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

Abstract

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.

1. Introduction

In literature, several statistical distributions are beneficial in analyzing and predicting different types of real lifetime data in many areas such as economics, finance, insurance, engineering, electronic sciences, hydrology, human mortality studies and biological surveys. Therefore, different shapes of lifetime distributions are required for fitting these types of lifetime data, but in many situations, classical distributions do not present suitable fits to real data. Thus, there is a necessity for a flexible family of distributions, and several generated classes of probability distributions have been recommended. Therefore, various approaches for generating new distributions, generalizing and extending existing distributions have been suggested to provide more flexibility than the existing distributions. Some of these approaches are methods of differential equation: the Pearson system and Burr system, the method of adding parameters, the method of transformation of variables and distribution functions, probability integral transforms, compound distributions, finite and infinite mixture distributions [1].

Another method for constructing new lifetime distributions is the competing risks approach. The competing risks approach depends on the concept of competing risks which appears in life-testing experiments, where the failure of the tested item may be ascribable to more than one cause or mode of failure. These failure modes in some sense compete to cause the failure of the tested item. Due to this reason, in statistical literature this is well known as competing risks. Competing risks often occurred in reliability studies, demographic, medical and biological sciences and engineering 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.

Based on the concept of the competing risks, there are many lifetime distributions that have been introduced in literature, such as the additive Weibull (AW) distribution that was introduced by [2]. The additive Burr XII (BXII) distribution was presented by [3]. Ref. [4] derived a new modified Weibull distribution. Ref. [5] constructed the exponential-Weibull distribution. Ref. [6] obtained the additive modified Weibull distribution. Ref. [7] proposed the log-logistic Weibull (LogL–W) distribution. Ref. [8] suggested the additive Perks–Weibull distribution. Ref. [9] derived the BXII modified Weibull distribution. Ref. [10] introduced Weibull–Chen distribution. Ref. [11] proposed the log-normal modified Weibull distribution. Ref. [12] constructed the Lomax–Weibull (L–W) distribution. Ref. [13] presented the flexible Weibull extension–BXII distribution. Ref. [14] introduced the additive Chen–Weibull distribution. Ref. [15] presented the flexible additive Weibull distribution by combining three Weibull distributions. Ref. [16] proposed the Lindley–BXII distribution. Ref. [17] introduced the flexible additive Chen–Gompertz distribution. Ref. [18] proposed the additive power-transformed half-logistic model. Ref. [19] presented the three-component additive Weibull distribution. Recently, Ref. [20] introduced the additive flexible Weibull extension–Lomax distribution. More recently, Ref. [21] presented the additive Perks distribution. Most recently, Ref. [22] proposed the additive Chen distribution. In addition, the additive Chen–Perks distribution was introduced by [23].

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, $X_1$, has the xgamma (Xg) distribution and the lifetime of the second component, $X_2$, has the BXII distribution. Therefore, the lifetime of the system is $X=\min \left\{X_1,X_2\right\}$ 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 xgamma (XG) distribution was derived by [24] using a finite gamma and exponential distribution mixture to model lifetime data sets. They investigated many useful features of the xgamma distribution since they studied some statistical properties and observed that in many cases the XG distribution has more flexibility than the exponential distribution. [See [25,26,27,28]]. The reliability function (rf) and the hrf of Xg distribution are given, respectively, by:

$$R_1\left(x;\alpha \right)=\frac{1+\alpha +\alpha x+\frac{{\alpha }^2{x}^2}{2}}{1+\alpha }{e}^{-\alpha x}, x>0, \alpha >0 ,$$

(1)

and

$$h_1\left(x;\alpha \right)=\frac{{\alpha }^2\left(1+\frac{\alpha x^2}{2}\right)}{1+\alpha +\alpha x+\frac{{\alpha }^2{x}^2}{2}}, x>0; \alpha ,>0.$$

(2)

The Burr XII distribution was proposed by [29] beside 11 other types of Burr distributions. The Burr XII is one of the important distributions which can be used for modeling real data due to its good statistical and reliability properties. It is applied in industry, medical sciences, business, finances, physics, hydrology, quality control, chemical engineering reliability, survival analysis and acceptance sampling plans. The rf and the hrf of BXII distribution are given, respectively, by:

$$R_2\left(x;c,k\right)={\left(1+x^c\right)}^{-k}, x>0; c,k>0,$$

(3)

and

$$h_2\left(x;c,k\right)=\frac{ck{x}^{c-1}}{\left(1+x^c\right)}, x>0; c,k>0.$$

(4)

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.

2. 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.

2.1. Model Construction

The rf of the Xg-BXII distribution can be obtained as follows:

$$R\left(x;\underset{¯}{\psi }\right)=\prod _{i=1}^2{R}_i\left(x\right)=\frac{1+\alpha +\alpha x+\frac{{\alpha }^2{x}^2}{2}}{\left(1+\alpha \right){\left(1+x^c\right)}^k}{e}^{-\alpha x},\hspace{1em}\hspace{1em}x>0, \underset{¯}{\psi }>\underset{¯}{0},$$

(5)

where $\underset{¯}{\psi }=\left(\mathsf{\alpha },c,k\right)$ is a parameter vector.

The corresponding cumulative distribution function (cdf) of the Xg-BXII distribution is:

$$F\left(x;\underset{¯}{\psi }\right)=1-R\left(x;\underset{¯}{\psi }\right)=1-\frac{1+\alpha +\alpha x+\frac{{\alpha }^2{x}^2}{2}}{\left(1+\alpha \right){\left(1+x^c\right)}^k}{e}^{-\alpha x}, x>0; \underset{¯}{\psi }>\underset{¯}{0}.$$

(6)

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:

$$\begin{array}{l}h\left(x;\underset{¯}{\psi }\right)=h_1\left(x\right)+h_2\left(x\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{{\alpha }^2\left(1+\frac{\alpha x^2}{2}\right)}{1+\alpha +\alpha x+\frac{{\alpha }^2{x}^2}{2}}+\frac{ck{x}^{c-1}}{1+x^c}, x>0; \underset{¯}{\psi }>\underset{¯}{0}.\end{array}$$

(7)

Therefore, the probability density function (pdf) of Xg-BXII distribution can be derived as:

$$\begin{array}{l}f\left(x;\underset{¯}{\psi }\right)=h\left(x;\underset{¯}{\psi }\right)R\left(x;\underset{¯}{\psi }\right)\\ =\left[{\alpha }^2\left(1+\frac{\alpha x^2}{2}\right)+\frac{ck{x}^{c-1}\left(1+\alpha +\alpha x+\frac{{\alpha }^2{x}^2}{2}\right)}{\left(1+x^c\right)}\right]\frac{{\left(1+x^c\right)}^{-k}{e}^{-\alpha x}}{1+\alpha },\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} x>0, \underset{¯}{\psi }>\underset{¯}{0}.\end{array}$$

(8)

In addition, the rhrf and the cumulative hrf (chrf) of the Xg-BXII distribution are obtained as:

$$\begin{gathered} r\left(x;\underset{¯}{\psi }\right)=\frac{f\left(x;\underset{¯}{\psi }\right)}{F\left(x;\underset{¯}{\psi }\right)}=\frac{\left[{\alpha }^2\left(1+\frac{\alpha x^2}{2}\right)+\frac{ck{x}^{c-1}\left(1+\alpha +\alpha x+\frac{{\alpha }^2{x}^2}{2}\right)}{1+x^c}\right]}{\left(1+\alpha \right){\left(1+x^c\right)}^k{e}^{\alpha x}-\left(1+\alpha +\alpha x+\frac{{\alpha }^2{x}^2}{2}\right)}, \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}x>0; \underset{¯}{\theta }>\underset{¯}{0}, \end{gathered}$$

(9)

and

$$H\left(x;\underset{¯}{\psi }\right)=-\ln R\left(x;\underset{¯}{\psi }\right)=\ln \left[\frac{\left(1+\alpha \right){\left(1+x^c\right)}^k}{1+\alpha +\alpha x+\frac{{\alpha }^2{x}^2}{2}}\right]+\alpha x, x>0; \underset{¯}{\psi }>\underset{¯}{0}.$$

(10)

2.2. 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.

2.3. Behavior of the Hazard Rate Function

Limiting behavior of the hrf, $h\left(x;\underset{¯}{\psi }\right)$, and its first derivative with respect to $x$ are considered for studying the behavior of the hrf of the Xg-BXII distribution, which is shown in Figure 2.

Using $\left(7\right)$, the first derivative of $h\left(x;\underset{¯}{\psi }\right)$ with respect to $x$ is given as:

$$\acute{h}\left(x;\underset{¯}{\psi }\right)=\frac{dh\left(x;\underset{¯}{\psi }\right)}{dx}={\acute{h}}_1\left(x;\alpha \right)+{\acute{h}}_2\left(x;c,k\right),$$

where

$${\acute{h}}_1\left(x;\alpha \right)={\alpha }^3\left[\frac{x\left(1+\frac{\alpha x}{2}\right)-1}{{\left(1+\alpha +\alpha x+\frac{{\alpha }^2{x}^2}{2}\right)}^2}\right],$$

and

$${\acute{h}}_2\left(x;c,k\right)=\frac{ck\left(c-1\right)x^{c-2}}{\left(1+x^c\right)}-\frac{c^{2}k{x}^{2c-2}}{{\left(1+x^c\right)}^2}.$$

The behavior of the hrf of the Xg-BXII distribution for different values of $c$ can be characterized as:

▪

For $c>1$, $\underset{x\to 0}{\lim }h\left(x;\underset{¯}{\psi }\right)=\frac{{\alpha }^2}{1+\alpha }$ and $\underset{x\to \infty }{\lim }h\left(x;\underset{¯}{\psi }\right)=\alpha$, in this case $h_1\left(x;\alpha \right)$ is an increasing hrf whereas $h_2\left(x;c,k\right)$ is unimodal hrf. Therefore, two shapes of the hrf can be demonstrated:

i.

