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

We propose a lifetime distribution with flexible hazard rate called cubic rank transmuted modified Burr III (CRTMBIII) distribution. We develop the proposed distribution on the basis of the cubic ranking transmutation map. The density function of CRTMBIII is symmetrical, right-skewed, left-skewed, exponential, arc, J and bimodal shaped. The flexible hazard rate of the proposed model can accommodate almost all types of shapes such as unimodal, bimodal, arc, increasing, decreasing, decreasing-increasing-decreasing, inverted bathtub and modified bathtub. To show the importance of proposed model, we present mathematical properties such as moments, incomplete moments, inequality measures, residual life function and stress strength reliability measure. We characterize the CRTMBIII distribution via techniques. We address the maximum likelihood method for the model parameters. We evaluate the performance of the maximum likelihood estimates (MLEs) via simulation study. We establish empirically that the proposed model is suitable for strengths of glass fibers. We apply goodness of fit statistics and the graphical tools to examine the potentiality and utility of the CRTMBIII distribution.


INTRODUCTION
In recent decades, many continuous univariate distributions have been developed, however, 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 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 X and 2 X be independent and identically distributed (...) random variables with the common cumulative distribution function   Gx.Then, the ranking quadratic transmutation map is Proof Let 1 X and 2 X be ...random variables with the common cumulative distribution function  .
Gx Now, consider the following order statistics:   the distribution in equation ( 2) is known as ranking quadratic transmutation map or transmuted distribution.

Cubic Ranking Transmutation Map
Theorem 1.1: Let  1 ,  2 and  3 be i.i.d.random variables with the common cumulative distribution function().Then, the cubic ranking transmutation map is (3) Proof Consider the following order statistics: min , , , max , , X X X X X and X X X X  .ii i where and where     If we take 1 1 2 2 33 and      the distribution in equation ( 4) is known as cubic ranking transmutation map or transmuted distribution of order 2.
Definition 1.1:The cdf and probability density function (pdf) for cubic rank transmuted distribution are given, respectively, by Burr III (Burr;1942) has wide range of applications in failure time modeling, reliability, business failure data, modeling finance, insurance data and quality control plans.Burr III (BIII) model accommodates only decreasing and inverted bathtub hazard rate functions (hrf).Transmuted Burr III (TBIII) accommodates only inverted bathtub hazard rate functions (Abdul-Moniem; 2015).The failure rate for modified Burr III (MBIII) can take only increasing, decreasing, inverted bathtub and modified bathtub shapes (Bhatti et al. 2019).Transmuted modified Burr III (TMBIII) accommodates only decreasing and inverted bathtub hazard rate functions (Ali and Ahmad;2016).The hrf for the CRTMBIII distribution accommodates almost all shapes such as bimodal, arc, increasing, decreasing, decreasing-increasing-decreasing, inverted bathtub (unimodal) and modified bathtub.Due to its flexible failure rate, it can be applicable to lifetime applications.
The basic motivations for proposing the CRTMBIII distribution are: (i) to generate distributions with symmetrical, right-skewed, left-skewed, exponential, arc, J and bimodal shaped; (ii) to obtain unimodal, bimodal, arc, increasing, decreasing, decreasing-increasingdecreasing, inverted bathtub and modified bathtub hazard rate function; (iii) to serve as the best alternative model for the current models to explore and modeling real data in economics, life testing, reliability, survival analysis manufacturing and other areas of research and (iv) to provide better fits than other sub-models.
This paper is sketched into the following sections.In Section 2, we develop and study the CRTMBIII distribution.We also present the basic structural properties and sub-models.We also study some plots of density and hazard rate functions.In Section 3, we derive mathematical properties such as moments, incomplete moments, inequality measures, residual and reverse residual life function and stress-strength reliability measure.In Section 4, two characterizations of the CRTMBIII distribution are studied.In Section 5, we address the parameters of the CRTMBIII distribution via maximum likelihood method.In Section 6, we evaluate the performance of the maximum likelihood estimates (MLEs) of the modal parameters via simulation study.In Section 7, we establish empirically that the proposed model is suitable for strengths of glass fibers.We apply goodness of fit statistics and graphical tools to examine the potentiality and utility of the CRTMBIII distribution.The concluding remarks are given in Section 8.

THE CRTMBIII DISTRIBUTION
Here, the CRTMBIII distribution is introduced with the help of ( 7) and (8).The cdf and pdf of the CRTMBIII distribution are given, respectively, by with 0, 0, 0 In future, the pdf in (11)       the survival, hazard, cumulative hazard, reverse hazard functions and the Mills ratio are given, respectively, by The elasticity     lnF( ) ln for the CRTMBIII distribution is The elasticity of the CRTMBIII distribution shows the behavior of the accumulation of probability in the domain of the random variable.
The quantile function of the CRTMBIII distribution is the solution of the following where The random number generator of the CRTMBIII distribution is the solution of the following where and the random variable Z has the uniform distribution on   0,1 .

Shapes of the CRTMBIII Density and Hazard Rate Functions
The following graphs show that shapes of CRTMBIII density are arc, exponential, positively skewed, negatively skewed and symmetrical (Fig. 1).The CRTMBIII distribution has unimodal, bimodal, arc, increasing, decreasing, decreasing-increasing-decreasing, inverted bathtub and modified bathtub hazard rate function (Fig. 2).

