The Additive Xgamma-Burr XII Distribution: Properties, Estimation and Applications

: This paper introduces a new four-parameter additive model, named xgamma-Burr XII distribution, by considering two competing risks: the former has the xgamma distribution and the latter has the Burr XII distribution. A graphical description of the xgamma-Burr XII distribution is presented, including plots of the probability density function, hazard rate and reversed hazard rate functions. The xgamma-Burr XII density has different shapes such as decreasing, unimodal, approximately symmetric and decreasing-unimodal. The main statistical properties of the proposed model are studied. The unknown model parameters, reliability, hazard rate and reversed hazard rate functions are estimated via the maximum likelihood method. The asymptotic confidence intervals of the parameters, reliability function, hazard rate function and reversed hazard rate function are also obtained. A simulation study is carried out to evaluate the performance of the maximum likelihood estimates. In addition, three real data are applied to show the superiority of the xgamma-Burr XII distribution over some known distributions in real-life applications.

Furthermore, the importance of the competing risks approach stems from the fact that the resulting model has a flexible hazard rate function (hrf), which accommodates a bathtub shape and more complex shapes.Based on the competing risks approach, the resulting model is known as a competing risks model, series model, additive model and multi-risk model.
In this paper a new competing risks model, called xgamma-BXII (Xg-BXII) distribution, is derived by considering a series system with two components functioning independently in series.The lifetime of the first component,  1 , has the xgamma (Xg) distribution and the lifetime of the second component,  2 , has the BXII distribution.Therefore, the lifetime of the system is  = min{ 1 ,  2 } has the Xg-BXII distribution.The importance of this distribution seems to be from its pdf and hrf which display high flexibility and diversity in shape.The Xg distribution hrf has two main shapes-bathtub and modified bathtub-that are very important in reliability analyses.These shapes increase the applicability of the proposed distribution for lifetime data modeling.Moreover, the proposed distribution has some new additive models as special cases, which have not been introduced in the statistical literature, as well as some special cases that are well-known models.
The rest of the paper is organized as follows: The construction of the proposed model and a graphical description of its main functions is given in Section 2. In Section 3, some important statistical properties are studied.The maximum likelihood (ML) approach is applied in Section 4 to estimate the unknown model parameters, the reliability and hazard rate function.In Section 5, a simulation study is performed to assess the efficiency of the derived estimators.Finally, four applications of real-life data are given in Section 6.

The Model
In this section, the construction of the proposed model based on the concept of competing risks is obtained.Graphical description of the pdf, hrf and reversed hrf (rhrf) is introduced.In addition, interpretation of the behavior of the hrf is presented.

Model Construction
The rf of the Xg-BXII distribution can be obtained as follows: where  = (α, , ) is a parameter vector.The corresponding cumulative distribution function (cdf) of the Xg-BXII distribution is: The hrf of the Xg-BXII distribution can be obtained as a sum of the two hrfs of the Xg and BXII distributions as follows: Therefore, the probability density function (pdf) of Xg-BXII distribution can be derived as: In addition, the rhrf and the cumulative hrf (chrf) of the Xg-BXII distribution are obtained as: and ] + ,  > 0;  > 0. (10)

Graphical Description
This subsection is devoted to present graphically the pdf, hrf and rhrf of the Xg-BXII distribution to show the model flexibility compared to the two distributions, namely Xg and BXII, of which the proposed model consists of them.
Figure 1 exhibits the pdf of the Xg-BXII distribution for different values of the parameters.It can be observed from Figure 1 that the pdf of the Xg-BXII distribution can be decreasing, unimodal, approximately symmetric and decreasing-unimodal. Figure 2 presents the hrf of the Xg-BXII distribution for some values of the parameters.The hrf of the Xg-BXII distribution displays two major failure shapes: bathtub shape and a generalized version of this, called the modified bathtub shape.Moreover, the hrf of the Xg-BXII distribution also displays an increasing and decreasing shape.Plots of the rhrf of the Xg-BXII distribution for some selected values of the parameters are given in Figure 3.The rhrf of the Xg-BXII distribution is decreasing.

Behavior of the Hazard Rate Function
Limiting behavior of the hrf, ℎ (; ), and its first derivative with respect to  are considered for studying the behavior of the hrf of the Xg-BXII distribution, which is shown in Figure 2. and The behavior of the hrf of the Xg-BXII distribution for different values of  can be characterized as: and lim →∞ ℎ (; ) = , in this case ℎ 1 (; ) is an increasing hrf whereas ℎ 2 (; , ) is unimodal hrf.Therefore, two shapes of the hrf can be demonstrated: i.

