The Exponentiated Burr XII Power Series Distribution: Properties and Applications

Abstract

In this work, we introduce a new Burr XII power series class of distributions, which is obtained by compounding exponentiated Burr XII and power series distributions and has a strong physical motivation. The new distribution contains several important lifetime models. We derive explicit expressions for the ordinary and incomplete moments and generating functions. We discuss the maximum likelihood estimation of the model parameters. The maximum likelihood estimation procedure is presented. We assess the performance of the maximum likelihood estimators in terms of biases, standard deviations, and mean square of errors by means of two simulation studies. The usefulness of the new model is illustrated by means of three real data sets. The new proposed models provide consistently better fits than other competitive models for these data sets.

1. Introduction and Motivation

Burr XII distribution is the most commonly used distribution for non-monotonic hazard rates. Burr XII distribution has algebraic tails useful for modeling failures that occur with lesser frequency than those with corresponding models based on exponential tails. Burr XII distribution finds its application in flood frequency, software reliability, structural and wind engineering. Burr XII distribution has two popular distributions as sub models i.e., Lomax distribution and Log logistic distributions. Weibull distribution is a limiting case of Burr XII distribution. Many extensions of the Burr XII distribution are found in literature, such as Kumaraswamy Burr XII distribution by [1], extended Marshall Olkin Burr XII distribution by [2], McDonald Burr XII distribution by [3] and a new generalized Burr family of distributions is proposed by [4].

The cumulative distribution function (cdf) and probability density function (pdf) of the two parameter Burr XII (BXII) distribution are given by

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

(1)

and

$$f_{c,k}\left(x\right)=ck{x}^{c-1}{(1+x^c)}^{-k-1}.$$

(2)

If $c=1$, the scaled BXII distribution reduces to Lomax distribution, if $k=1$, then Burr XII distribution reduces to log logistic distribution (see [5]). Furthermore, Burr XII distribution finds its application in various areas such as flood frequency, software reliability, structural engineering, wind engineering, renewable energy and climatology (see Garcia et al. [6]; Harris [7], Kantar and Usta [8]; Carta et al. [9]; Panteli and Mancarella [10]; Chiodo and De Falco [11]; Feijóo [12]). On the other hand, Noack [13] proposed and studied the power series class of distributions; this class of distribution includes binomial, geometric, logarithmic and Poisson distributions as special cases. However, these distributions may not be useful when a random variable (rv) takes the value of zero with high probability i.e., zero inflated. In such situations, it is more appropriate to consider the distribution which is truncated at zero.

The cdf of the G-power series (G-PS) distributions is

$$F_{\lambda ,\xi }\left(x\right)=1-C\left[\lambda \overline{G}(x;\xi )\right]/C\left(\lambda \right).$$

(3)

A discrete rv N is a member of power series distributions (truncated at the zero), if the probability mass function (pmf) is given by

$$P(N=n|\lambda )=a_n{\lambda }^n{/C\left(\lambda \right)|}_{\left(n=1,2,\dots \right)},$$

(4)

where $a_n\ge 0$, $C\left(\lambda \right)={\displaystyle \sum _{n=1}^{\infty }}{a}_n{\lambda }^n$ and $\lambda \in (0,s)$ is chosen such that $C\left(\lambda \right)$ is finite and its first, second and third derivative are defined also

$$C^{\prime }\left(\lambda \right)={\displaystyle \sum _{n=1}^{\infty }}n{a}_n{\lambda }^{n-1}.$$

Some example of $C\left(\lambda \right)$ functions are given by

Table 1. Useful quantities for power series distributions.

Distribution	$\mathit{C}\left(\mathit{\lambda }\right)$	${\mathit{C}}^{-1}\left(\mathit{\lambda }\right)$	${\mathit{a}}_{\mathit{n}}$	Parameter Space
Poisson	$e^{\lambda }-1$	$log(1+\lambda )$	${(n!)}^{-1}$	$(0,\infty )$
Geometric	$\lambda {(1-\lambda )}^{-1}$	$\lambda {(1+\lambda )}^{-1}$	1	(0,1)
Binomial	${(1+\lambda )}^m-1$	${(\lambda +1)}^{m\frac{1}{}}-1$	$\left(\genfrac{}{}{0pt}{}{n}{m}\right)$	$(0,\infty )$
Logarithmic	$-log(1-\lambda )$	$1-e^{-\lambda }$	$n^{-1}$	(0,1)

. The pdf, survival function (sf) and hazard rate function (hrf) corresponding to Equation (3) are

$$f_{\lambda ,\xi }\left(x\right)=\lambda g\left(x\right)C^{\prime }\left[\lambda \overline{G}(x;\xi )\right]/C\left(\lambda \right),$$

(5)

$${\overline{F}}_{\lambda ,\xi }\left(x\right)=C\left[\lambda \overline{G}(x;\xi )\right]/C\left(\lambda \right),$$

(6)

and

$$h_{\lambda ,\xi }\left(x\right)=\lambda g(x;\xi )C^{\prime }\left[\lambda \overline{G}(x;\xi )\right]/C\left[\lambda \overline{G}(x;\xi )\right].$$

(7)

The quantile function can be obtained by using Lambert function.