The modified bathtub shape: the hrf of the Xg-BXII distribution can exhibit the modified bathtub shape. In this case let $x_0^{\ast }$ and $x_0^{\ast \ast }$ be two critical values of $h\left(x;\underset{¯}{\psi }\right)$, then for $x<x_0^{\ast }$, ${\acute{h}}_1\left(x;\alpha \right)>0$ and ${\acute{h}}_2\left(x;c,k\right)>0$. So, $h\left(x;\underset{¯}{\psi }\right)$ is increasing for $x<x_0^{\ast }$, that is $\acute{h}\left(x;\underset{¯}{\psi }\right)>0$. For $x_0^{\ast }<x\le x_0^{\ast \ast }$, ${\acute{h}}_1\left(x;\alpha \right)>0$ and ${\acute{h}}_2\left(x;c,k\right)<0$, so that the negative one dominates the positive one, therefore $\acute{h}\left(x;\underset{¯}{\psi }\right)<0$, that is $h\left(x;\underset{¯}{\theta }\right)$ is a decreasing hrf. For $x_0^{\ast \ast }<x$, $h\left(x;\underset{¯}{\psi }\right)$ is increasing hrf again because ${\acute{h}}_1\left(x;\alpha \right)>0$ is positive and dominates the negative one ${\acute{h}}_2\left(x;c,k\right)<0$. In brief, the hrf of the Xg-BXII distribution exhibits the modified bathtub shape, that is, the hrf is increasing on $\left[\left.0,x_0^{\ast }\right)\right.$, decreasing on $\left(x_0^{\ast },x_0^{\ast \ast }\right)$ and finally takes an increasing pattern on $\left(\left.x_0^{\ast \ast },\infty \right]\right.$.

ii.

An increasing shape: when ${\acute{h}}_1\left(x;\alpha \right)$ dominates the negative term for ${\acute{h}}_2\left(x;c,k\right)$ and $\acute{h}\left(x;\underset{¯}{\psi }\right)>0$. $h\left(x;\underset{¯}{\psi }\right)$ is an increasing hrf.

▪

For $c\le 1$, $\underset{x\to 0}{\lim }h\left(x;\underset{¯}{\psi }\right)=\infty$ and $\underset{x\to \infty }{\lim }h\left(x;\underset{¯}{\psi }\right)=\alpha$, in this case $h_1\left(x;\alpha \right)$ is an increasing hrf whereas $h_2\left(x;c,k\right)$ is a decreasing hrf. In addition, two shapes of the hrf can be demonstrated:

i.

The bathtub shape: in the present case, ${\acute{h}}_1\left(x;\alpha \right)>0$ and ${\acute{h}}_2\left(x;c,k\right)<0$. For $x<x_0$, where $x_0$ is a critical point at which $\acute{h}\left(x;\underset{¯}{\psi }\right)=0$, $\acute{h}\left(x;\underset{¯}{\psi }\right)<0$. For $x>x_0$, $\acute{h}\left(x;\underset{¯}{\psi }\right)>0$. Therefore, the hrf of the Xg-BXII distribution is a bathtub hrf.

ii.

A decreasing shape: since $h_1\left(x;\alpha \right)$ is an increasing hrf and $h_2\left(x;c,k\right)$ is a decreasing hrf. Thus, $h_2\left(x;c,k\right)$ dominates $h_1\left(x;\alpha \right)$.

2.4. 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, $r^{th}$ 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.

2.4.1. Quantile Function and the Mode

The quantile function of the Xg-BXII distribution can be obtained by inverting:

$$R\left(x;\underset{¯}{\psi }\right)=1-\mathcal{q}.$$

(11)

So, the quantile function of the Xg-BXII distribution can be obtained by solving the following nonlinear equation:

$$\mathrm{l}\mathrm{n}\left[\frac{1+\alpha +\alpha x_{\mathcal{q}}+\frac{{\alpha }^2{x}_{\mathcal{q}}^2}{2}}{\left(1+\alpha \right){\left(1+x_{\mathcal{q}}^c\right)}^k}\right]-\alpha x_{\mathcal{q}}-\ln \left(1-\mathcal{q}\right)=0.$$

(12)

As special cases of the quantile function the median of the Xg-BXII distribution, denoted by $x_m$, the first quartile, denoted by $x_{0.25}$, and the third quartile, denoted by $x_{0.75}$, can be obtained, respectively, by setting $\mathcal{q}$ = 0.5, $\mathcal{q}$ = 0.25 and $\mathcal{q}$ = 0.75 into $\left(12\right)$ and solving numerically.

The mode of a random variable X has the Xg-BXII distribution is the value of $x_0$ which maximizes $f\left(x;\underset{¯}{\psi }\right)$. Hence, the mode of the Xg-BXII distribution can be obtained by solving the following nonlinear equation:

$$\begin{gathered} \left\{\frac{{\alpha }^3}{1+\alpha +\alpha x_0+\frac{{\alpha }^2{x}_0^2}{2}}\left[x-\frac{\left(1+\alpha x_0\right)\left(1+\frac{\mathsf{\alpha }{x}_0^2}{2}\right)}{1+\alpha +\alpha x_0+\frac{{\alpha }^2{x}_0^2}{2}}\right]+\frac{ck{x}_0^{c-2}}{1+x_0^c}\left[c-1-\frac{c{x}_0^c}{1+x_0^c}\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-{\left[\frac{{\alpha }^2\left(1+\frac{\alpha x_0^2}{2}\right)}{1+\alpha +\alpha x_0+\frac{{\alpha }^2{x}_0^2}{2}}+\frac{ck{x}_0^{c-1}}{1+x_0^c}\right]}^2\right\}\frac{1+\alpha +\alpha x_0+\frac{{\alpha }^2{x}_0^2}{2}}{\left(1+\alpha \right){\left(1+x_0^c\right)}^k}{e}^{-\alpha x_0}=0. \end{gathered}$$

(13)

Derivatives of the mode of the Xg-BXII distribution are given in Appendix A.

Some numerical results of the first quartile, $x_{0.25}$, median, $x_m$, and the third quartile, $x_{0.75}$, as special cases of the quantile function and the mode of the Xg-BXII distribution for different parameter values $\underset{¯}{\psi }=\left(\alpha ,c,k\right)$ using R software are presented in

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

α	c	k	x0.25	xm	x0.75	Mode
2.6	5	6	0.1610	0.3807	0.5899	0.5563
2	0.5	0.5	0.1115	0.3748	0.9107	-
0.5	3	1.5	0.5275	0.7756	1.0800	0.6789
0.25	3	5	0.3810	0.5229	0.6775	0.4955
0.15	5	0.8	0.8346	1.0571	1.3498	0.9533
0.05	4	2	0.6263	0.8016	0.9994	0.7595

. 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.

2.4.2. Moments

The $r^{th}$ non-central moment of a random variable X has the Xg-BXII distribution is:

$$\begin{gathered} {\acute{\mu }}_r=\frac{r}{c}\sum _{i=0}^{\mathrm{\infty }}\frac{{\left(-1\right)}^i{\alpha }^i}{i!}\left[\mathsf{{\rm B}}\left(\frac{r+i}{c},k-\frac{r+i}{c}\right)+\frac{\alpha \mathsf{{\rm B}}\left(\frac{r+i+1}{c},k-\frac{r+i+1}{c}\right)}{1+\alpha } \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{{\alpha }^2\mathsf{{\rm B}}\left(\frac{r+i+2}{c},k-\frac{r+i+2}{c}\right)}{2\left(1+\alpha \right)}\right],\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}r=\mathrm{1,2},..., \end{gathered}$$

(14)

where

$$0<\frac{r+i+j}{c}<k,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}j=0,1,2.$$

By substituting $r=1$ and $r=2$ into $\left(14\right)$, the mean and the second non-central moment of the Xg-BXII distribution can be obtained as follows:

$$\begin{gathered} \mu =\frac{1}{c}\sum _{i=0}^{\infty }\frac{{\left(-1\right)}^i{\alpha }^i}{i!}\left[\mathsf{{\rm B}}\left(\frac{i+1}{c},k-\frac{i+1}{c}\right)+\frac{\alpha \mathsf{{\rm B}}\left(\frac{i+2}{c},k-\frac{i+2}{c}\right)}{1+\alpha } \\ +\frac{{\alpha }^2\mathsf{{\rm B}}\left(\frac{i+3}{c},k-\frac{i+3}{c}\right)}{2\left(1+\alpha \right)}\right], \end{gathered}$$

(15)

where

$$0<\frac{i+j+1}{c}<k, j=0,1,2,$$

and

$$\begin{gathered} {\acute{\mu }}_2=\frac{2}{c}\sum _{i=0}^{\mathrm{\infty }}\frac{{\left(-1\right)}^i{\alpha }^i}{i!}\left[\mathsf{{\rm B}}\left(\frac{i+2}{c},k-\frac{i+2}{c}\right)+\frac{\alpha \mathsf{{\rm B}}\left(\frac{i+3}{c},k-\frac{i+3}{c}\right)}{1+\alpha } \\ +\frac{{\alpha }^2\mathsf{{\rm B}}\left(\frac{i+4}{c},k-\frac{i+4}{c}\right)}{2\left(1+\alpha \right)}\right], \end{gathered}$$

(16)

where

$$0<\frac{i+j+2}{c}<k,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}j=0,1,2.$$

The variance of the Xg-BXII distribution can be obtained using $\left(15\right)$ and $\left(16\right)$ in the following equation:

$$V\left(x\right)={\mu }_2={\acute{\mu }}_2-{\mu }^2.$$

(17)

The coefficient of variation (CV), the coefficient of skewness (CS) and the coefficient of kurtosis (CK) are given, respectively, by

$$CV=\frac{\sqrt{{\mu }_2}}{\mu }=\frac{\sqrt{{\acute{\mu }}_2-{\mu }^2}}{\mu }=\sqrt{\frac{{\acute{\mu }}_2}{{\mu }^2}-1},$$

(18)

$$CS=\frac{{\mu }_3}{{\mu }_2^{3/2}}=\frac{{\mu }_3^{’}-3\mu {\acute{\mu }}_2+2{\mu }^3}{{\left({\acute{\mu }}_2-{\mu }^2\right)}^{3/2}},$$

(19)

and

$$CK=\frac{{\mu }_4}{{\mu }_2^2}=\frac{{\acute{\mu }}_4-4\mu {\acute{\mu }}_3+6{\mu }^2{\acute{\mu }}_2-3{\mu }^4}{{\left({\acute{\mu }}_2-{\mu }^2\right)}^2},$$

(20)

where $\mu ,$ ${\acute{\mu }}_2$ and ${\mu }_2$ are obtained, respectively, in $\left(15\right),$ $\left(16\right)$ and $\left(17\right)$ and ${\acute{\mu }}_3$ and ${\acute{\mu }}_4$ can be obtained, respectively, by setting $r=3$ and $r=4$ into $\left(14\right)$.