Statistical Properties
In this subsection some main properties of the Xg-BXII distribution are studied including: the quantile function, mode, central and non-central moments, moment generating function,  ℎ incomplete moment and inequality curves, mean residual life (MRL) and mean inactivity time (MPL), mean time to failure (MTTF), mean time between failures (MTBF) and availability (Av), the order statistics and some new and well-known sub-models of the proposed distribution.

Quantile Function and the Mode
The quantile function of the Xg-BXII distribution can be obtained by inverting: So, the quantile function of the Xg-BXII distribution can be obtained by solving the following nonlinear equation: As special cases of the quantile function the median of the Xg-BXII distribution, denoted by   , the first quartile, denoted by  0.25 , and the third quartile, denoted by  0.75 , can be obtained, respectively, by setting  = 0.5,  = 0.25 and  = 0.75 into (12) and solving numerically.
The mode of a random variable  has the Xg-BXII distribution is the value of  0 which maximizes  (; ).Hence, the mode of the Xg-BXII distribution can be obtained by solving the following nonlinear equation: Derivatives of the mode of the Xg-BXII distribution are given in Appendix A. Some numerical results of the first quartile,  0.25 , median,   , and the third quartile,  0.75 , as special cases of the quantile function and the mode of the Xg-BXII distribution for different parameter values  = (, , ) using R software are presented in Table 1.From this table, it is obvious that the Xg-BXII distribution has a unimodal or non-modal pdf, which is shown clearly in Figure 1.] , = 1,2, . .., where By substituting  = 1 and  = 2 into (14), the mean and the second non-central moment of the Xg-BXII distribution can be obtained as follows: ], where ], where The variance of the Xg-BXII distribution can be obtained using (15) and (16) in the following equation: The coefficient of variation (CV), the coefficient of skewness (CS) and the coefficient of kurtosis (CK) are given, respectively, by and where , ́2 and  2 are obtained, respectively, in (15), ( 16) and ( 17) and ́3 and ́4 can be obtained, respectively, by setting  = 3 and  = 4 into (14).
Numerical results of the first four non-central moments, variance, CV, CS and CK of the Xg-BXII distribution for some parameter values are listed in Table 2. Furthermore, the moment generating function, denoted by   (), of a random variable  has the Xg-BXII distribution can be obtained as given below: where ́ is given in (14).

Incomplete Moments and Inequality Curves
The  ℎ incomplete moment of a random variable  has the Xg-BXII distribution is given by: where  (  ) (. , . ) is a lower incomplete beta function and Lorenz and Bonferroni curves are well-known inequality curves that have been extensively used in different fields such as economics, demography, insurance, reliability analysis and life testing.These curves are important applications of the first incomplete moment.Lorenz and Bonferroni curves are denoted, respectively, by   () and   () which are defined by: and where  is obtained from (15), () is the first incomplete moment which can be obtained by substituting  = 1 and  =  into (22) and  =  −1 () for 0 <  < 1.

The Mean Residual Life and the Mean Inactivity Time
The MRL function or the life expectation at age , denoted by (), which represents the expected additional life length for a system or a unit which is alive at age , is defined by: For the Xg-BXII distribution the MRL is given by: where  (  ) (. , . ) is a lower incomplete beta function and The MPL, mean inactivity time or mean waiting time, also called the mean reversed residual life function, denoted by (), which represents the waiting time elapsed since the failure of a system or a unit on the condition that this failure had occurred in (0, ), is defined by: For a random variable  has the Xg-BXII distribution, the MPL is given by: where  (  ) (. , . ) is a lower incomplete beta function and

Mean Time to Failure, Mean Time between Failures and Availability
The MTTF, MTBF and the Av are reliability terms for predicting the lifecycle of products.They are ways for providing numeric results to quantify a failure rate and the resulting time of expected performance based on a set of data.In addition, for designing and manufacturing a maintainable system, it is necessary to predict the MTTF, MTBF and Av.Additionally, customers can use these reliability terms when deciding what product to buy.
For Xg-BXII the MTTF and MTBF are defined, respectively, by: and The Av is the probability that a product is successful at time  0 and is defined as: [See, [30]].

Sub-Models of the Xg-BXII Distribution
In this subsection several new and well-known distributions are obtained as special cases of the Xg-BXII distribution.Table 3 summarizes these sub-models.Table 3. Sub-models of the Xg-BXII distribution.

