Truncated Cauchy Power Weibull-G Class of Distributions: Bayesian and Non-Bayesian Inference Modelling for COVID-19 and Carbon Fiber Data

: The Truncated Cauchy Power Weibull-G class is presented as a new family of distributions. Unique models for this family are presented in this paper. The statistical aspects of the family are explored, including the expansion of the density function, moments, incomplete moments (IMOs), residual life and reversed residual life functions, and entropy. The maximum likelihood (ML) and Bayesian estimations are developed based on the Type-II censored sample. The properties of Bayes estimators of the parameters are studied under different loss functions (squared error loss function and LINEX loss function). To create Markov-chain Monte Carlo samples from the posterior density, the Metropolis–Hasting technique was used with posterior density. Using non-informative and informative priors, a full simulation technique was carried out. The maximum likelihood estimator was compared to the Bayesian estimators using Monte Carlo simulation. To compare the performances of the suggested estimators, a simulation study was carried out. Real-world data sets, such as strength measured in GPA for single carbon fibers and impregnated 1000-carbon fiber tows, maximum stress per cycle at 31,000 psi, and COVID-19 data were used to demonstrate the relevance and flexibility of the suggested method. The suggested models are then compared to comparable models such as the Marshall–Olkin alpha power exponential, the extended odd Weibull exponential, the Weibull–Rayleigh, the Weibull–Lomax, and the exponential Lomax distributions.


Introduction
In recent years, many authors have made a great effort to construct new families to extend existing well-known distributions and provide flexible classes to model data in various fields such as medical sciences and environmental, demographic, actuarial, and economic studies.Many generalized families were generated and extended to explain various phenomena in real data.Some examples of these families are beta-generated [1], gamma-generated [2], generalized Kumaraswamy [3], Marshall-Olkin alpha power-G [4], and odd Lomax-G [5], Type II half logistic-G in [6], transmuted odd Fréchet-G in [7], Type II exponentiated half logistic-G in [8], exponentiated M-G by [9], exponentiated truncated inverse Weibull-G in [10] among others.
A new wider family of distributions, called the odd Weibull-G (OW-G) family was created by [11] where  is a shape parameter and  is the vector of parameters of the parent distributions, (; ) and (; ) are the CDF and PDF of a baseline continuous distribution with  as parameter vector, respectively.Establishing new distributions is a hot topic in contemporary research.It offers more flexible distributions capable of modelling complicated data structures.
The Cauchy distribution is symmetric and has a very heavy tail, it is used in many applications such as econometrics, biological, spectroscopy, engineering studies, and theory of reliability.Many authors have shown the different aspects of generalization and extension of Cauchy distribution, for example, [12] provided a truncated Cauchy distribution, [13] introduced a Kumaraswamy-half-Cauchy distribution, and [14] presented the power-Cauchy negative-binomial distribution.
The main objective of this paper is: • To present a new, wider, and flexible, family of distributions based on the W-G family and TCP family.

•
The PDF of submodels of the suggested family can be decreasing, unimodal, right skewness, and symmetrically shaped.Additionally, the hazard function can be unimodal, U-shaped, J-shaped, increasing, decreasing, and constant.

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To investigate some of its statistical features, such as the quantile function, moments, incomplete moments and Rényi entropy.

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To discuss the statistical inference of the TCPW-G family by using the ML and Bayesian approaches.

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To conduct a simulation study to demonstrate the behavior of the parameters model.

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To provide better fits than some known models with favorable results for the TCPWE, TCPWR, and TCPWL models.
This family called the Truncated Cauchy Power Weibull-G (TCPW-G) family of distributions, which is created by inserting (1) into (3), with CDF and PDF, respectively, provided by (; , , ) = arctan 1 −  ( ; ) ( ; ) , ,  ∈ , ,  > 0, , ,  ∈ , ,  > 0. ( We will denote a random variable  with PDF (7) by ~(, , ).The reliability and hazard rate functions for the  −  family are, respectively, given by (; , , ) = 1 − F(; , , ), and (; , , ) = ( ; , , ) ( ; , , ) .The outline for this paper is as follows.Section 2 includes a valuable extension of the TCPW density as well as three sub-models.Section 3 investigates the basic statistical properties of the TCPW-G family.In Section 4, the family's unknown parameters are estimated using Bayesian and non-Bayesian approaches.Section investigates Type-II censoring techniques.In Section 6, Monte Carlo simulation is used to compare the MLE and Bayesian parameters of the TCPWE.In Section 7, we present two real-world examples to show the versatility of the TCPW-G family.
Thus, Equation (13) shows that the TCPW-G density is a mixture of the Exp-G densities with the parameter (( + 1) + ).Consequently, we can derive the statistical properties of the TCPW-G family from those of Exp-G.Similarly, the CDF TCPW families can also be represented as a linear combination as follows: where  ( ( ) ) () is the Exp-G CDF with power parameter (( + 1) + ).