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

α	c	k	μ	${\acute{\mu }}_2$	${\acute{\mu }}_3$	${\acute{\mu }}_4$	μ2	CV	CS	CK
2.6	5	6	0.3878	0.2135	0.1352	0.0929	0.0630	0.6474	0.2193	2.0180
2	0.5	0.5	0.6330	0.9187	1.9520	5.3099	0.5181	1.1371	1.9165	7.8042
0.5	3	1.5	0.8529	0.9717	1.4616	3.0361	0.2444	0.5796	1.7888	11.7837
0.25	3	5	0.5391	0.3405	0.2430	0.1926	0.0499	0.4142	0.5131	3.5599
0.15	5	0.8	1.1585	1.6443	3.1835	10.8832	0.3022	0.4745	3.4822	43.457
0.05	4	2	0.8321	0.7841	0.8313	0.9974	0.0917	0.3639	0.9458	5.9130

.

Furthermore, the moment generating function, denoted by $M_X\left(t\right)$, of a random variable $X$ has the Xg-BXII distribution can be obtained as given below:

$$M_X\left(t\right)=E\left(e^{tx}\right)={\int }_0^{\mathrm{\infty }}{e}^{tx}f\left(x;\underset{¯}{\psi }\right)dx=\sum _{r=0}^{\mathrm{\infty }}\frac{t^r}{r!}{\acute{\mu }}_r,$$

(21)

where ${\acute{\mu }}_r$ is given in $\left(14\right)$.

2.4.3. Incomplete Moments and Inequality Curves

The $r^{th}$ incomplete moment of a random variable $X$ has the Xg-BXII distribution is given by:

$$\begin{gathered} {\mu }_r\left(t\right)={\int }_0^t{x}^{r}f\left(x;\underset{¯}{\psi }\right)dx=-t^{r}R\left(t;\underset{¯}{\psi }\right)+{\int }_0^{t}r{x}^{r-1}R\left(x;\underset{¯}{\psi }\right)dx \\ \hspace{1em}\hspace{1em}=\frac{r{t}^{r-1}\left(1+\alpha +\alpha t+\frac{{\alpha }^2{t}^2}{2}\right)}{\left(1+\alpha \right){\left(1+t^c\right)}^k}{e}^{-\alpha t} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{r}{c}\sum _{i=0}^{\infty }\frac{{\left(-1\right)}^i{\alpha }^i}{i!}\left[{\mathbf{I{\rm B}}}_{\left(t^c\right)}\left(\frac{r+i}{c},k-\frac{r+i}{c}\right) \\ \hspace{1em}+\frac{\alpha {\mathbf{I{\rm B}}}_{\left(t^c\right)}\left(\frac{r+i+1}{c},k-\frac{r+i+1}{c}\right)}{1+\alpha } \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{{\alpha }^2{\mathbf{I{\rm B}}}_{\left(t^c\right)}\left(\frac{r+i+2}{c},k-\frac{r+i+2}{c}\right)}{2\left(1+\alpha \right)}\right],\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}r=\mathrm{1,2},\dots , \end{gathered}$$

(22)

where ${\mathbf{I{\rm B}}}_{\left(t^c\right)}\left(.,.\right)$ is a lower incomplete beta function and

$$0<\frac{r+i+j}{c}<k, j=0, 1, 2.$$

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 $L_F\left(p\right)$ and $B_F\left(p\right)$ which are defined by:

$$L\left(p\right)=\frac{1}{\mu }{\int }_0^{q}xf\left(x\right)dx=\frac{\mu \left(q\right)}{\mu },$$

(23)

and

$$B\left(p\right)=\frac{1}{p\mu }{\int }_0^{q}xf\left(x\right)dx=\frac{L\left(p\right)}{p},$$

(24)

where $\mu$ is obtained from $\left(15\right)$, $\mu \left(q\right)$ is the first incomplete moment which can be obtained by substituting $r=1$ and $t=q$ into $\left(22\right)$ and $q=F^{-1}\left(p\right)$ for $0<p<1$.

2.4.4. The Mean Residual Life and the Mean Inactivity Time

The MRL function or the life expectation at age $t$, denoted by $m\left(t\right)$, which represents the expected additional life length for a system or a unit which is alive at age $t$, is defined by:

$$m\left(t\right)=E\left(X-t|X>t\right)=\frac{1}{R\left(t\right)}{\int }_t^{\mathrm{\infty }}R\left(x\right)dx.$$

For the Xg-BXII distribution the MRL is given by:

$$\begin{gathered} m\left(t\right)=\frac{{\left(1+t^c\right)}^k{e}^{\alpha t}}{c\left(1+\alpha +\alpha t+\frac{{\alpha }^2{t}^2}{2}\right)}\sum _{i=0}^{\mathrm{\infty }}\frac{{(-1)}^i{a}^i}{i!}\left[{\mathbf{IB}}^{\left(t^c\right)}\left(\frac{i+1}{c},k-\frac{i+1}{c}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\alpha {\mathbf{IB}}^{\left(t^c\right)}\left(\frac{i+2}{c},k-\frac{i+2}{c}\right)+\frac{{\alpha }^2}{2}{\mathbf{IB}}^{\left(t^c\right)}\left(\frac{i+3}{c},k-\frac{i+3}{c}\right)\right], \end{gathered}$$

(25)

where ${\mathbf{IB}}^{\left(t^c\right)}\left(.,.\right)$ is a lower incomplete beta function and

$$0<\frac{i+\tau }{c}<k,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\tau =1, 2, 3.$$

The MPL, mean inactivity time or mean waiting time, also called the mean reversed residual life function, denoted by $M\left(t\right)$, which represents the waiting time elapsed since the failure of a system or a unit on the condition that this failure had occurred in $\left(0,t\right)$, is defined by:

$$M\left(t\right)=E\left[\left(t-X\right)\left|X\le t\right.\right]=\frac{1}{F\left(t\right)}{\int }_0^{t}F\left(x\right)dx.$$

For a random variable $X$ has the Xg-BXII distribution, the MPL is given by:

$$\begin{gathered} M\left(t\right)=\frac{{\left(1+t^c\right)}^k}{\left(1+\alpha \right){\left(1+t^c\right)}^k-\left(1+\alpha +\alpha t+\frac{{\alpha }^2{t}^2}{2}\right)e^{-\alpha t}}\left\{\left(1+\alpha \right)t \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\frac{1}{c}\sum _{i=0}^{\infty }\frac{{\left(-1\right)}^i{\alpha }^i}{i!}\left[\left(1+\alpha \right){\mathbf{I{\rm B}}}_{\left(t^c\right)}\left(\frac{i+1}{c},k-\frac{i+1}{c}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\alpha {\mathbf{I{\rm B}}}_{\left(q^c\right)}\left(\frac{i+2}{c},k-\frac{i+2}{c}\right)+\frac{{\alpha }^2}{2}{\mathbf{I{\rm B}}}_{\left(q^c\right)}\left(\frac{i+3}{c},k-\frac{i+3}{c}\right)\right]\right\}, \end{gathered}$$

(26)

where ${\mathbf{I{\rm B}}}_{\left(t^c\right)}\left(.,.\right)$ is a lower incomplete beta function and

$$0<\frac{i+\tau }{c}<k,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\tau =1, 2, 3.$$

2.4.5. 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:

$$\mathrm{M}\mathrm{T}\mathrm{T}\mathrm{F}={\int }_0^{\infty }R\left(x;\underset{¯}{\psi }\right)dx=\mu$$

(27)

and

$$\mathrm{M}\mathrm{T}\mathrm{B}\mathrm{F}=\frac{-x}{\ln R\left(x;\underset{¯}{\psi }\right)}=\frac{x}{H\left(x;\underset{¯}{\psi }\right)}=\frac{x}{\ln \left[\frac{\left(1+\alpha \right){\left(1+x^c\right)}^k}{1+\alpha +\alpha x+\frac{{\alpha }^2{x}^2}{2}}\right]+\alpha x}.$$

(28)

The Av is the probability that a product is successful at time $x_0$ and is defined as:

$$\mathrm{A}\mathrm{v}=\frac{MTTF}{MTBF}.$$

(29)

[See, [30]].

2.4.6. 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. Sub-models of the Xg-BXII distribution.