Maximum Likelihood Estimation
In this subsection, the ML estimators of the parameters, rf and hrf are derived.In addition, ACIs (asymptotic confidence intervals) of the parameters, rf and the hrf are obtained.a. Point estimation Suppose that  1 ,  2 , . . .,   is a random sample of size  from the Xg-BXII distribution with parameter vector  = (, , ), then the likelihood function is given by: ) The natural logarithm of the likelihood function is 2 ) By differentiating the log-likelihood function in (37) with respect to the parameters ,  and  as follows: ) ) and ) ) The ML estimates of the parameters  = (, , ) can be obtained by equating (38) − (40) to zeros then solving numerically.
The ML estimators of  (; ) and ℎ (; ) can be obtained, using the invariance property of the ML estimators, by replacing the parameters  = (, , ) in ( 5) and (7) with their ML estimators, then the ML estimators of  (; ) and ℎ (; ) can be given, respectively, as: and b. Asymptotic confidence intervals To obtain the confidence intervals for the parameters  = (, , ) of the Xg-BXII distribution, the distributions of the ML estimators  ̂= ( ̂, ,  ̂) are needed.Since the ML estimators  ̂= ( ̂, ,  ̂) do not have closed form, their exact distribution cannot be obtained.Therefore, the ACIs can be derived by using the asymptotic distribution of the ML estimators.The ML estimators are asymptotically normal with mean (, , ) and the asymptotic variance-covariance matrix of the estimators is obtained depending on the inverse asymptotic Fisher information matrix.The asymptotic Fisher information matrix can be written as given below: where  1 = ,  2 =  and  3 = .Therefore, the (1 − )100% bounds of the ACIs of the parameters  = (, , ) are as follows: where  ̂ in this paper is  ̂, ,  ̂,  ̂(;  ̂) or ℎ ̂(;  ̂), and  ̂ ̂ is the standard deviation.

Simulation
This section is devoted to evaluating the performance of the ML estimates of the parameters, rf, hrf and rhrf of the Xg-BXII distribution through a simulation study as follows: a.The simulation study is conducted using two sets of parameters: f.Tables 4-9 are graphically displayed in Figures 4-9.

Concluding remarks:
Based on the simulation tables and figures, one can conclude that:

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The ML averages of the estimates for the parameters of the Xg-BXII distribution are stabilized as the sample size  increases.

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The ERs and REs of the ML estimates of the parameters  = (, , ), rf, hrf and rhrf decrease, as the sample size increases.

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In most cases, the RABs of the ML estimates of the parameters  = (, , ), rf, hrf and rhrf decrease as the sample size increases.

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As the sample size increases, the variances of the parameters, rf, hrf and rhrf decrease.

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The lengths of the 95% ACIs of the parameters, rf, hrf and rhrf decrease as the sample size increases.

Applications
In this section the applicability of the Xg-BXII distribution is conferred.Three applications are used to demonstrate the superiority of the Xg-BXII distribution over some existing distributions namely, LogL-W, L-W, AW, two-parameter Xg (TXg), Xg and BXII distributions.
The ML estimates of the parameters and their standard errors (SE), Kolmogorov-Smirnov (K-S) statistic and its corresponding p-value, the −2 log likelihood statistic (−2ℒ), Akaike information criterion (AIC), Bayesian information criterion (BIC) and corrected Akaike information criterion (CAIC), are used to compare the fit of the competitor distributions, where where ℒ is the natural logarithm of the value of the likelihood function evaluated at the ML estimates,  is the number of the observations and  is the number of the estimated parameters.
The best distribution corresponds to the lowest values of AIC, BIC and CAIC, also the highest p-values.

Application 1:
This application is given by [3] and represents the time to failure data of 18 electronic devices.The data are tabulated in Table 10 and a summary of measures for these data is given in Table 11.