We provide three motivations of the G-PS class of distribution, which can be applied in some interesting situation: due to the Stochastic reorientation

$$Z=min(X_1,X_2,\dots ,Z_N),$$

the G-PS class of distributions can arise in many industrial applications and biological organisms. The G-PS class of distributions can be used to model approximately the time to the first failure of a system of identical components that are in a series. The G-PS class of distributions exhibit some interesting behaviors with non-monotonic failure rates such as bathtub, upside bathtub and increasing-decreasing failure rates which are more likely to be encountered in real life situations. The justification for the practicality of the new model is based on the wider use of the BXII model. In addition, we are motivated to introduce the exponentiated Burr XII power series (EBXIIPS) model because it exhibits increasing, decreasing, bathtub and reversed J shaped hazard rates as illustrated in Figures 1–4. It can be viewed as a suitable model for fitting the right skewed and unimodal data.

The rest of the paper is outlined as follows. Construction of the new family is given in Section 2. Four special models of the new family are presented in Section 3. In Section 4, we derive some mathematical properties of this family. Maximum likelihood estimation of the parameters is addressed in Section 5. Two simulation studies are given by Section 6 to see performance of the maximum likelihood estimators for two special members of this family. In Section 7, the potentiality of the proposed models is illustrated by means of two real data sets. In Section 8, we offer some concluding remarks.

2. Construction of the New Family

Let $X_{\left(1\right)}=min(X_1,X_2,\dots ,X_n)$. The conditional cdf of $X_{\left(1\right)}|N=n$ is given by

$$G_{X_{\left(1\right)}|N=n}\left(x\right)=1-{\left[\overline{G}\left(x\right)\right]}^n.$$

(8)

If $\overline{G}(.)$ is the survival function of exponentiated Burr XII (EBXII) distribution, with cdf and pdf as under

$$G_{c,k,\alpha }\left(x\right)={\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }{|}_{\left(x>0,c,k,\alpha >0\right)},$$

(9)

and

$$g_{c,k,\alpha }\left(x\right)=\alpha ck{x}^{c-1}{(1+x^c)}^{-k-1}{\left[1-{(1+x^c)}^{-k}\right]}^{\alpha -1}.$$

(10)

Then, Equation (8) becomes

$$G_{X_{\left(1\right)}|N=n}\left(x\right)=1-{\left[1-{\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right]}^n.$$

(11)

Then, the cdf and pdf of the EBXIIPS distribution are, respectively, given by

$$F_{\Theta }\left(x\right)=1-C\left[\lambda -\lambda {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right]/C\left(\lambda \right),$$

(12)

and

$$f_{\Theta }\left(x\right)=\lambda \alpha ck{x}^{c-1}\frac{{\left[1-{(1+x^c)}^{-k}\right]}^{\alpha -1}}{{(1+x^c)}^{k+1}}\frac{C^{\prime }\left[\lambda -\lambda {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right]}{C\left(\lambda \right)},$$

(13)

where $\Theta ={(\lambda ,c,k,\alpha )}^T$. For $\alpha =1$, the EBXIIPS family is reduced to a Burr XII power series family that has been introduced by [14]. The hrf of EBXIIPS distribution is obtained with

$$h_{\Theta }\left(x\right)=\lambda \alpha ck{x}^{c-1}\frac{{\left[1-{(1+x^c)}^{-k}\right]}^{\alpha -1}}{{(1+x^c)}^{k+1}}\frac{C^{\prime }\left[\lambda -\lambda {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right]}{C\left[\lambda -\lambda {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right]}.$$

(14)

The quantile function of EBXIIPS distribution, $Q_x\left(p\right)$, can be defined as $F\left(Q_x\left(p\right)\right)=p$

$$x=Q_x\left(p\right)={\left\{-1+{\left[1-Z(\lambda ,p)\right]}^{-\frac{1}{k}}\right\}}^{\frac{1}{c}},$$

(15)

where

$$Z(\lambda ,p)={\left\{\frac{\lambda -C^{-1}\left[(1-p)C\left(\lambda \right)\right]}{\lambda }\right\}}^{\frac{1}{\alpha }},$$

and $C^{-1}(.)$ is the inverse function of $C(.)$.

Proposition 1.

The exponentiated Burr XII distribution is the limiting case of the EBXIIPS family of distributions when $\lambda \to 0^{+}$.

Proof. 