Parameter	The Resulting Distribution	Pdf
c = 1	Xg-compound exponential distribution	$f\left(x;\alpha ,k\right)=\left[{\alpha }^2\left(1+\frac{\alpha x^2}{2}\right)+\frac{k}{(1+x)}\right]\frac{{\left(1+x\right)}^{-k}{e}^{-\alpha x}}{1+\alpha }, x,\alpha ,k>0.$
c = 2	Xg-compound Rayleigh distribution	$f\left(x;\alpha ,k\right)=\left[{\alpha }^2\left(1+\frac{\alpha x^2}{2}\right)+\frac{2xk}{(1+x^2)}\right]\frac{{\left(1+x^2\right)}^{-k}{e}^{-\alpha x}}{1+\alpha },x,\alpha ,k>0.$
k = 1	Xg-log logistic	$f\left(x;\alpha , c\right)=\left[{\alpha }^2\left(1+\frac{\alpha x^2}{2}\right)+\frac{c{x}^{c-1}}{(1+x^c)}\right]\frac{e^{-\alpha x}}{\left(1+x^c\right)\left(1+\alpha \right)},x, \alpha , c>0.$
$\mathit{\alpha }\mathbf{\to }\mathbf{0}$	BXII distribution	$f\left(x;c,k\right)=\frac{ck{x}^{c-1}}{{\left(1+x^c\right)}^{k+1}},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}x>0; c,k>0.$
$\mathit{\alpha }\mathbf{\to }\mathbf{0}$ and $\mathit{c}=1$	Lomax distribution	$f\left(x;k\right)=\frac{k}{{\left(1+x\right)}^{k+1}},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}x>0; k>0.$
$\mathit{\alpha }\mathbf{\to }\mathbf{0}$ and $\mathit{c}\mathbf{=}\mathbf{2}$	Compound Rayleigh distribution	$f\left(x;k\right)=\frac{2kx}{{\left(1+x^2\right)}^{k+1}},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}x>0; k>0.$
$\mathit{\alpha }\mathbf{\to }\mathbf{0}$ and $\mathit{k}\mathbf{=}\mathbf{1}$	Log logistic distribution	$f\left(x;c\right)=\frac{c{x}^{c-1}}{{\left(1+x^c\right)}^2},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}x>0; c>0.$
$\mathit{k}\mathbf{\to }{\mathbf{0}}^{\mathbf{+}}$	Xgamma distribution	$f\left(x;\alpha \right)=\frac{{\alpha }^2}{\left(1+\alpha \right)}\left(1+\frac{\alpha x^2}{2}\right)e^{-\alpha x},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}x,\mathsf{\alpha }>0.$

summarizes these sub-models.

2.4.7. Order Statistics

Suppose that $X_1, X_2,\dots ,X_n$ are i.i.d. random variables have the Xg-BXII distribution. Then $X_{\left(1\right)}\le X_{\left(2\right)}\le \dots \le X_{\left(n\right)}$ are the corresponding order statistics, and the pdf of the $s^{th}$ order statistic is given by:

$$f_{s:n}\left(x;\underset{¯}{\psi }\right)=C_{s,n}f\left(x;\underset{¯}{\psi }\right){\left[F\left(x;\underset{¯}{\psi }\right)\right]}^{s-1}{\left[1-F\left(x;\underset{¯}{\psi }\right)\right]}^{n-s}, x_{\left(s\right)}>0,$$

(30)

where

$$C_{s,n}=\frac{n!}{\left(s-1\right)!\left(n-s\right)!}.$$

Since

$$R\left(x;\underset{¯}{\psi }\right)=e^{-H\left(x;\underset{¯}{\psi }\right)}, {\left[1-F\left(x;\underset{¯}{\psi }\right)\right]}^{n-s}=e^{-\left(n-s\right)H\left(x;\underset{¯}{\psi }\right)},$$

$${\left[F\left(x;\underset{¯}{\psi }\right)\right]}^{s-1}={\left[1-e^{-H\left(x;\underset{¯}{\psi }\right)}\right]}^{s-1},$$

and

$$f\left(x;\underset{¯}{\psi }\right)=h\left(x;\underset{¯}{\psi }\right)e^{-H\left(x;\underset{¯}{\psi }\right)},$$

and by using the binomial expansion of ${\left[F\left(x;\underset{¯}{\psi }\right)\right]}^{s-1}$, then

$${\left[F\left(x;\underset{¯}{\psi }\right)\right]}^{s-1}={\left[1-e^{-H\left(x;\underset{¯}{\psi }\right)}\right]}^{s-1}=\sum _{j=0}^{s-1}\left(\genfrac{}{}{0pt}{}{s-1}{j}\right){\left(-1\right)}^j{e}^{-jH\left(x;\underset{¯}{\psi }\right)}.$$

Therefore, the pdf of the $s^{th}$ order statistic of the Xg-BXII distribution can be rewritten as

$$f_{s:n}\left(x;\underset{¯}{\psi }\right)=\sum _{j=0}^{s-1}{C}_{s,n,j}h\left(x;\underset{¯}{\psi }\right)e^{-\left(j+n-s+1\right)H\left(x;\underset{¯}{\psi }\right)},\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{x}_{\left(s\right)}>0,$$

(31)

where

$$C_{s,n,j}=\frac{n!{\left(-1\right)}^j}{j!\left(s-j-1\right)!\left(n-s\right)!}.$$

(32)

Substituting $\left(7\right)$ and $\left(10\right)$ into $\left(31\right)$, then the pdf of the $s^{th}$ order statistic of the Xg-BXII distribution is given by:

$$\begin{gathered} f_{s:n}\left(x;\underset{¯}{\psi }\right)=\sum _{j=0}^{s-1}{C}_{s,n,j}\left[\frac{{\alpha }^2\left(1+\frac{\alpha x_{\left(s\right)}^2}{2}\right)}{1+\alpha +\alpha x_{\left(s\right)}+\frac{{\alpha }^2{x}_{\left(s\right)}^2}{2}}+\frac{ck{x}_{\left(s\right)}^{c-1}}{1+x_{\left(s\right)}^c}\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\left[\frac{\left(1+\alpha \right){\left(1+x_{\left(s\right)}^c\right)}^k}{1+\alpha +\alpha x_{\left(s\right)}+\frac{{\alpha }^2{x}_{\left(s\right)}^2}{2}}\right]}^{-\left(j+n-s+1\right)}{e}^{-\left(j+n-s+1\right)\alpha x_{\left(s\right)}},x_{\left(s\right)}>0. \end{gathered}$$

(33)

Special cases:

a.

The pdf of the smallest order statistics can be obtained when $s=1$ as:

$$\begin{gathered} f_{1:n}\left(x;\underset{¯}{\psi }\right)=n\left[\frac{{\alpha }^2\left(1+\frac{\alpha x_{\left(1\right)}^2}{2}\right)}{1+\alpha +\alpha x_{\left(1\right)}+\frac{{\alpha }^2{x}_{\left(1\right)}^2}{2}}+\frac{ck{x}_{\left(1\right)}^{c-1}}{1+x_{\left(1\right)}^c}\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\left[\frac{\left(1+\alpha \right){\left(1+x_{\left(1\right)}^c\right)}^k}{1+\alpha +\alpha x_{\left(1\right)}+\frac{{\alpha }^2{x}_{\left(1\right)}^2}{2}}\right]}^{-n}{e}^{-n\alpha x_{\left(1\right)}}, x_{\left(1\right)}>0. \end{gathered}$$

(34)

b.

The pdf of the largest order statistics can be obtained if $s=n$ as:

$$\begin{gathered} f_{n:n}\left(x;\underset{¯}{\psi }\right)=n\sum _{j=0}^{n-1}\left(\genfrac{}{}{0pt}{}{n-1}{j}\right){\left(-1\right)}^j\left[\frac{{\alpha }^2\left(1+\frac{\alpha x_{\left(n\right)}^2}{2}\right)}{1+\alpha +\alpha x_{\left(n\right)}+\frac{{\alpha }^2{x}_{\left(n\right)}^2}{2}}+\frac{ck{x}_{\left(n\right)}^{c-1}}{1+x_{\left(n\right)}^c}\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\left[\frac{\left(1+\alpha \right){\left(1+x_{\left(n\right)}^c\right)}^k}{1+\alpha +\alpha x_{\left(n\right)}+\frac{{\alpha }^2{x}_{\left(n\right)}^2}{2}}\right]}^{-\left(j+1\right)}{e}^{-\left(j+1\right)\alpha x_{\left(n\right)}}, x_{\left(n\right)}>0. \end{gathered}$$

(35)

3. 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 $x_1, x_2,\dots , x_n$ is a random sample of size $n$ from the Xg-BXII distribution with parameter vector $\underset{¯}{\psi }=\left(\alpha ,c,k\right)$, then the likelihood function is given by:

$$\begin{array}{l} L\left(\underset{¯}{\psi };\underset{¯}{x}\right)=\left[{\displaystyle \prod _{i=1}^n}f\left(x_{\left(i\right)};\underset{¯}{\psi }\right)\right]\\ ={\displaystyle \prod _{i=1}^n}\left[{\alpha }^2\left(1+\frac{\alpha x_{\left(i\right)}^2}{2}\right)+\frac{ck{x}_{\left(i\right)}^{c-1}\left(1+\alpha +\alpha x_{\left(i\right)}+\frac{{\alpha }^2{x}_{\left(i\right)}^2}{2}\right)}{\left(1+x_{\left(i\right)}^c\right)}\right]\\ \times {\displaystyle \prod _{i=1}^n}\left[\frac{{\left(1+x_{\left(i\right)}^c\right)}^{-k}{e}^{-\alpha x_{\left(i\right)}}}{1+\alpha }\right] .\end{array}$$

(36)

The natural logarithm of the likelihood function is

$$\begin{array}{l}\mathcal{l}\equiv \ln L\left(\underset{¯}{\psi };\underset{¯}{x}\right)\\ ={\displaystyle \sum _{i=1}^n}\ln \left[{\alpha }^2\left(1+\frac{\alpha x_{\left(i\right)}^2}{2}\right)+\frac{ck{x}_{\left(i\right)}^{c-1}\left(1+\alpha +\alpha x_{\left(i\right)}+\frac{{\alpha }^2{x}_{\left(i\right)}^2}{2}\right)}{\left(1+x_{\left(i\right)}^c\right)}\right]\\ +{\displaystyle \sum _{i=1}^n}\mathrm{l}\mathrm{n}\left[\frac{{\left(1+x_{\left(i\right)}^c\right)}^{-k}{e}^{-\alpha x_{\left(i\right)}}}{1+\alpha }\right]\end{array}$$

(37)