Empirical scaled TTT-transform Boxplot Histogram
Table 12 displays the ML estimates of the parameters along with their SEs, K-S statistic and its corresponding p-value, AIC, BIC and CAIC of Wang's data for the competitor distributions.The results of this table show that the Xg-BXII distribution has the lowest K -S statistic and its corresponding p-value is the highest one.This indicates that the proposed distribution provides the best fit for Wang's data in comparison to the other competitors.Furthermore, the Xg-BXII distribution has the lowest values of AIC, BIS, and CAIC compared to the other competitor distributions.
In Figure 12, the histogram of Wang's data and the fitted pdfs and empirical cdf versus the fitted cdfs of the competitor models are presented.In addition, P-P plots of the competitor models are given in Figure 13.From these figures, it can be concluded that the Xg-BXII distribution provides the best fit for these data.This application is given by [32] and represents the time to failure data of 30 devices.The data are tabulated in Table 13 and a summary of measures for these data is obtained in Table 14.The empirical scaled TTT-transform plot, boxplot and the histogram of Meeker and Escober's data are presented in Figure 14.The empirical scaled TTT-transform plot indicates that the data have a bathtub hrf.Furthermore, Figure 15   ML estimates of the parameters along with their SEs, K-S statistic and its corresponding p-value, AIC, BIC and CAIC of Meeker and Escober's data for the competitor distributions are tabulated in Table 15.From this, it is obvious that the Xg-BXII distribution has the lowest K-S statistic and the highest p-value compared to the other competitors.Moreover, the Xg-BXII distribution has the lowest values of AIC, BIS, and CAIC in comparison with the competitor distributions.The results of Table 15   The estimated asymptotic variance-covariance matrix of the ML estimates for the Xg-BXII distribution for Meeker and Escober's data is as follows:   This application represents the lifetime of 20 electronic components given by [33].The data are tabulated in Table 16, while Table 17 presents a summary of measures for these data.Table 18 presents the ML estimates of the parameters along with their SEs, K-S statistic and its corresponding p-value, AIC, BIC and CAIC of Murthy's data for the competitor distributions.From Table 18, the Xg-BXII distribution has the lowest K-S statistic and the highest p-value compared to the other competitors.In addition, the Xg-BXII distribution has the lowest values of AIC, BIS, and CAIC in comparison with the competitor distributions.A graphical comparison of the Xg-BXII distribution and the other competitor distributions is presented in Figures 20 and 21 19 and Table 20 presents a summary of measures for these data.

Conclusions
In this paper, a new three-parameter competing risks model, called Xg-BXII distribution, is introduced by combining Xg and BXII distributions in a series system with two components functioning independently.The pdf of the proposed model displays unimodal and decreasing-unimodal shapes, whereas the hrf exhibits important shapes: bathtub and modified bathtub shapes.These shapes increase the applicability of the proposed distribution for lifetime data modeling.Moreover, the proposed distribution has some new additive models as special cases these models have not been introduced in the statistical literature.In addition, it has some well-known models as special cases.Several statistical properties of the proposed model are derived.The ML estimators of the parameters, rf, hrf and rhrf are presented.Moreover, The ACIs of the parameters, rf, hrf and rhrf are obtained.The performance the ML estimates is evaluated through a simulation study.Furthermore, four applications are used to demonstrate the applicability of Xg-BXII distribution over some existing distributions.Xg-BXII distribution provides the best fitting compared with the used competitor distributions.(1 + )(1 +   )   −  ∞ 0 .
Since the power series expansion of  − is as follows:

Figure 10 presents
Figure10presents the empirical scaled TTT-transform plot, boxplot and the histogram of Wang's data.From the empirical scaled TTT-transform plot the data have a bathtub hrf and from the boxplot and the histogram one can say that the data have a rightskewed pdf.Moreover, Figure11exhibits the fitted pdf, cdf and hrf plots of the Xg-BXII distribution and P-P plot of Wang's data.It is obvious that the fitted pdf of the Xg-BXII distribution has a decreasing-unimodal and right-skewed shape and the fitted hrf is a bathtub shape.From the empirical and theoretical cdf plot and the P-P plot one can conclude that the proposed distribution fits Wang's data very well.

Figure 11 .
Figure 11.Empirical and theoretical density and cdf plots, fitted hrf and P-P plot of Wang's data.

Figure 12 .
Figure 12.Histogram of Wang's data and the fitted pdfs and empirical cdf versus the fitted cdfs of the competitor models.

Figure 13 .
Figure 13.P-P plot of the competitor models for Wang's data.The estimated asymptotic variance-covariance matrix of the ML estimates for the Xg-BXII distribution for Wang's data is as follows:

Figure 14 .
Figure 14.The empirical scaled TTT-transform, boxplot and histogram of Meeker and Escober's data.

Figure 15 .
Figure 15.Empirical and theoretical density and cdf plots, fitted hrf and P-P plot of Meeker and Escober's data.
are confirmed by a visual comparison of the histogram of Meeker and Escober's data and the fitted pdfs and empirical cdf versus the fitted cdfs and the P-P plot of the competitor models are presented in Figures 16 and 17.From these figures, it can be concluded that the Xg-BXII distribution provides a better fit for these data compared to the other considered distributions.

Figure 16 .
Figure 16.Histogram of Meeker and Escober's data and the fitted pdfs and Empirical cdf versus the fitted cdfs of the competitor models.