For $X>0$, we have

$$\begin{array}{ccc}\hfill \underset{\lambda \to 0^{+}}{lim}F\left(x\right)& =& 1-\underset{\lambda \to 0^{+}}{lim}\frac{{\displaystyle \sum _{n=1}^{\infty }}{a}_n{\left[\lambda \left\{1-{\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right\}\right]}^n}{{\displaystyle \sum _{n=1}^{\infty }}{a}_n{\lambda }^n}\hfill \\ & =& 1-\underset{\lambda \to 0^{+}}{lim}\frac{a_1\lambda \left\{1-{\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right\}+{\displaystyle \sum _{n=2}^{\infty }}{a}_n{\left[\lambda \left\{1-{\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right\}\right]}^n}{a_1\lambda +{\displaystyle \sum _{n=2}^{\infty }}{a}_n{\lambda }^n}\hfill \\ & =& 1-\underset{\lambda \to 0^{+}}{lim}\frac{a_1\lambda {\left[\left\{1-{\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right\}+a_1^{-1}{\displaystyle \sum _{n=2}^{\infty }}{a}_n{\lambda }^{n-1}\left\{1-{\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right\}\right]}^n}{a_1\lambda \left[1+a_1^{-1}{\displaystyle \sum _{n=2}^{\infty }}{a}_n{\lambda }^n\right]}.\hfill \end{array}$$

Now, applying the limit on the right side, we have

$$\underset{\lambda \to 0^{+}}{lim}F\left(x\right)={\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }.$$

(16)

 □

Proposition 2.

The infinite mixture representation of EBXIIPS can be expressed in terms of EBXII distribution with parameters $c,k$ and $n\alpha$.

Proof. 

The pdf of EBXIIPS distribution can be expressed as an infinite mixture representation of order distribution i.e.,

$$f\left(x\right)={\displaystyle \sum _{n=1}^{\infty }}P(N=n)g_{\left(1\right)}(x;n),$$

(17)

where $P(N=n)$ is defined in (4). Now, (17) becomes

$$f\left(x\right)={\displaystyle \sum _{n=1}^{\infty }}\frac{a_n{\lambda }^n}{C\left(\lambda \right)}{g}_{\left(1\right)}(x;n),$$

(18)

where $g_{\left(1\right)}(x;n)$ is the pdf of $Y_{\left(i\right)}=min(Y_1,\dots ,Y_n)$, is given by

$$g_{\left(1\right)}(x;n)=ng\left(x\right){\left[1-G\left(x\right)\right]}^n.$$

(19)

Using (9) and (10), we have

$$g_{\left(1\right)}(x;n)=n\alpha ck\frac{x^{c-1}}{{(1+x^c)}^{k+1}}{\left[1-G\left(x\right)\right]}^n{\left[1-{(1+x^c)}^{-k}\right]}^{n\alpha -1}.$$

(20)

Now, (21) becomes

$$f\left(x\right)=n\alpha ck{\displaystyle \sum _{n=1}^{\infty }}\frac{a_n{\lambda }^n}{C\left(\lambda \right)}\frac{x^{c-1}}{{(1+x^c)}^{k+1}}{\left[1-G\left(x\right)\right]}^n{\left[1-{(1+x^c)}^{-k}\right]}^{n\alpha -1}={\displaystyle \sum _{n=1}^{\infty }}{w}_{n}g(x;c,k,n\alpha ),$$

where $w_n=n\alpha ck\frac{a_n{\lambda }^n}{C\left(\lambda \right)}$ and $g(x;c,k,n\alpha )=\frac{x^{c-1}}{{(1+x^c)}^{k+1}}{\left[1-G\left(x\right)\right]}^n{\left[1-{(1+x^c)}^{-k}\right]}^{n\alpha -1}$ is the EBXII density function with parameters $c,k$ and $n\alpha$.  □

3. Special Sub Models

In this section, we will discuss some special models of the power series distributions such as logarithmic, binomial, Poisson and geometric distributions.

3.1. EBXII Logarithmic Distribution

Consider the logarithmic distribution to be the zero truncated power series distribution with $a_n=\frac{1}{n}$ and $C\left(\theta \right)=-log(1-\theta )$. Then, the cdf and pdf and hrf of EBXII logarithmic (EBXII-L) distribution are respectively given by

$$F_{\theta ,c,k,\alpha }\left(x\right)=1-\frac{log\left[1-\theta +\theta {\left\{1-{(1+x^c)}^{-k}\right\}}^{\alpha }\right]}{log(1-\theta )},$$

(21)

$$f_{\theta ,c,k,\alpha }\left(x\right)=\frac{\theta \alpha ck{x}^{c-1}{(1+x^c)}^{-k-1}{\left[1-{(1+x^c)}^{-k}\right]}^{\alpha -1}}{log(1-\theta )\left\{\theta -\theta {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }-1\right\}},$$

and

$$h_{\theta ,c,k,\alpha }\left(x\right)=\frac{\theta \alpha ck{x}^{c-1}{(1+x^c)}^{-k-1}{\left[1-{(1+x^c)}^{-k}\right]}^{\alpha -1}}{\left\{\theta -\theta {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }-1\right\}log\left\{1-\theta +\theta {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right\}}.$$

(22)

For the EBXII-L model, we can say that $\theta$ parameter is also valid on $(-\infty ,1)$. In Figure 1, the plots of density and hazard rate functions are displayed. The density is right skewed, reversed J and symmetrical, while hrf is decreasing and upside-down bathtub.