By differentiating the log-likelihood function in $\left(37\right)$ with respect to the parameters $\alpha ,c$ and $k$ as follows:

$$\begin{gathered} \frac{\partial \mathcal{l}}{\partial \alpha }=\left\{\sum _{i=1}^n{\left[\frac{\left(1+x_{\left(i\right)}^c\right)\times {\alpha }^2\left(1+\frac{\alpha x_{\left(i\right)}^2}{2}\right)+ck{x}_{\left(i\right)}^{c-1}\left(1+\alpha +\alpha x_{\left(i\right)}+\frac{{\alpha }^2{x}_{\left(i\right)}^2}{2}\right)}{\left(1+x_{\left(i\right)}^c\right)}\right]}^{-1} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \left[\alpha \left[\alpha \left(\frac{x_{\left(i\right)}^2}{2}\right)+\left(2+\frac{\alpha x_{\left(i\right)}^2}{2}\right)\right]\right]+\left[\frac{ck{x}_{\left(i\right)}^{c-1}}{\left(1+x_{\left(i\right)}^c\right)}\times \left(1+x_{\left(i\right)}+\alpha x_{\left(i\right)}^2\right)\right]\right\} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left\{\sum _{i=1}^n\frac{\left[\left(1+\alpha \right)\left(-x_{\left(i\right)}\right)\left\{{\left(1+x_{\left(i\right)}^c\right)}^{-k}{e}^{-\alpha x_{\left(i\right)}}\right\}\right]-\left[\left\{{\left(1+x_{\left(i\right)}^c\right)}^{-k}{e}^{-\alpha x_{\left(i\right)}}\right\}\right]}{{\left(1+x_{\left(i\right)}^c\right)}^{-k}{e}^{-\alpha x_{\left(i\right)}}\times \left(1+\alpha \right)}\right\}, \end{gathered}$$

(38)

$$\begin{array}{l}\frac{\partial \mathcal{l}}{\partial c}={\displaystyle \sum _{i=1}^n}{\left[{\alpha }^2\left(1+\frac{\alpha x_{\left(i\right)}^2}{2}\right)+\frac{ck{x}_{\left(i\right)}^{c-1}\left(1+\alpha +\alpha x_{\left(i\right)}+\frac{{\alpha }^2{x}_{\left(i\right)}^2}{2}\right)}{\left(1+x_{\left(i\right)}^c\right)}\right]}^{-1}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \frac{\left(1+x_{\left(i\right)}^c\right)\left[x_{\left(i\right)}^{c-1}k\left(1+\alpha +\alpha x_{\left(i\right)}+\frac{{\alpha }^2{x}_{\left(i\right)}^2}{2}\right)\right]-\left[ck{x}_{\left(i\right)}^{c-1}\left(1+\alpha +\alpha x_{\left(i\right)}+\frac{{\alpha }^2{x}_{\left(i\right)}^2}{2}\right)\left[x_{\left(i\right)}^{c}ln\left[x_i\right]\right]\right]}{{\left(1+x_{\left(i\right)}^c\right)}^2}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{i=1}^n}{\left[\frac{{\left(1+x_{\left(i\right)}^c\right)}^{-k}{e}^{-\alpha x_{\left(i\right)}}}{\left(1+\alpha \right)}\right]}^{-1}\left(\frac{e^{-\alpha x_{\left(i\right)}}}{1+\alpha }\right)\left(-k\right){\left(1+x_{\left(i\right)}^c\right)}^{-\left(k+1\right)}\left[x_{\left(i\right)}^{c}ln\left(x_i\right)\right],\end{array}$$

(39)

and

$$\begin{gathered} \frac{\partial \mathcal{l}}{\partial k}=\sum _{i=1}^n{\left[{\alpha }^2\left(1+\frac{\alpha x_{\left(i\right)}^2}{2}\right)+\frac{ck{x}_{\left(i\right)}^{c-1}\left(1+\alpha +\alpha x_{\left(i\right)}+\frac{{\alpha }^2{x}_{\left(i\right)}^2}{2}\right)}{\left(1+x_{\left(i\right)}^c\right)}\right]}^{-1}\times \left[\frac{c{x}_{\left(i\right)}^{c-1}\left(1+\alpha +\alpha x_{\left(i\right)}+\frac{{\alpha }^2{x}_{\left(i\right)}^2}{2}\right)}{\left(1+x_{\left(i\right)}^c\right)}\right] \\ -\sum _{i=1}^n{\left[\frac{{\left(1+x_{\left(i\right)}^c\right)}^{-k}{e}^{-\alpha x_{\left(i\right)}}}{1+\alpha }\right]}^{-1}\left(\frac{e^{-\alpha x_{\left(i\right)}}}{1+\alpha }\right)\frac{ln\left(1+x_{\left(i\right)}^c\right)}{{\left(1+x_{\left(i\right)}^c\right)}^k}. \end{gathered}$$

(40)

The ML estimates of the parameters $\underset{¯}{\psi }=\left(\alpha ,c,k\right)$ can be obtained by equating $\left(38\right)-\left(40\right)$ to zeros then solving numerically.

The ML estimators of $R\left(x;\underset{¯}{\psi }\right)$ and $h\left(x;\underset{¯}{\psi }\right)$ can be obtained, using the invariance property of the ML estimators, by replacing the parameters $\underset{¯}{\psi }=\left(\alpha ,c,k\right)$ in $\left(5\right)$ and $\left(7\right)$ with their ML estimators, then the ML estimators of $R\left(x;\underset{¯}{\psi }\right)$ and $h\left(x;\underset{¯}{\psi }\right)$ can be given, respectively, as:

$$\widehat{R}\left(x;\widehat{\underset{¯}{\psi }}\right)=\frac{1+\widehat{\alpha }+\widehat{\alpha }x+\frac{{\widehat{\alpha }}^2{x}^2}{2}}{\left(1+\widehat{\alpha }\right){\left(1+x^{\widehat{c}}\right)}^{\widehat{k}}}{e}^{-\widehat{\alpha }x},x>0,$$

(41)

and

$$\widehat{h}\left(x;\widehat{\underset{¯}{\psi }}\right)=\frac{\left(1+\frac{\widehat{\alpha }{x}^2}{2}\right)}{1+\widehat{\alpha }(1+x)+\frac{{\widehat{\alpha }}^2{x}^2}{2}}+\frac{\widehat{c}\widehat{k}{x}^{\widehat{c}-1}}{\left(1+x^{\widehat{c}}\right)},x>0.$$

(42)

b.

Asymptotic confidence intervals

To obtain the confidence intervals for the parameters $\underset{¯}{\psi }=\left(\alpha ,c,k\right)$ of the Xg-BXII distribution, the distributions of the ML estimators $\underset{¯}{\widehat{\psi }}=\left(\widehat{\alpha },\widehat{c},\widehat{k}\right)$ are needed. Since the ML estimators $\underset{¯}{\widehat{\psi }}=\left(\widehat{\alpha },\widehat{c},\widehat{k}\right)$ 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 $\left(\alpha ,c,k\right)$ 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:

$$\tilde{I} \simeq \left(\hspace{1em}\frac{\partial \mathcal{l}}{\partial {\psi }_i \partial {\psi }_j}\right), i, j=\mathrm{1,2},3,$$

(43)

where ${\psi }_1=\alpha ,{\psi }_2=c\mathrm{a}\mathrm{n}\mathrm{d}{\psi }_3=k$.

Therefore, the $\left(1-\omega \right)100\mathrm{\%}$ bounds of the ACIs of the parameters $\underset{¯}{\psi }=\left(\alpha ,c,k\right)$ are as follows:

$$L_w=\widehat{w}-Z_{\frac{\omega }{2}}{ \widehat{\sigma }}_{\widehat{w}}\hspace{1em}\hspace{1em}\mathrm{and}\hspace{1em}\hspace{1em}{U}_w=\widehat{w}-Z_{\frac{\omega }{2}}{\widehat{\sigma }}_{\widehat{w}} ,$$

where $\widehat{w}$ in this paper is $\widehat{\alpha },\widehat{c},\widehat{k}, \widehat{R}\left(x;\widehat{\underset{¯}{\psi }}\right)$ or $\widehat{h}\left(x;\widehat{\underset{¯}{\psi }}\right)$, and ${ \widehat{\sigma }}_{\widehat{w}}$ is the standard deviation.

4. 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:

$$I:\left(\alpha =2,c=0.5,k=0.3\right),$$

$$II:\left(\alpha =0.5,c=3,k=1.5\right),$$

and

$$III:\left(\alpha =0.15,c=1.5,k=0.8\right).$$

b.

For different sample sizes $\left(n=30, 60, 100, 200, 500\right)$ random samples from the Xg-BXII distribution are generated.

c.

The simulation study is performed using the number of replications $NR=1000$ using Mathematica 11 [31].

d.