Sub-Models
The CRTMBIII distribution has the following sub models (Table 1

MATHEMATICAL PROPERTIES
We derive theoretically some mathematical properties such as the r th ordinary moments, s th incomplete moments, and inequality measures, residual and reverse residual life function and reliability measures in this section.

Ordinary Moments
The moments are significant tools for statistical analysis in pragmatic sciences.The descriptive measures such as central tendency   kurtosis ( 2  ) can be calculated from the moments.
For X~CRTMBIII(, , ,  1 ,  2 ),the th ordinary moment is , then The Mellin transformation is applied to get the moments of a probability distribution.

Order Statistics and their moments
Order statistics (OS) have wide applications in climatology, life testing and reliability.Moments of OS are also designed for replacement policy with the prediction of failure of future items determined from few early failures.Let The q th ordinary moment of : in X say

Incomplete Moments
Mean inactivity life; mean waiting time and inequality measures can be obtained from incomplete moments.For X~CRTMBIII   12 , , , , ,      the incomplete lower moments are is the incomplete beta function.
For X~CRTMBIII   12 , , , , ,      the incomplete upper moments are , we arrive at where r   and   .,.
is the incomplete beta function.
The mean deviation about the mean  

Residual Life functions
For X~CRTMBIII (, , ,  1 ,  2 ), the n th moment of the residual life,       the nth moment of the reverse residual life,   () = [( − ) 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 with CRTMBIII distribution is defined by

Stress-strength Reliability for CRTMBIII Distribution
, , , , , , , , ,      such that 1 X represents strength and 2 X represents stress.Then reliability of the component is: Therefore the stress-strength reliability parameter R is independent of parameters 12 ,, and     .

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.Here, we present two characterizations of the CRTMBIII distribution (i) ratio of the truncated moments and (ii) double truncated moments.

Ratio of Truncated Moments
We characterize the CRTMBIII distribution on the basis of a simple relationship between two truncated moments of functions of X [Theorem G (Glänzel;1987)].
The pdf of X is (10), if and only if   qx(in Theorem G) has the form   ,0 q x x x   .
Proof.If X has pdf (10), then and Therefore according to theorem G, X has pdf (11).
The pdf of X is (10) if and only if functions () and ℎ 1 () satisfy the equation Remark 4.1.1:The general solution of the above differential equation is where D is a constant.

Proof:
For random variable X with pdf (10), we have Differentiating with respect to y, we have which is pdf of the CRTMBIII distribution.

MAXIMUM LIKELIHOOD ESTIMATION
Here, we adopt maximum likelihood estimation technique for the CRTMBIII parameters.Let      by solving equations 32-36 either directly or using quasi-Newton procedure, computer packages/ softwares such as R, SAS, Ox, MATLAB, MAPLE and MATHEMATICA,.

SIMULATION STUDY
In this section, we survey the performance of the MLEs of the parameters of the CRTMBIII distribution with respect to sample size n.This performance is done based on the following simulation study: Step 1: Generate 1000 samples of size n from the CRTMBIII distribution based on the inverse cdf method.
Step 3: Compute the biases, mean squared errors and coverage probability of MLEs.For this purpose, we have selected different arbitrarily parameter values and n=50,100,150,200 sample sizes.All codes are written in R and the results are summarized in Table 3.The result clearly shows that when the sample size increases, the mean square errors (MSEs) decrease.This shows the consistency of MLE estimators.

APPLICATIONS
We consider an application to data set such as strengths of 1.5 cm glass fibers for authentication of the flexibility, utility and potentiality of the CRTMBIII distribution.We compare the CRTMBIII distribution with TMBIII, MBIII, BIII, LL distributions.For selection of the optimum distribution, we compute the estimate of likelihood ratio statistics ( 2  ), Akaike information criterion (AIC), corrected Akaike information criterion (CAIC), Bayesian information criterion (BIC), Hannan-Quinn information criterion (HQIC), Cramervon Mises (W*), Anderson Darling (A*), and Kolmogorov-Smirnov [K-S] statistics with pvalues for all competing and sub distributions.We compute the MLEs and their standard errors (in parentheses).We also compute goodness of fit statistics (GOFs) values for the CRTMBIII, TMBIII, MBIII, BIII, LL models.

3(a))
for strengths of glass fibers data is negatively skewed.The TTT (total time on test) plot (Fig. 3(b)) for strengths of glass fibers data is concave, which infers increasing failure rate.So, the BIII-ME distribution is suitable to model these data.From the tables 4 and 5, it is clear that our proposed model is best fitted, with smallest values for all GOFs and maximum p-value.
Ali et al. (2015) studied modified Burr III (MBIII) distribution with its properties.Ali and Ahmad (2015) studied transmuted MBIII (TMBIII) distribution and its properties.The cdf and pdf of MBIII distribution are given, respectively, by

Fig. 1 Fig: 2
Fig.1 Plots of pdf of the CRTMBIII distribution for the selected parameter values

1 𝑆
The average remaining lifetime of a component at time z, say   1 mz, or life expectancy is known as mean residual life (MRL) function is given by  be a continuous random variable and let  be a continuous random variable and let

Figure 3
Figure 3 Boxplot (a) and TTT plot (b) for glass fiber data