3.2. EBXII Binomial Distribution

Consider the binomial distribution to be the zero truncated power series distribution with $a_n=\left(\genfrac{}{}{0pt}{}{m}{n}\right)$ and $C\left(\theta \right)={(\theta +1)}^m-1$, where $m(n\le m)$ is the number of replicas. Then, the cdf and pdf and hrf of EBXII binomial (EBXII-B) distribution are respectively given by

$$F_{\theta ,c,k,\alpha }\left(x\right)=1-\frac{-1+{\left\{1+\theta -\theta {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right\}}^m}{-1+{(\theta +1)}^m},$$

(23)

$$\begin{array}{ccc}\hfill f_{\theta ,c,k,\alpha }\left(x\right)& =& m\theta \alpha ck{x}^{c-1}{(1+x^c)}^{-k-1}{\left[1-{(1+x^c)}^{-k}\right]}^{\alpha -1}\hfill \\ & & \times \frac{{\left\{1+\theta -\theta {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right\}}^{m-1}}{-1+{(\theta +1)}^m},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill h_{\theta ,c,k,\alpha }\left(x\right)& =& m\theta \alpha ck{x}^{c-1}{(1+x^c)}^{-k-1}{\left\{1-{(1+x^c)}^{-k}\right\}}^{\alpha -1}\hfill \\ & & \times \frac{{\left\{1+\theta -\theta {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right\}}^{m-1}}{{\left\{1+\theta -\theta {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right\}}^m-1}.\hfill \end{array}$$

In Figure 2, the plots of density and hazard rate functions are displayed. The density is right skewed and reversed J, while hrf is decreasing and upside-down bathtub.

3.3. EBXII Poisson Distribution

Consider the Poisson distribution to be the zero truncated power series distribution with $a_n=\frac{1}{n!}$ and $C\left(\theta \right)=e^{\theta }-1$. Then, the cdf and pdf and hrf of EBXII Poisson (EBXII-P) distribution are respectively given by

$$F_{\theta ,c,k,\alpha }\left(x\right)=\frac{e^{\theta }-e^{\theta -\theta {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }}}{e^{\theta }-1},$$

(24)

$$f_{\theta ,c,k,\alpha }\left(x\right)=\frac{\theta \alpha ck{x}^{c-1}{\left[1-{(1+x^c)}^{-k}\right]}^{\alpha -1}{e}^{\theta -\theta {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }}}{{(1+x^c)}^{k+1}\left(e^{\theta }-1\right)},$$

and

$$h_{\theta ,c,k,\alpha }\left(x\right)=\theta \alpha ck{x}^{c-1}{\left[1-{(1+x^c)}^{-k}\right]}^{\alpha -1}\frac{e^{\theta -\theta {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }}}{e^{\theta -\theta {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }}-1}.$$

We note that EBXII-P distribution has been introduced by [15]. In Figure 3, the plots of density and hazard rate functions are displayed. The density is reversed J, symmetrical and left skewed, while hrf is decreasing, increasing and bathtub shaped.

3.4. EBXII Geometric Distribution

Consider the geometric distribution to be the zero truncated power series distribution with $a_n=1$ and $C\left(\theta \right)=\frac{\theta }{1-\theta }$. Then, the cdf and pdf and hrf of EBXII geometric (EBXII-G) distribution are respectively given by

$$F_{\theta ,c,k,\alpha }\left(x\right)=\frac{{\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }}{1-\theta +\theta {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }},$$

(25)

$$f_{\theta ,c,k,\alpha }\left(x\right)=\frac{\alpha ck(1-\theta )x^{c-1}{\left[1-{(1+x^c)}^{-k}\right]}^{\alpha -1}}{{\left\{1-\theta +\theta {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right\}}^2},$$

and

$$h_{\theta ,c,k,\alpha }\left(x\right)=\frac{\alpha ck{x}^{c-1}{\left[1-{(1+x^c)}^{-k}\right]}^{\alpha -1}}{\left\{1-{\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right\}\left\{1-\theta +\theta {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right\}}.$$

For the EBXII-G model, we can say that $\theta$ parameter is also valid on $(-\infty ,1)$. In Figure 4, the plots of density and hrfs are displayed. The density is reversed J, symmetrical and right skewed, while hrf is decreasing, bathtub and upside-down bathtub.

4. Mathematical Properties

Moments

The $r^{th}$ moments of EBPS distribution can be obtained by using the following expressions:

$${\mu }_r^{\prime }={\displaystyle \underset{0}{\overset{\infty }{\int }}}{x}^{r}f\left(x\right)dx.$$

Using the infinite mixture representation in (21), we have

$${\mu }_r^{\prime }={\displaystyle \sum _{n=1}^{\infty }}{w}_n{\displaystyle \underset{0}{\overset{\infty }{\int }}}{x}^{r}g(x;c,k,n\alpha )dx={\displaystyle \sum _{n=1}^{\infty }}{w}_n\left({\tau }_r{|}_0^{\infty }\right),$$

where ${\tau }_r{|}_0^{\infty }={\displaystyle \underset{0}{\overset{\infty }{\int }}}{x}^{r}g(x;c,k,n\alpha )dx$. Consider

$${\tau }_r{|}_0^{\infty }={\displaystyle \underset{0}{\overset{\infty }{\int }}}{x}^{r+c-1}\frac{{\left[{(1+x^c)}^k\right]}^{n\alpha -1}}{{(1+x^c)}^{k+1}}dx.$$

