponds to smaller W, A, K-S (p-value), AIC, CAIC, BIC, HQIC and  value.The maximum likelihood estimates (MLEs) of unknown parameters and values of goodness of fit measures are computed for CRTMBIII-P distribution and its sub-models.The MLEs, their standard errors (in parentheses) and goodness-of-fit statistics like W, A, K-S (p-values) are given in table 4 and 6.Table 5 and 7 displays goodness-of-fit values.

Concluding Remarks
We have developed a more flexible distribution on the basis of the cubic transmuted mapping that is suitable for applications in survival analysis, reliability and actuarial science.The important properties of the proposed CRTMBIII-P distribution such as survival function, hazard function, reverse hazard function, cumulative hazard function, mills ratio, elasticity, quantile function, moments about the origin, incomplete moments, inequality measures and stress-strength reliability measures are presented.The proposed distribution is characterized via ratio of truncated moments and doubly truncated moment.Maximum likelihood estimates are computed.The simulation study for the performance of the MLEs of parameters for the new distribution is carried out.Applications of the proposed model to tensile strength of carbon fibers and strengths of glass fibers are presented to show its significance and flexibility.Goodness of fit shows that the new distribution is a better fit.We have demonstrated that proposed distribution is empirically better for tensile strength of carbon fibers and strengths of glass fibers data.

Appendix A
Theorem 1: Let   , F, P  be a probability space for given interval 1 2 [d ,d ] [d ,d ] and F is two times continuously differentiable and strictly monotone function on 1 2 [d ,d ]   1 2 [d ,d ] .
Finally, assume that the equation   q t h t s t q t h t h t and K is a constant, adopted such that

Figure 1 .
Figure 1.Plots of pdf of CRTMBIII-P distribution

Figure 3 .
Figure 3. Fitted pdf, cdf, survival and pp plots of the CRTMBIII-P distribution for carbon fibers

Figure 4 .
Figure 4. Fitted pdf, cdf, survival and pp plots of the CRTMBIII-P distribution ) Z be i.i.d.random variables with the common cumulative distribution function  

Table 2 .
Median, mean, standard deviation, skewness and Kurtosis of the CRTMBIII-P Distribution

Table 4 .
MLEs and their standard errors (in parentheses) and Goodness-of-fit statistics for data set I

Table 5 .
Goodness-of-fit statistics for data set I

Table 6 .
MLEs and their standard errors (in parentheses) and Goodness-of-fit statistics for data set II

Table 7 .
Goodness-of-fit statistics for data set II