Table 4. ML averages, estimated risks, relative errors, Relative absolute biases, variances and 95% ACI bounds and their lengths of the parameters of the Xg-BXII distribution for different $n$, $NR=1000$ and $\left(\alpha =2,c=0.5,k=0.3\right)$.

n	$\underset{¯}{\psi }$	Average	ER	RE	RAB	Variance	UL	LL	Length
30	$\alpha$	1.9502	0.1022	0.1599	0.0249	0.0998	2.5692	1.3311	1.2381
	$c$	0.6009	0.0556	0.4717	0.2017	0.0455	1.0186	0.1830	0.8358
	$k$	0.3736	0.0557	0.7865	0.2453	0.0503	0.8130	0	0.8130
60	$\alpha$	1.9442	0.0760	0.1378	0.0279	0.0728	2.4732	1.4152	1.0579
	$c$	0.5770	0.0413	0.4065	0.1541	0.0354	0.9456	0.2084	0.7372
	$k$	0.3806	0.0513	0.7552	0.2687	0.0448	0.7956	0	0.7956
100	$\alpha$	1.9443	0.0477	0.1092	0.0279	0.0446	2.3582	1.5304	0.8278
	$c$	0.5518	0.0347	0.3724	0.1036	0.0320	0.9023	0.2013	0.7011
	$k$	0.3742	0.0452	0.7090	0.2472	0.0397	0.7649	0	0.7649
200	$\alpha$	1.9418	0.0356	0.0943	0.0291	0.0322	2.2936	1.5900	0.7036
	$c$	0.5395	0.0265	0.3253	0.0789	0.0249	0.8488	0.2302	0.6186
	$k$	0.3637	0.0355	0.6276	0.2124	0.0314	0.7110	0.0164	0.6946
500	$\alpha$	1.9679	0.0176	0.0664	0.0160	0.0166	2.2204	1.7155	0.5049
	$c$	0.5163	0.0136	0.2331	0.0326	0.0133	0.7425	0.2901	0.4524
	$k$	0.3373	0.0199	0.4699	0.1244	0.0185	0.6038	0.0709	0.5329

,

Table 5. ML averages, estimated risks, relative errors, relative absolute biases, variances and 95% ACI bounds and their lengths of the parameters of the Xg-BXII distribution for different $n$, $NR=1000$ and $\left(\alpha =0.5,c=3,k=1.5\right)$.

n	$\underset{¯}{\psi }$	Average	ER	RE	RAB	Variance	UL	LL	Length
30	$\alpha$	0.5110	0.1166	0.6830	0.0219	0.1165	1.1799	0	1.1799
	$c$	3.1666	0.2807	0.1766	0.0555	0.2530	4.1524	2.1808	1.9716
	$k$	1.4544	0.0955	0.2060	0.0304	0.0934	2.0534	0.8554	1.1981
60	$\alpha$	0.4926	0.0881	0.5935	0.0148	0.0880	1.0740	0	1.0740
	$c$	3.1551	0.2316	0.1604	0.0517	0.2075	4.0480	2.2622	1.7858
	$k$	1.4780	0.0720	0.1789	0.0147	0.0715	2.0022	0.9538	1.0483
100	$\alpha$	0.4754	0.0583	0.4828	0.0491	0.0577	0.9461	0.0047	0.9414
	$c$	3.0700	0.1330	0.1216	0.0233	0.1281	3.7715	2.3684	1.4032
	$k$	1.5032	0.0536	0.1543	0.0021	0.0536	1.9568	1.0495	0.9073
200	$\alpha$	0.4757	0.0277	0.3328	0.0485	0.0271	0.7984	0.1531	0.6453
	$c$	3.0194	0.0743	0.0909	0.0065	0.0740	3.5524	2.4864	1.0660
	$k$	1.5075	0.0326	0.1203	0.0050	0.0325	1.8609	1.1541	0.7069
500	$\alpha$	0.4786	0.0107	0.2073	0.0428	0.0103	0.6773	0.2799	0.3975
	$c$	2.9967	0.0324	0.0600	0.0011	0.0324	3.3497	2.6438	0.7058
	$k$	1.5114	0.0119	0.0729	0.0076	0.0118	1.7244	1.2984	0.4260

and

Table 6. ML averages, estimated risks, relative errors, relative absolute biases, variances and 95% ACI bounds and their lengths of the parameters of the Xg-BXII distribution for different $n$, $NR=1000$ and $\left(\alpha =0.15,c=1.5,k=0.8\right)$.

n	$\underset{¯}{\psi }$	Average	ER	RE	RAB	Variance	UL	LL	Length
30	$\alpha$	0.1574	0.0120	0.7288	0.0495	0.0119	0.3712	0	0.3712
	$c$	1.5322	0.0539	0.1548	0.0215	0.0529	1.9829	1.0815	0.9014
	$k$	0.8033	0.0268	0.2048	0.0041	0.0268	1.1243	0.4822	0.6421
60	$\alpha$	0.1624	0.0110	0.7003	0.0827	0.0109	0.3669	0	0.3669
	$c$	1.5095	0.0332	0.1214	0.0063	0.0331	1.8659	1.1531	0.7128
	$k$	0.7839	0.0177	0.1662	0.0202	0.0174	1.0426	0.5252	0.5174
100	$\alpha$	0.1667	0.0089	0.6272	0.1116	0.0086	0.3482	0	0.3482
	$c$	1.5018	0.0207	0.0960	0.0012	0.0207	1.7840	1.2196	0.5644
	$k$	0.7834	0.0120	0.1366	0.0208	0.0117	0.9951	0.5716	0.4235
200	$\alpha$	0.1674	0.0054	0.4876	0.1161	0.0051	0.3066	0.0282	0.2785
	$c$	1.5014	0.0123	0.0740	0.0010	0.0123	1.7189	1.2840	0.4349
	$k$	0.7848	0.0067	0.1021	0.0191	0.0064	0.9421	0.6274	0.3147
500	$\alpha$	0.1594	0.0025	0.3328	0.0627	0.0024	0.2555	0.0633	0.1922
	$c$	1.4980	0.0048	0.0462	0.0014	0.0048	1.6339	1.3621	0.2718
	$k$	0.7953	0.0026	0.0635	0.0059	0.0026	0.8945	0.6961	0.1984

display The ML averages, estimated risks (ER), relative errors (RE), relative absolute biases (RAB), the variances and the ACI bounds of the parameters with their lengths, where the ER and the RE are computed as follows:

$$ER\left(\widehat{\psi }\right)=\frac{\sum _{i=1}^{NP}{\left({\widehat{\psi }}_i-\psi \right)}^2}{NR},$$

$$RE\left(\widehat{\psi }\right)=\frac{\sqrt{ER\left(\widehat{\psi }\right)}}{\psi },$$

and

$$RAB\left(\widehat{\psi }\right)=\frac{\left|bias\left(\widehat{\psi }\right)\right|}{\psi },$$

where

$$bias\left(\widehat{\psi }\right)={\widehat{\psi }}_i-\psi .$$

e.

Table 7. ML averages, estimated risks, relative errors, relative absolute biases, variances and 95% ACI bounds of the rf, hrf and rhrf of the Xg-BXII distribution for different $n$, $NR=1000$, $\left(\alpha =2,c=0.5,k=0.3\right)$ and $x_0=0.5$.

$\mathit{n}$	Function	Average	ER	RE	RAB	Variance	UL	LL	Length
30	$R\left(x_0;\underset{¯}{\psi }\right)$	0.4664	0.0027	0.1092	0.0078	0.0025	0.5672	0.3655	0.2017
	$h\left(x_0;\underset{¯}{\psi }\right)$	1.2808	0.0400	0.1597	0.0367	0.0379	1.6624	0.8992	0.7632
	$r\left(x_0;\underset{¯}{\psi }\right)$	1.1119	0.0169	0.1825	0.0149	0.0167	1.3649	0.8590	0.5059
60	$R\left(x_0;\underset{¯}{\psi }\right)$	0.4681	0.0016	0.0832	0.0042	0.0016	0.5472	0.3889	0.1583
	$h\left(x_0;\underset{¯}{\psi }\right)$	1.2599	0.0233	0.1219	0.0198	0.0227	1.5551	0.9647	0.5905
	$r\left(x_0;\underset{¯}{\psi }\right)$	1.1042	0.0106	0.1393	0.0079	0.0105	1.3050	0.9035	0.4015
100	$R\left(x_0;\underset{¯}{\psi }\right)$	0.4678	0.0008	0.0734	0.0048	0.0008	0.5237	0.4118	0.1120
	$h\left(x_0;\underset{¯}{\psi }\right)$	1.2506	0.0130	0.1063	0.0124	0.0127	1.4716	1.0296	0.4420
	$r\left(x_0;\underset{¯}{\psi }\right)$	1.0971	0.0069	0.1039	0.0014	0.0069	1.2601	0.9342	0.3259
200	$R\left(x_0;\underset{¯}{\psi }\right)$	0.4696	0.0006	0.0526	0.0010	0.0006	0.5169	0.4223	0.0946
	$h\left(x_0;\underset{¯}{\psi }\right)$	1.2400	0.0080	0.0908	0.0038	0.0079	1.4147	1.0654	0.3493
	$r\left(x_0;\underset{¯}{\psi }\right)$	1.0966	0.0044	0.0814	0.0009	0.0044	1.2273	0.9660	0.2613
500	$R\left(x_0;\underset{¯}{\psi }\right)$	0.4690	0.0003	0.0513	0.0021	0.0003	0.4999	0.4382	0.0617
	$h\left(x_0;\underset{¯}{\psi }\right)$	1.2383	0.0035	0.0722	0.0024	0.0035	1.3534	1.1232	0.2302
	$r\left(x_0;\underset{¯}{\psi }\right)$	1.0934	0.0020	0.0537	0.0020	0.0020	1.1811	1.0057	0.1754

,

Table 8. ML averages, estimated risks, relative errors, relative absolute biases, variances and 95% ACI bounds of the rf, hrf and rhrf of Xg-BXII distribution for different $n$, $NR=1000$, $\left(\alpha =0.5,c=3,k=1.5\right)$ and $x_0=0.5$.