Using the generalized binomial theorem and, after some algebra, we have

$${\tau }_r{|}_0^{\infty }={\displaystyle \sum _{j=0}^{\infty }}\left(\begin{array}{c}n\alpha -1\\ j\end{array}\right){(-1)}^{j}B\left(1+\frac{r}{c},(1+j)k-\frac{r}{c}\right),$$

where $B\left(·,·\right)$ is the ordinary beta function. Similarly, the r^th incomplete moment is

$$T_r^{\prime }\left(x\right)={\displaystyle \sum _{n=1}^{\infty }}{w}_n{\displaystyle \underset{0}{\overset{\infty }{\int }}}{x}^{r}g(x;c,k,n\alpha )dx={\displaystyle \sum _{n=1}^{\infty }}{w}_n{\Delta }_r^{\prime }\left(X\right),$$

where

$${\Delta }_r^{\prime }\left(X\right)={\displaystyle \sum _{j=0}^{\infty }}\left(\begin{array}{c}n\alpha -1\\ j\end{array}\right){(-1)}^j{B}_{x^c}\left(1+\frac{r}{c},(1+j)k-\frac{r}{c}\right)$$

and $B_z\left(·,·\right)$ is the incomplete beta function. The moment generating function of EBPS distribution can be obtained as

$$M_0\left(t\right)={\displaystyle \sum _{n=1}^{\infty }}{w}_n{\displaystyle \underset{0}{\overset{\infty }{\int }}}{e}^{tx}g(x;c,k,n\alpha )dx={\displaystyle \sum _{n=1}^{\infty }}{w}_n{M}_{EB}\left(t\right),$$

where

$$M_{EB}\left(t\right)={\displaystyle \underset{0}{\overset{\infty }{\int }}}{e}^{tx}\frac{x^{c-1}}{{(1+x^c)}^{k+1}}{\left[{(1+x^c)}^k\right]}^{n\alpha -1}dx.$$

Using generalize binomial theorem and after some algebra, we have

$$M_{EB}\left(t\right)={\displaystyle \sum _{j=0}^{\infty }}{\displaystyle \sum _{i=0}^{\infty }}\left(\begin{array}{c}n\alpha -1\\ j\end{array}\right)\left(\begin{array}{c}(1+j)k+i\\ i\end{array}\right){(-1)}^{i+j}\Gamma \left((1+i)c\right){\left(\frac{-1}{t}\right)}^{(1+i)c}$$

The mean deviations about mean and median of EBPS distribution can be obtained as

$$D_{\mu }=2\mu F\left(\mu \right)-2T_1^{\prime }\left(\mu \right),$$

and

$$D_M=\mu -2T_1^{\prime }\left(M\right),$$

where $\mu =E\left(x\right)={\mu }_1^{\prime }$ can be obtained from (12), $F(.)$ can be obtained from (15) and $T_1^{\prime }\left(z\right)$ is the first incomplete moment.

5. Estimation

The maximum likelihood estimators (MLEs) has a desirable properties and can be used for constructing confidence intervals and regions and also in test statistics. The normal approximation for these estimators in large samples can be easily handled either analytically or numerically. Thus, we consider the estimation of the unknown parameters of this family from complete samples only by maximum likelihood. Let $x_1,\dots ,x_n$ be a random sample from the EBXIIPS distribution with parameters $\lambda ,\alpha ,k$ and c. Let $\Theta$ = ${\left(\lambda ,\alpha ,k,c\right)}^T$ be the 4 × 1 parameter vector. For determining the MLE of parameters, we have the log-likelihood function

$$\begin{array}{ccc}\hfill \ell =\ell (\Theta )& =& nlogc+nlog\lambda +nlogk+nlog\alpha -nlogC\left[\lambda \right]+(c-1){\displaystyle \sum _{i=1}^n}log{x}_i-(k+1){\displaystyle \sum _{i=1}^n}log\left(1+x_i^c\right)\hfill \\ & +& (n\alpha -1){\displaystyle \sum _{i=1}^n}log\left[1-{\left(1+x_i^c\right)}^{-k}\right]+log\left(C^{\prime }\left\{\lambda -\lambda {\left[1-{\left(1+x_i^c\right)}^{-k}\right]}^{\alpha }\right\}\right).\hfill \end{array}$$