Figure 17 .
Figure 17.P-P plot of the competitor models for Meeker and Escober's data.Application 3:This application represents the lifetime of 20 electronic components given by[33].The data are tabulated in Table16, while Table17presents a summary of measures for these data.

Figure 19 .
Figure 19.Empirical and theoretical density and cdf plots, fitted hrf and P-P plot of Murthy's data. .
Figure 20 displays the histogram of Murthy's data and the fitted pdfs and empirical cdf versus the fitted cdfs and the P-P plot of the competitor models is shown in Figures 21.From these figures, it can be concluded that the Xg-BXII distribution presents a very good fit for these data compared to the other considered distributions.

Figure 20 .
Figure 20.Histogram of Murthy's data and the fitted pdfs and empirical cdf versus the fitted cdfs of the competitor models.

Figure 21 .
Figure 21.P-P plot of the competitor models for Murthy's data.Application 4:This application is from[33].The application represents COVID-19 data belonging to the United Kingdom of 76 days, from 15 April to 30 June 2020.The data are formed of

Figure 22 Figure 22 .
Figure 22 exhibits the empirical scaled TTT-transform plot, boxplot and the histogram of COVID-19 data in the United Kingdom.The empirical scaled TTT-transform plot indicates that the data have a modified bathtub hrf and from the boxplot and the histogram it is shown that the data have a right-skewed pdf.Empirical scaled TTT-transform Boxplot Histogram

Figure 23 .
Figure 23.Empirical and theoretical density and cdf plots, fitted hrf and P-P plot of the United Kingdom's data.Table21exhibits the ML estimates of the parameters along with their SEs, K-S statistic and its corresponding p-value, AIC, BIC and CAIC of COVID-19 data in the United Kingdom for the competitor distributions.From Table21, the Xg-BXII distribution has the lowest K-S statistic and the highest p-value compared to the other competitors.In addition, the Xg-BXII distribution has the lowest values of AIC, BIS, and CAIC in comparison with the competitor distributions.A graphical comparison of the Xg-BXII distribution and the other competitor distributions is presented in Figures24 and 25.Figure24displays the histogram of the United Kingdom's data and the fitted pdfs and empirical cdf versus the fitted cdfs, and the P-P plot of the competitor models is shown in Figures 25.From these figures, it can be concluded that the Xg-BXII distribution provides a better fit for these data compared to the other considered distributions.The estimated asymptotic variance-covariance matrix of the ML estimates for the Xg-BXII distribution for the United Kingdom's data is as follows: .
Figure 24 displays the histogram of the United Kingdom's data and the fitted pdfs and empirical cdf versus the fitted cdfs, and the P-P plot of the competitor models is shown in Figures 25.From these figures, it can be concluded that the Xg-BXII distribution provides a better fit for these data compared to the other considered distributions.The estimated asymptotic variance-covariance matrix of the ML estimates for the Xg-BXII distribution for the United Kingdom's data is as follows: 0045 0.0284 −0.0029 −0.0095 −0.0029 0.0211 ) Therefore, the 95% ACI bounds of  = (, , ) are, respectively: 1.6585 ± 1.96√0.2356,1.4737 ± 1.96√0.1463,and 0.7792 ± 196√0.3247.

Figure 24 .
Figure 24.Histogram of the United Kingdom's data and the fitted pdfs and empirical cdf versus the fitted cdfs of the competitor models.

Figure 25 .
Figure 25.P-P plot of the United Kingdom's data.

Table 1 .
Some quartiles and the mode of the Xg-BXII distribution for different parameters values.

Table 2 .
Moments of the Xg-BXII distribution for different parameters values.

Table 10 .
Failure times of Wang's data.

Table 11 .
Summary measures of Wang's data.

Table 12 .
ML estimates and their relevant SEs, K-S statistics, p-values, AIC, BIC and CAIC of the fitted models of Wang's data.

Table 13 .
Failure times of Meeker and Escober's data.

Table 14 .
Summary measures of Meeker and Escober's data.

Table 15 .
ML estimates and their relevant SEs, K-S statistics, p-values, AIC, BIC and CAIC of the fitted models of Meeker and Escober's data.

Table 16 .
Failure times of Murthy's data.

Table 17 .
Summary of measures of Murthy's data.

Table 18 .
ML estimates and their relevant SEs, K-S statistics, p-values, AIC, BIC and CAIC of the fitted models of Murthy's data.

Table 19 .
Drought mortality rates of the United Kingdom's data.

Table 20 .
Summary measures of the United Kingdom's data.

Table 21 .
ML estimates and their relevant SEs, K-S statistics, p-values, AIC, BIC and CAIC of the fitted models of the United Kingdom's data.