$\mathit{n}$	Function	Average	ER	RE	RAB	Variance	UL	LL	Length
30	$R\left(x_0;\underset{¯}{\psi }\right)$	0.7728	0.0037	0.0761	0.0030	0.0037	0.8915	0.6540	0.2376
	$h\left(x_0;\underset{¯}{\psi }\right)$	1.1087	0.0484	0.1888	0.0352	0.0468	1.5327	0.6846	0.8480
	$r\left(x_0;\underset{¯}{\psi }\right)$	3.9365	0.8867	0.0556	0.0058	0.8862	5.7816	2.0914	3.6902
60	$R\left(x_0;\underset{¯}{\psi }\right)$	0.7762	0.0022	0.0795	0.0015	0.0022	0.8690	0.6835	0.1856
	$h\left(x_0;\underset{¯}{\psi }\right)$	1.1148	0.0323	0.1954	0.0299	0.0311	1.4605	0.7691	0.6913
	$r\left(x_0;\underset{¯}{\psi }\right)$	3.9736	0.5880	0.0454	0.0036	0.5877	5.4762	2.4710	3.0052
100	$R\left(x_0;\underset{¯}{\psi }\right)$	0.7746	0.0014	0.0611	0.0006	0.0014	0.8482	0.7011	0.1471
	$h\left(x_0;\underset{¯}{\psi }\right)$	1.1383	0.0235	0.1564	0.0094	0.0234	1.4381	0.8386	0.5995
	$r\left(x_0;\underset{¯}{\psi }\right)$	3.9758	0.3862	0.0387	0.0042	0.3860	5.1935	2.7582	2.4353
200	$R\left(x_0;\underset{¯}{\psi }\right)$	0.7747	0.0007	0.0484	0.0004	0.0007	0.8265	0.7230	0.1035
	$h\left(x_0;\underset{¯}{\psi }\right)$	1.1458	0.0154	0.1334	0.0029	0.0154	1.3887	0.9028	0.4859
	$r\left(x_0;\underset{¯}{\psi }\right)$	3.9639	0.1742	0.0313	0.0012	0.1741	4.7818	3.1460	1.6358
500	$R\left(x_0;\underset{¯}{\psi }\right)$	0.7754	0.0003	0.0341	0.0005	0.0003	0.8104	0.7404	0.0700
	$h\left(x_0;\underset{¯}{\psi }\right)$	1.1506	0.0069	0.1079	0.0013	0.0069	1.3137	0.9876	0.3261
	$r\left(x_0;\underset{¯}{\psi }\right)$	3.9813	0.0601	0.0210	0.0056	0.0596	4.4599	3.5027	0.9572

and

Table 9. ML averages, estimated risks, relative errors, relative absolute biases, variances and 95% ACI bounds of the rf, hrf and rhrf of the Xg-BXII distribution for different $n$, $NR=1000$, $\left(\alpha =0.15,c=1.5,k=0.8\right)$ and $x_0=0.5$.

$\mathit{n}$	Function	Average	ER	RE	RAB	Variance	UL	LL	Length
30	$R\left(x_0;\underset{¯}{\psi }\right)$	0.7757	0.0024	0.0550	0.0023	0.0024	0.8721	0.6794	0.1927
	$h\left(x_0;\underset{¯}{\psi }\right)$	0.6493	0.0126	0.4097	0.0058	0.0126	0.8689	0.4297	0.4392
	$r\left(x_0;\underset{¯}{\psi }\right)$	2.3043	0.1900	0.0497	0.0217	0.1876	3.1532	1.4554	1.6979
60	$R\left(x_0;\underset{¯}{\psi }\right)$	0.7779	0.0016	0.0622	0.0005	0.0016	0.8566	0.6991	0.1576
	$h\left(x_0;\underset{¯}{\psi }\right)$	0.6362	0.0079	0.1696	0.0145	0.0078	0.8091	0.4633	0.3458
	$r\left(x_0;\underset{¯}{\psi }\right)$	2.2673	0.1195	0.0393	0.0053	0.1193	2.9444	1.5903	1.3542
100	$R\left(x_0;\underset{¯}{\psi }\right)$	0.7774	0.0011	0.0517	0.0001	0.0011	0.8419	0.7129	0.1290
	$h\left(x_0;\underset{¯}{\psi }\right)$	0.6371	0.0051	0.1374	0.0131	0.0051	0.7765	0.4978	0.2788
	$r\left(x_0;\underset{¯}{\psi }\right)$	2.2514	0.0754	0.0318	0.0018	0.0754	2.7897	1.7132	1.0765
200	$R\left(x_0;\underset{¯}{\psi }\right)$	0.7781	0.0007	0.0423	0.0008	0.0007	0.8289	0.7273	0.1016
	$h\left(x_0;\underset{¯}{\psi }\right)$	0.6381	0.0030	0.1109	0.0116	0.0029	0.7441	0.5321	0.2120
	$r\left(x_0;\underset{¯}{\psi }\right)$	2.2535	0.0462	0.0242	0.0009	0.0462	2.6749	1.8320	0.8429
500	$R\left(x_0;\underset{¯}{\psi }\right)$	0.7768	0.0003	0.0333	0.0009	0.0003	0.8081	0.7455	0.0626
	$h\left(x_0;\underset{¯}{\psi }\right)$	0.6442	0.0011	0.0846	0.0021	0.0011	0.7095	0.5790	0.1305
	$r\left(x_0;\underset{¯}{\psi }\right)$	2.2481	0.0182	0.0148	0.0032	0.0181	2.5119	1.9843	0.5276

present the ML averages, ERs, REs, RABs variances of the rf, hrf and rhrf along with their ACI bounds and the lengths of the ACIs at time $x_0=0.5$.

f.

Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9 are graphically displayed in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9.

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

• The ML averages of the estimates for the parameters of the Xg-BXII distribution are stabilized as the sample size $n$ increases.
• The ERs and REs of the ML estimates of the parameters $\underset{¯}{\psi }=\left(\alpha ,c,k\right)$, rf, hrf and rhrf decrease, as the sample size increases.
• In most cases, the RABs of the ML estimates of the parameters $\underset{¯}{\psi }=\left(\alpha ,c,k\right)$, rf, hrf and rhrf decrease as the sample size increases.
• As the sample size increases, the variances of the parameters, rf, hrf and rhrf decrease.
• The lengths of the 95% ACIs of the parameters, rf, hrf and rhrf decrease as the sample size increases.

5. 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 $\left(-2\mathcal{L}\right)$, 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

$$AIC=2m-2\mathcal{L},$$

$$BIC=m\ln \left(n\right)-2\mathcal{L},$$

and

$$CAIC=AIC+2\left(\frac{m\left(m+1\right)}{n-m-1}\right),$$

where $\mathcal{L}$ is the natural logarithm of the value of the likelihood function evaluated at the ML estimates, $n$ is the number of the observations and $m$ 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. Failure times of Wang’s data.

Failure Times
5	11	21	31	46	75	98	122	145
165	196	224	245	293	321	330	350	420

and a summary of measures for these data is given in

Table 11. Summary measures of Wang’s data.

Min	${\mathit{x}}_{0.25}$	${\mathit{x}}_{\mathit{m}}$	$\overline{\mathit{x}}$	${\mathit{x}}_{0.75}$	Max
5	53.25	155	172.11	281	420

.

Figure 10 presents 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 right-skewed pdf. Moreover, Figure 11 exhibits 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.

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.

Model	$\underset{¯}{\mathit{\psi }}$	MLE	SE	K-S	p-Value	AIC	BIC	CAIC
Xg-BXII	$\alpha$	0.0134	0.0024	0.0980	0.9883	227.1598	229.8309	228.8741
	$c$	0.9894	0.7263
	$k$	0.0565	0.0511
LogL-W	$\alpha$	0	3.6511 × 102	0.1132	0.9551	228.8923	232.4538	231.9691
	$\beta$	0.8331	6.5613 × 101
	$\theta$	0.0026	8.3356 × 10−4
	$\lambda$	1.1462	7.0112 × 10−2
L-W	$\alpha$	0.0390	0.1127	0.1144	0.9512	228.6825	232.244	231.7594
	$\beta$	0.0375	0.1227
	$\theta$	0.0023	0.0008
	$\lambda$	1.1538	0.0690
AW	$\alpha$	1.1424	0.2285	0.1177	0.9539	228.8920	232.4535	231.9689
	$\beta$	0.0056	0.0012
	$\theta$	3.8917	3.1294 × 102
	$\lambda$	0	1.7617 × 102
TXg	$\alpha$	0.0168	0.0023	0.2292	0.2586	236.7577	238.5384	237.5577
	$\theta$	1.6744	0.0159
Xg	$\alpha$	0.0168	0.0023	0.2263	0.2720	233.3785	234.2689	233.6285
BXII	$c$	3.4236	7.0650	0.3679	0.0103	262.1132	263.8939	262.9132
	$k$	0.0363	0.2104

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.

The estimated asymptotic variance–covariance matrix of the ML estimates for the Xg-BXII distribution for Wang’s data is as follows:

$${\left.{\tilde{I}}^{-1}\left(\underset{¯}{\psi }\right)\right|}_{\underset{¯}{\widehat{\psi }}}\simeq \left(\begin{array}{ccc}5.6861\times {10}^{-6}& -8.2406\times {10}^{-5}& -1.6949\times {10}^{-5}\\ -8.2406\times {10}^{-5}& 5.2757\times {10}^{-1}& -2.8599\times {10}^{-2}\\ -1.6949\times {10}^{-6}& -2.8599\times {10}^{-2}& 2.6101\times {10}^{-3}\end{array}\right),$$

Therefore, the 95% ACI bounds of $\underset{¯}{\psi }=\left(\alpha ,c,k\right)$ are, respectively:

$$0.0134\pm 1.96\sqrt{5.6861e-6}, 0.9894\pm 1.96\sqrt{5.2757e-1},$$

and

$$0.0565\pm 196\sqrt{2.6101e-3}.$$

• Application 2:

This application is given by [32] and represents the time to failure data of 30 devices. The data are tabulated in

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

Failure Times
0.1	0.2	1	2	3	6	7	11	12	18
21	32	36	40	45	46	47	50	55	60
63	67	72	75	79	82	83	84	85	86

and a summary of measures for these data is obtained in

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

Min	${\mathit{x}}_{0.25}$	${\mathit{x}}_{\mathit{m}}$	$\overline{\mathit{x}}$	${\mathit{x}}_{0.75}$	Max
0.1	11.25	45.50	42.28	70.75	86

.

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 displays the fitted pdf, cdf and hrf plots of the Xg-BXII distribution and P-P plot of Meeker and Escober’s data. This figure shows that the fitted pdf of the Xg-BXII distribution has a decreasing-unimodal shape and the fitted hrf is a bathtub shape. The empirical and theoretical cdf plot and the P-P plot indicate that the proposed distribution fits Meeker and Escober’s data well.

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. 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.