The components of score vector are

$$U_{\lambda }=\frac{n}{\lambda }-\frac{n{C}^{\prime }\left[\lambda \right]}{C\left[\lambda \right]}+{\displaystyle \sum _{i=1}^n}\frac{{\left[1-{\left(1+x_i^c\right)}^{-k}\right]}^{\alpha }{C}^{″}\left\{\lambda -\lambda {\left[1-{\left(1+x_i^c\right)}^{-k}\right]}^{\alpha }\right\}}{C^{\prime }\left\{\lambda -\lambda {\left[1-{\left(1+x_i^c\right)}^{-k}\right]}^{\alpha }\right\}},$$

$$\begin{array}{ccc}\hfill U_{\alpha }& =& \frac{n}{\alpha }-\lambda {\displaystyle \sum _{i=1}^n}\frac{log\left[1-{\left(1+x_i^c\right)}^{-k}\right]{\left[1-{\left(1+x_i^c\right)}^{-k}\right]}^{\alpha }{C}^{″}\left\{\lambda -\lambda {\left[1-{\left(1+x_i^c\right)}^{-k}\right]}^{\alpha }\right\}}{C^{\prime }\left\{\lambda -\lambda {\left[1-{\left(1+x_i^c\right)}^{-k}\right]}^{\alpha }\right\}}\hfill \\ & & +n{\displaystyle \sum _{i=1}^n}log\left[1-{\left(1+x_i^c\right)}^{-k}\right],\hfill \end{array}$$

$$\begin{array}{ccc}\hfill U_k& =& \frac{n}{k}-{\displaystyle \sum _{i=1}^n}log\left(1+x_i^c\right)+(n\alpha -1){\displaystyle \sum _{i=1}^n}\frac{log\left(1+x_i^c\right){\left(1+x_i^c\right)}^{-k}}{1-{\left(1+x_i^c\right)}^{-k}}\hfill \\ & & -\alpha \lambda {\displaystyle \sum _{i=1}^n}\frac{log\left(1+x_i^c\right){\left(1+x_i^c\right)}^{-k}{\left[1-{\left(1+x_i^c\right)}^{-k}\right]}^{\alpha -1}{C}^{″}\left\{\lambda -\lambda {\left[1-{\left(1+x_i^c\right)}^{-k}\right]}^{\alpha }\right\}}{C^{\prime }\left\{\lambda -\lambda {\left[1-{\left(1+x_i^c\right)}^{-k}\right]}^{\alpha }\right\}},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill U_c& =& \frac{n}{c}-{\displaystyle \sum _{i=1}^n}log{x}_i-(k+1){\displaystyle \sum _{i=1}^n}\frac{x_i^{c}log{x}_i}{1+x_i^c}+k(n\alpha -1){\displaystyle \sum _{i=1}^n}\frac{x_i^{c}log{x}_i{(1+x_i^c)}^{-k-1}}{-{(1+x_i^c)}^{-k}1}\hfill \\ & & -k\alpha {\displaystyle \sum _{i=1}^n}\frac{x_i^{c}log{x}_i{(1+x_i^c)}^{-k-1}{\left[1-{\left(1+x_i^c\right)}^{-k}\right]}^{\alpha -1}{C}^{″}\left\{\lambda -\lambda {\left[1-{\left(1+x_i^c\right)}^{-k}\right]}^{\alpha }\right\}}{C^{\prime }\left\{\lambda -\lambda {\left[1-{\left(1+x_i^c\right)}^{-k}\right]}^{\alpha }\right\}}.\hfill \end{array}$$

Setting the nonlinear system of equations $U_{\lambda }=U_{\alpha }=U_k=U_c=0$ above and solving them simultaneously yields the MLE $\widehat{\Theta }={(\widehat{\lambda },\widehat{\alpha },\widehat{k},\widehat{c})}^T$. To solve these equations, it is usually more convenient to use nonlinear optimization methods such as the quasi-Newton algorithm to numerically maximize ℓ. The above log-likelihood function also can be maximized numerically by using R (optim), SAS (PROC NLMIXED) or Ox program (sub-routine MaxBFGS), among others.

For interval estimation of the parameters, we obtain the $4\times 4$ observed information matrix $J_{rs}=J(\Theta )=\left\{\frac{{\partial }^2\ell }{\partial r\partial s}\right\}$ (for $r,s=\lambda ,\alpha ,k,c$), whose elements are available upon request from the authors. Its elements also can be computed numerically by the packet program. Under standard regularity conditions when $n\to \infty$, the distribution of $\widehat{\Theta }$ can be approximated by a multivariate normal $N_4(0,J{\left(\widehat{\Theta }\right)}^{-1})$ distribution to construct approximate confidence intervals for the parameters. Here, $J\left(\widehat{\Theta }\right)$ is the total observed information matrix evaluated at $\widehat{\Theta }$. Then, approximate $100(1-\delta )\%$ confidence intervals for $\alpha$, $\lambda$, c and k can be determined by: $\widehat{\alpha }\pm z_{\delta /2}\sqrt{{\widehat{J}}_{\alpha \alpha }^{-1}}$, $\widehat{\lambda }\pm z_{\delta /2}\sqrt{{\widehat{J}}_{\lambda \lambda }^{-1}}$, $\widehat{c}\pm z_{\delta /2}\sqrt{{\widehat{J}}_{cc}^{-1}}$ and $\widehat{k}\pm z_{\delta /2}\sqrt{{\widehat{J}}_{kk}^{-1}}$, where $z_{\delta /2}$ is the upper $\delta$th percentile of the standard normal model and ${\widehat{J}}_{ii}^{-1}$ are diagonal elements of $J{\left(\widehat{\Theta }\right)}^{-1}$ for $i=\alpha$, $\lambda$, c and k.