Some Special Models of the TCPW-G Family
Among the many different distributions arising from the TCPW-G family of distributions, we now present three special cases using classic distributions as baselines.

Truncated Cauchy Power Weibull Lomax (TCPWL) Distribution
The TCPW family has a Lomax distribution as the baseline distribution, i.e., with and () = (1 + ) and the CDF and PDF of TCPWL and , , , , ,  > 0, (15) where  and  are two shape parameters.

Truncated Cauchy Power Weibull Exponential (TCPWE) Distribution
The TCPW family has an exponential distribution as the baseline distribution, i.e., with () = 1 −   () =  .The CDF and the PDF of the TCPWE model are provided through and where  is a scale parameter.
where  is a scale parameter.
From the Figures 1-3 we can note the PDF of TCPWL, TCPWE, and TCPWR can be decreasing, unimodal, right skewness and symmetric shaped.Additionally, the hazard function can be unimodal, U-shaped, J-shaped, increasing, decreasing, and constant.

Statistical Features
In this section, we will discuss some statistical features of the TCPW class of distributions.

Moments
This sub-section derives the ordinary-moment and moment-generating functions of the  family.The skewness, kurtosis, and expected lifetime of a device can be obtained from the moments.
We have now shown two formulas for the moment generating function.From Equation (13), we may obtain the first of the following formulas: where  ( ( ) ) () is the moment generating function of  ( ( ) ) .Consequently, we can easily calculate  () from the Exp-G generating function.The second formula for the moment generating function of the TCPW class of distributions is based on the baseline quantile function as , where  ( , ) =    ( ; )  .

Conditional Moments
Incomplete moments are commonly used to describe the Bonferroni and Lorenz curves, which are both essential in many domains such as reliability theory, finance, economics, and demography.The incomplete moments  of X denoted by  () for any real  > 0 is defined from (13) as where  , () =   ()  ( ) and  , () can be calculated numerically.The mean deviations provide essential information on population characteristics and have also been used in income fields and economics.If  has the TCPW family of distribution, the mean deviations about the mean () and the mean deviations about the median are defined by and respectively, where  () is the first complete moment given by ( 23) with  = 1.The Lorenz and Bonferroni curves of the TCPW class of distributions for a given probability  can be written as () = ̀  () and () = ̀  () , respectively, where ̀ = () and  = () is the QF of  at .
The  moment of the residual life is provided by where  , * =  , ∑   (−) .
The  moment of the reversed residual life is provided by

Rényi Entropy
The Rényi entropy is provided by ( > 0,  ≠ 1) Utilizing ( 7), the same procedure as the beneficial expansion ( 13) and some simplifications, we acquire where Thus, Rényi entropy of  family is provided by

Maximum Likelihood Estimation
Let  , . . .,  be a random sample of size  from the TCPW distribution.The loglikelihood function for  =(θ, , ) is the  × 1 vector of parameters, given by where  = The maximum likelihood estimation (MLEs) can be obtained numerically by solving the non-linear equations simultaneously: = = = 0. Iterative methods such as the Newton-Raphson type algorithms can be used.

Bayesian Estimation
The Bayesian estimation method depends on an informative prior; let  ,  , . . .,  be a random-sample, distributed TCPW-G family with unknown parameters for the TCPWE or TCPWR distribution.To calculate the Bayesian estimation, we choose a prior distribution that is the independent from the gamma distributions.The independent joint prior density function of , θ and  is supplied by The joint posterior density function of , θ, and  is obtained from likelihood and joint prior functions as follows:

𝑒 (
) . (34) Under the symmetric and asymmetric loss functions, the loss function is important in the Bayesian estimation method.Square error (SE) is a well-known symmetric loss function, which can also be called the posterior mean, and it is defined as LINEX loss is well-known as an asymmetric loss function, and it can be defined as See [16,17] for further details on Bayesian estimation.We obviously cannot obtain explicit formulations for the expectation of the marginal posterior distribution for each and every parameter.As a result, the Markov Chain Monte Carlo (MCMC) approach is used to approximate the value of integrals.Gibbs sampling and, more broadly, Metropolis within Gibbs samplers are key subclasses of MCMC approaches.The Gibbs approach reduces the combined posterior distribution into a whole conditional distribution for each parameter.From PBLJ, the posterior conditional PDF of  given ,  can be obtained as .
In a similar way, the posterior conditional PDF of given ,  are provided by .
The standard techniques of generating random numbers, however, make it difficult to produce random samples directly from the conditional Posterior distributions of , , .As a result, the Metropolis-Hastings (M-H) method with a normal proposal distribution may be used to generate random samples from these distributions.See [18] for further details on this algorithm.In addition, [19] explains how to extract the informative prior algorithms' hyper-parameters.

Censored Scheme
Reference [20] covers the two most frequent censoring techniques, which are referred to as Type-I and Type-II censoring schemes.A life test is ended after a given number of failures in Type-II censoring, where n and r are fixed and preset, but  =  is a random variable.For further information, see [21].The data consist of observations  <  <. . .<  <. . .<  and the information that ( − ) items survive beyond the time of termination T, where r is the number of the uncensored items.The likelihood function of the TCPWE distribution under Type-II censored sample can be written as This function can be used in the upper section to estimate parameters of this model under the Type-II censored sample.For more examples, see [22][23][24][25].

Numerical Outcomes
In this section, a Monte Carlo simulation is used to examine the MLE and Bayesian parameters of the TCPWE under full and Type-II censored data.We produced 10,000 random samples from the TCPWE distribution for various parameter settings.
For various sample sizes  = 25, 50 and 100, different ratios of censored sample of failures were obtained, i.e., 72% and 92%.We can define the best scheme as the scheme that minimizes the bias (A1) and the mean squared error (A2) of the estimator.We consider A1 and A2 in order to perform the necessary comparison between various methods of point estimation.The code of numerical results is in the Appendix A.
Tables 1-3 illustrate the simulated results of the point estimation techniques given inside this research, from which we can draw the following conclusions:

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As the sample size increases, the A1 and A2 of the parameters under consideration decrease.

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As the ratio of censored sample of failures increase, the value of the A1 and A2 are also decrease.1.
Figure 8. Plot A2 of  by results in Table 1.

Applications
We offer three real-world data applications in this section to demonstrate the significance of the TCPWE, TCPWL, and TCPWR models given in Section 2. We produce the MLEs of the model parameters, as well as goodness-of-fit statistics for any of these models, especially in comparison to those of other competing distributions, such as the Kumaraswamy-generalized Lomax (KGL) distribution, which was introduced by [26]; the Weibull-Lomax (WL), which was introduced by [27]; the Marshall-Olkin alpha power exponential (MOAPE), which was introduced by [4]; the extended odd Weibull exponential (EOWE), which was introduced by [28]; the Exponential Lomax (EL) distribution, which was introduced by [29]; and the Weibull-Rayleigh (WR) distribution, which was introduced by [30].To compare the fitted models, several measures are taken into account.The Akaike information criterion (AIC), the consistent AIC (CAIC), the Hannan-Quinn information criterion (HQIC), the Bayesian information criterion (BIC), and the accompanying p-value are recommended measurements.The smaller the value of these statistics, the better the match.The results of this models have been shown in Tables 4-6.

The First Data Set
The first true data set represents the strength of single carbon fibers impregnated at gauge lengths of 1, 10, 20, and 50 mm in GPa.Tows impregnated with 100 fibers were tested at gauge lengths of 20, 50, 150, and 300 mm.The second data set depicts the fatigue times of 6061-T6 aluminum coupons comprising 101 observations with maximum stress per cycle of 31,000 psi.See [4,31].The numerical values of the AIC, CAIC, BIC, HQIC, A*, W*, and K-S statistics are listed in Table 4.These results show that the TCPWE, TCPWL, and TCPWR models have the lowest KS values and p-values among all fitted models.

The Second Data Set
The second data set depicts the fatigue times of 6061-T6 aluminum coupons comprising 101 observations with maximum stress per cycle of 31,000 psi.See [4,31].The numerical values of the AIC, CAIC, BIC, HQIC, A*, W*, and K-S statistics are listed in Table 5.These results show that the TCPWE, TCPWL, and TCPWR models have the lowest KS values and p-values among all fitted models.
Figures 9-11 show estimated PDF with histogram, estimated CDF with empirical CDF, P-P plot and Q-Q plot for TCPWE, TCPWL, and TCPWR distributions, which these figures confirmed that the TCPWE, TCPWL, and TCPWR distributions have been fitted for strength data, fatigue times data and COVID-19 data, respectively.To confirm these estimates are maximum, we plot likelihood profile for parameters of TCPWE distribution, where estimate has maximum log-likelihood value, see Figures 12-14.To also check existence, we plot the contour plot by two parameters for log-likelihood of the TCPWE distribution, see Figures 15-17.We plot likelihood profile for parameters of TCPWL distribution, where the estimate has the maximum log-likelihood value in Figure 18.