Model	$\underset{¯}{\mathit{\psi }}$	MLE	SE	K-S	p-value	AIC	BIC	CAIC
Xg-BXII	$\alpha$	0.0506	0.0073	0.1258	0.6859	279.5354	283.7389	280.4585
	$c$	0.7605	0.3349
	$k$	0.1183	0.0618
LogL-W	$\alpha$	0.0721	0.0485	0.4869	0.0000	323.1017	328.7065	324.7017
	$\beta$	0.2094	0.2475
	$\theta$	0.0077	0.0068
	$\lambda$	1.2239	0.2054
L-W	$\alpha$	0.0037	0.0169	0.1930	0.1875	292.5615	298.1663	294.1615
	$\beta$	0.1799	0.2451
	$\theta$	0.0283	0.0171
	$\lambda$	0.9544	0.1451
AW	$\alpha$	1.6777	0.5348	0.1469	0.4912	282.8124	288.4172	284.4124
	$\beta$	0.0158	0.0027
	$\theta$	0.0028	0.0029
	$\lambda$	0.4609	0.1411
TXg	$\alpha$	0.0515	0.0079	0.1446	0.5112	283.6904	286.4928	284.1348
	$\theta$	0.0738	0.0467
Xg	$\alpha$	0.0646	0.0071	0.2260	0.0793	296.3653	297.7665	296.5082
BXII	$c$	1.1138	0.3116	0.3026	0.0062	327.2089	330.0113	327.6533
	$k$	0.2804	0.0905

. 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 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 Figure 16 and Figure 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.

The estimated asymptotic variance–covariance matrix of the ML estimates for the Xg-BXII distribution for Meeker and Escober’s data is as follows:

$${\left.{\tilde{I}}^{-1}\left(\underset{¯}{\psi }\right)\right|}_{\underset{¯}{\widehat{\psi }}}\simeq \left(\begin{array}{ccc}5.3082\times {10}^{-5}& -0.0003& -0.0001\\ -3.1560\times {10}^{-4}& 0.1122& -0.0128\\ -5.8770\times {10}^{-5}& -0.0128& 0.0038\end{array}\right),$$

Therefore, the 95% ACI bounds of $\underset{¯}{\psi }=\left(\alpha ,c,k\right)$ are, respectively:

$$0.0506\pm 1.96\sqrt{5.3082\times {10}^{-5}}, 0.7605\pm 1.96\sqrt{0.1122},$$

and

$$0.1183\pm 196\sqrt{0.0038}.$$

• Application 3:

This application represents the lifetime of 20 electronic components given by [33]. The data are tabulated in

Table 16. Failure times of Murthy’s data.

Failure Times
0.03	0.12	0.22	0.35	0.73	0.79	1.25	1.41	1.25	1.79
1.80	1.94	2.38	2.40	2.87	2.99	3.14	3.19	4.72	5.09

, while

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

Min	${\mathit{x}}_{0.25}$	${\mathit{x}}_{\mathit{m}}$	$\overline{\mathit{x}}$	${\mathit{x}}_{0.75}$	Max
0.03	0.775	1.795	1.923	2.900	5.090

presents a summary of measures for these data.

In Figure 18 the empirical scaled TTT-transform plot, boxplot and the histogram of Murthy’s data are given. The empirical scaled TTT-transform plot implies that the data have a bathtub hrf and from the boxplot and the histogram it is observed that the data have a right-skewed pdf. In addition, Figure 19 exhibits the fitted pdf, cdf and hrf plots of the Xg-BXII distribution and P–P plot of Murthy’s data. From which the fitted pdf of the Xg-BXII distribution has a decreasing-unimodal shape and the fitted hrf is a bathtub shape. The empirical and theoretical cdf plot and the P-P plot indicate that the Xg-BXII distribution fits well 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.

Model	$\underset{¯}{\mathit{\psi }}$	MLE	SE	K-S	p-Value	AIC	BIC	CAIC
Xg-BXII	$\alpha$	1.6585	0.4854	0.0793	0.9985	68.2122	71.1994	69.7122
	$c$	1.4737	0.3825
	$k$	0.7792	0.5698
LogL-W	$\alpha$	0.7653	0.3680	0.0943	0.9866	70.6350	74.6179	73.3017
	$\beta$	3.5675	4.4914
	$\theta$	0.0618	0.0962
	$\lambda$	2.3888	0.9450
L-W	$\alpha$	0.8038	0.9719	0.0928	0.9888	71.1862	75.1646	73.8483
	$\beta$	0.4535	0.6129
	$\theta$	0.0658	0.0906
	$\lambda$	2.2808	0.8595
AW	$\alpha$	2.4874	1.1855	0.0949	0.9858	70.6935	74.6764	73.3602
	$\beta$	0.2950	0.0954
	$\theta$	0.2214	0.2859
	$\lambda$	0.7303	0.3590
TXg	$\alpha$	1.1999	0.2288	0.0939	0.9873	69.4791	71.4706	70.1850
	$\theta$	2.2659	2.0125
Xg	$\alpha$	1.0484	0.1611	0.1447	0.7439	71.2099	72.2056	71.4321
BXII	$c$	1.4535	0.3065	0.2328	0.1952	77.0271	79.0185	77.7329
	$k$	0.8763	0.2231

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 Figure 20 and Figure 21. 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 Figure 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.

The estimated asymptotic variance–covariance matrix of the ML estimates for the Xg-BXII distribution for Murthy’s data is as follows:

$${\left.{\tilde{I}}^{-1}\left(\underset{¯}{\psi }\right)\right|}_{\underset{¯}{\widehat{\psi }}}\simeq \left(\begin{array}{ccc}0.2356& 0.0051& -0.2596\\ 0.0051& 0.1463& 0.0396\\ -0.2596& 0.0396& 0.3247\end{array}\right).$$

Therefore, the 95% ACI bounds of $\underset{¯}{\psi }=\left(\alpha ,c,k\right)$ are, respectively:

$$1.6585\pm 1.96\sqrt{0.2356}, 1.4737\pm 1.96\sqrt{0.1463},$$

and

$$0.7792\pm 196\sqrt{0.3247}.$$

• Application 4:

This application is from [34]. 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 drought mortality rates. The data are tabulated in

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

Drought Mortality Rates
0.0587	0.0863	0.1165	0.1247	0.1277	0.1303	0.1652	0.2079	0.2395	0.2751	0.2845
0.2992	0.3188	0.3317	0.3446	0.3553	0.3622	0.3926	0.3629	0.4110	0.4633	0.4690
0.4954	0.5139	0.5696	0.5837	0.6197	0.6365	0.7096	0.7193	0.7444	0.8590	1.0438
1.0602	1.1305	1.1468	1.1533	1.2260	1.2707	1.3423	1.4149	1.5709	1.6017	1.6083
1.6324	1.6998	1.8164	1.8392	1.8721	1.9844	2.1360	2.3987	2.4153	2.5225	2.7087
2.7946	3.3609	3.3715	3.7840	3.9042	4.1969	4.3451	4.4627	4.6477	5.3664	5.4500
5.7522	6.4241	7.0657	7.4456	8.2307	9.6315	10.1870	11.1429	11.2019	22.4584

and

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

Min	${\mathit{x}}_{0.25}$	${\mathit{x}}_{\mathit{m}}$	$\overline{\mathit{x}}$	${\mathit{x}}_{0.75}$	Max
0.0587	0.4064	1.2484	2.4372	3.3636	11.4584

presents a summary of measures for these data.

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.

Moreover, Figure 23 displays the fitted pdf, cdf and hrf plots of the Xg-BXII distribution and P–P plot of COVID-19 data in the United Kingdom. It is obvious that the fitted pdf of the Xg-BXII distribution has a unimodal shape and the fitted hrf is a modified bathtub shape. The empirical and theoretical cdf plot and the P-P plot indicate that the Xg-BXII distribution fits the given data very well.

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.

Model	$\underset{¯}{\mathit{\psi }}$	MLE	SE	K-S	p-Value	AIC	BIC	AICc
Xg-BXII	$\alpha$	0.3454	0.1043	0.0609	0.9242	283.9968	290.9890	284.3301
	$c$	1.2184	0.1685
	$k$	0.6902	0.1452
LogL-W	$\alpha$	0.8673	0.2271	0.0800	0.6847	291.1339	300.4569	291.6973
	$\beta$	9.6291	0.9726
	$\theta$	0.3803	0.1367
	$\lambda$	0.8697	0.1023
L-W	$\alpha$	2.5131	23.8014	0.0797	0.6702	291.4390	300.7620	292.0024
	$\beta$	0	0.0602
	$\theta$	0.5547	0.3752
	$\lambda$	0.8610	0.1268
AW	$\alpha$	0.8466	0.1787	0.0806	0.6764	291.5037	300.8266	292.0671
	$\beta$	0.1649	6.4435
	$\theta$	0.2333	6.7957
	$\lambda$	0.8467	0.1421
TXg	$\alpha$	0.5412	0.0890	0.1279	0.1523	288.5941	293.2556	288.7585
	$\theta$	0.1028	0.0864
Xg	$\alpha$	0.8154	0.0643	0.1904	0.0069	302.7674	305.0982	302.8215
BXII	$c$	1.3326	0.1420	0.0686	0.8431	288.9744	293.6359	289.1388
	$k$	0.9017	0.1171

exhibits 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 Table 21, 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 Figure 24 and Figure 25. 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 Figure 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:

$${\left.{\tilde{I}}^{-1}\left(\underset{¯}{\psi }\right)\right|}_{\underset{¯}{\widehat{\psi }}}\simeq \left(\begin{array}{ccc}0.0109& -0.0045& -0.0095\\ -0.0045& 0.0284& -0.0029\\ -0.0095& -0.0029& 0.0211\end{array}\right)$$

Therefore, the 95% ACI bounds of $\underset{¯}{\psi }=\left(\alpha ,c,k\right)$ are, respectively:

$$1.6585\pm 1.96\sqrt{0.2356}, 1.4737\pm 1.96\sqrt{0.1463},$$

and

$$0.7792\pm 196\sqrt{0.3247}.$$

6. 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.