6. Simulation Studies

In this section, we perform two simulation studies by using the EBXII binomial and EBXII logarithmic distributions to see the performance of MLEs corresponding to these distribution. The random numbers generation is obtained by the inverse of their cdfs. Thus, we use the following steps:

• Generate $u\sim uniform(0,1),$
• Solve the following nonlinear equation for given parameters values,

  $$F(x;\Theta )=1-C\left[\lambda -\lambda {\left[1-{(1+x^c)}^{-k}\right]}^{\alpha }\right]/C\left(\lambda \right)-u=0.$$

The uniroot routine of the R packet program can be used to solve the nonlinear equation given in the second item. All results related to MLEs were obtained using the optim-CG routine in the R program.

6.1. Simulation Study 1

In the first simulation study, we obtain the graphical results. We generate $n=1000$ samples of size $n=20,25,\dots ,400$ from EBXII binomial distribution with true parameters’ values $\alpha =1.5$, $\theta =10$, $c=8$ and $k=0.5$. We assume that the parameter m, number of replicas, is 3. In this simulation study, we empirically calculate the mean, standard deviations (SD), bias and mean square error (MSE) of the MLEs. The bias and MSE are calculated by (for $h=\alpha ,\theta ,c,k$)

$$\widehat{Bia{s}_h}=\frac{1}{1000}{\sum }_{i=1}^{1000}\left({\widehat{h}}_i-h\right),$$

and

$$\widehat{MS{E}_h}=\frac{1}{1000}{\sum }_{i=1}^{1000}{\left({\widehat{h}}_i-h\right)}^2,$$

respectively. We give results of this simulation study in Figure 5. From Figure 5, we observe that, when the sample size increases, the empirical means approach the true parameter value, whereas all biases, SDs and MSEs approach 0 in all cases.

6.2. Simulation Study 2

In the second simulation study, we generate 1000 samples of sizes 50, 100 and 200 from selected EBXII logarithmic distributions. For this simulation study, we obtain the empirical means and SDs of the MLEs. The results of this simulation study are reported in

Table 2. Empirical means and standard deviations (in parentheses) for the special EBXII logarithmic distributions.

Parameters	$\mathit{n}=50$				$\mathit{n}=100$				$\mathit{n}=200$
$\mathit{\alpha },\mathit{c},\mathit{k},\mathit{\theta }$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{c}}$	$\widehat{\mathit{k}}$	$\widehat{\mathit{\theta }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{c}}$	$\widehat{\mathit{k}}$	$\widehat{\mathit{\theta }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{c}}$	$\widehat{\mathit{k}}$	$\widehat{\mathit{\theta }}$
2, 0.5, 0.5, 0.5	2.1928	0.5310	0.3114	2.1858	2.1284	0.5115	0.3696	2.0686	2.0594	0.5008	0.4321	2.0351
	(0.3739)	(0.1468)	(0.2383)	(0.5159)	(0.2951)	(0.1055)	(0.2005)	(0.4276)	(0.1405)	(0.0677)	(0.1259)	(0.2414)
1, 0.5, 3, 0.75	1.0621	0.6203	2.9386	0.6986	1.0569	0.5545	2.9642	0.7734	0.9722	0.5219	2.9723	0.7567
	(0.4210)	(0.2078)	(0.3679)	(0.3324)	(0.3474)	(0.1529)	(0.2411)	(0.2188)	(0.2352)	(0.0731)	(0.1793)	(0.1538)
3, 0.1, 3, 0.1	2.9904	0.1044	3.0617	0.1154	2.9902	0.1002	2.9978	0.0785	3.0087	0.1005	3.0027	0.1022
	(0.1310)	(0.0117)	(0.1578)	(0.1754)	(0.1053)	(0.0078)	(0.1181)	(0.1633)	(0.0805)	(0.0053)	(0.0942)	(0.1170)
5, 0.9, 2, 0.9	4.9810	0.9545	2.1140	0.8496	5.0138	0.9169	2.0942	0.8699	4.9976	0.9104	1.9875	0.8711
	(0.7592)	(0.1832)	(0.6073)	(0.9044)	(0.2847)	(0.0882)	(0.4431)	(0.4365)	(0.1819)	(0.0703)	(0.2970)	(0.1816)
5, 5, 5, 0.99	4.9483	4.9838	5.0854	0.9572	4.9415	5.0126	5.1212	0.9842	4.9743	4.9785	5.0246	0.9954
	(0.3309)	(0.2440)	(0.4827)	(0.1362)	(0.3526)	(0.2423)	(0.5678)	(0.1036)	(0.1308)	(0.1446)	(0.2212)	(0.0209)
10, 0.7, 0.5, 0.3	10.0795	0.7367	0.4899	0.2615	10.0686	0.7103	0.4901	0.2925	10.0355	0.7077	0.4974	0.3026
	(0.3178)	(0.2787)	(0.1950)	(0.5675)	(0.2825)	(0.2398)	(0.1558)	(0.4701)	(0.1563)	(0.2049)	(0.1341)	(0.2793)
0.8, 2, 0.2, 0.5	1.0035	1.8689	0.2907	0.4027	0.8862	1.9841	0.2169	0.5169	0.8488	1.9911	0.2104	0.5121
	(0.3488)	(0.4761)	(0.2087)	(0.4911)	(0.1938)	(0.3719)	(0.0883)	(0.2861)	(0.1275)	(0.2608)	(0.0616)	(0.2586)

. Table 2 shows that, when the sample size increases, the empirical means approach true parameter values, whereas the SDs decrease, as expected.