Conclusions
This article formulated a new family that has more attractive and flexible distributions and greatly improves the modelling ability of their baseline distributions in the description of real-life events such as carbon and COVID-19 data.A new, flexible extension to the odd Weibull-G family based on a truncated Cauchy power generator was provided.Based on Truncated Cauchy Power Weibull and some classic distribution baselines such as Lomax, exponential, and Rayleigh distributions, this paper introduced three new distributions.Some statistical properties of the new family were provided.Complete and censored samples were considered under this model.In censored samples, the greater the number of censored sample units, the lower the MSE and the bias.Bayesian and MLE methods of estimation were also considered.LINEX and SE loss functions were deduced for Bayesian estimators based on different loss functions.A Monte Carlo simulation study was designed to assess the performance of estimates.Generally, we concluded that the Bayesian estimates are preferable to the corresponding other estimates in most of the situations.A new distribution based on the Lomax distribution was the best model to fit different data sets such as those for carbon fibers and COVID-19.

Figure 1 .
Figure 1.Plots of PDF and the hazard function of TCPWL distribution.

Figure 2 .
Figure 2. Plots of PDF and the hazard function of TCPWE distribution.

Figure 3 .
Figure 3. Plots of PDF and the hazard function of TCPWR distribution.

Figures 4 and 5
show the skewness and kurtosis plots for TCPWE distribution with different values of parameters when  = 2 and  = 5.

Figure 4 .
Figure 4. Skewness plot for TCPWE distribution with different values of parameters when  = 2 and  = 5.

Figure 5 .
Figure 5. Kurtosis plot for TCPWE distribution with different values of parameters when  = 2 and  = 5.

Figure 9 .
Figure 9.The estimated CDF, fitted PDF, PP-plot, and QQ-plot of the TCPWE and TCPWL and TCPWR distributions for strength data.

Figure 10 .
Figure 10.The estimated CDF, fitted PDF, PP-plot, and QQ-plot of the TCPWE and TCPWL and TCPWR distributions for fatigue times.

Figure 12 .
Figure 12.Likelihood profile for parameters of TCPWE distribution for strength data.

Figure 13 .
Figure 13.Likelihood profile for parameters of TCPWE distribution for fatigue times.

Figure 14 .
Figure 14.Likelihood profile for parameters of TCPWE distribution for COVID-19 data.

Figure 15 .
Figure 15.Contour plot for log-likelihood of the TCPWE distribution for strength data.

Figure 16 .
Figure 16.Contour plot for log-likelihood of the TCPWE distribution for fatigue times.

Figure 17 .
Figure 17.Contour plot for log-likelihood of the TCPWE distribution for COVID-19 data.

Figure 18 .
Figure 18.Likelihood profile for parameters of TCPWL distribution for COVID-19 data.

•
For most TCPWE distribution parameters, Bayesian estimates are more efficient than MLE.• Bayesian estimates based on linex (1.5) loss function have more relative efficiency than other loss functions for most parameters of TCPWE distribution.• when actual value of  increases the A1 and A2 of the considered parameters decreases for , .• when actual value of  increases the A1 and A2 of the considered parameters decreases for ,  and increase for .• when actual value of  increases the A1 and A2 of the considered parameters decreases for  and increases for , .

Table 1 .
MLE and Bayesian estimate for TCPWE model at complete sample.

Table 2 .
MLE and Bayesian estimate for TCPWE model at Type-II censored sample at 72%.

Table 3 .
MLE and Bayesian estimate for TCPWE model at Type-II censored sample at 92%.
shows the numerical values of A2 for the parameters , ,  in Table1.Figure 6.Plot A2 of  by results in Table1.

Table 4 .
Estimates, SE, goodness-of-fit test by using KS and different criteria measures for strength data.

Table 5 .
Estimates, SE, goodness-of-fit test by using KS and different criteria measures for fatigue times.

Table 6 .
Estimates, SE, goodness-of-fit test by using KS and different criteria measures for COVID-19 data.