7. Data Analysis

In this section, we provide applications to three real data sets to prove empirically the potentiality of some members of the EBXIIPS distributions. We also compare the fits of these models with Burr XII geometric (BXII-G) model, which is studied by [14,16]. To determine the optimum model, we also compute the estimated log-likelihood values $\widehat{\ell }$, Akaike Information Criteria (AIC), Kolmogorov–Smirnov (KS), Cramer–von Mises ($W^{\ast }$) and Anderson–Darling ($A^{\ast }$) goodness of-fit statistics for all models. The statistics $W^{\ast }$ and $A^{\ast }$ are described in detail in [17]. In general, it can be chosen as the best model which has the smaller values of the AIC, KS, $W^{\ast }$ and $A^{\ast }$ statistics and the larger values of $\widehat{\ell }$.

All computations of the MLEs are performed by the maxLik routine and all goodness-of-fits statistics are calculated by the goftest routine in the R program. The details are given by followings.

7.1. Stress Data

The first real data set introduces the stress–rupture life of kevlar 49/epoxy strands which are subjected to constant sustained pressure at the 90% stress level until all had failed such that we obtain complete data with exact failure times. This data set was studied by [1,18,19]. The data are: 0.01, 0.01, 0.02, 0.02, 0.02, 0.03, 0.03, 0.04, 0.05, 0.06, 0.07, 0.07, 0.08, 0.09, 0.09, 0.1, 0.1, 0.11, 0.11, 0.12, 0.13, 0.18, 0.19, 0.2, 0.23, 0.24, 0.24, 0.29, 0.34, 0.35, 0.36, 0.38, 0.4, 0.42, 0.43, 0.52, 0.54, 0.56, 0.6, 0.6, 0.63, 0.65, 0.67, 0.68, 0.72, 0.72, 0.72, 0.73, 0.79, 0.79, 0.8, 0.8, 0.83, 0.85, 0.9, 0.92, 0.95, 0.99, 1, 1.01, 1.02, 1.03, 1.05, 1.1, 1.1, 1.11, 1.15, 1.18, 1.2, 1.29, 1.31, 1.33, 1.34, 1.4, 1.43, 1.45, 1.5, 1.51, 1.52, 1.53, 1.54, 1.54, 1.55, 1.58, 1.60, 1.63, 1.64, 1.8, 1.8, 1.81, 2.02, 2.05, 2.14, 2.17, 2.33, 3.03, 3.03, 3.34, 4.2, 4.69, 7.89.

Table 3. MLEs, standard errors of the estimates (in parentheses), $\widehat{\ell }$ and goodness-of-fits statistics for the first data set.

Model	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\theta }}$	$\widehat{\mathit{c}}$	$\widehat{\mathit{k}}$	$-\widehat{\mathit{\ell }}$	$\mathit{AIC}$	$\mathit{KS}$	${\mathit{A}}^{\ast }$	${\mathit{W}}^{\ast }$
EBXII-G	0.1837	−2.3736	2.8794	0.7734	102.2356	212.4712	0.0835	0.8597	0.1276
	(0.0531)	(1.2987)	(0.0852)	(0.0573)
BXII-G		−6.5779	0.7905	3.8292	103.7589	213.5178	0.0907	1.3308	0.1920
		(9.3669)	(0.2420)	(1.6013)
EBXII-L	0.1466	−16.6902	3.5208	0.7453	101.0149	210.0298	0.0761	0.6064	0.0821
	(0.0244)	(5.1283)	(0.0233)	(0.1067)
EBXII-P	0.2237	−1.6098	2.8432	0.6581	103.4967	214.9934	0.0850	1.2765	0.2155
	(0.0592)	(0.8678)	(0.0996)	(0.1445)
BurrXII			1.1736	1.6327	108.5477	221.0955	0.1143	2.8873	0.3072
			(0.0983)	(0.1638)

lists the MLEs, their standard errors of the parameters, $\widehat{\ell }$ and goodness-of-fits statistics from the fitted models. Table 3 shows that the EBXII-L model could be chosen as the best model among the fitted models since these models have the lowest values of the AIC, KS, $W^{\ast }$ and $A^{\ast }$ statistics and have the biggest $\widehat{\ell }$ values.

The plots of the fitted densities and cdfs are displayed in Figure 6. We also draw probability–probability (P–P) plot of all models in Figure 7. These plots shows that the EBXII-L provides the good fit to these data compared to the other models.

7.2. Service Times Data

The second real data set represents the data on service times of 63 aircraft windshield given in [20]. The data are: 0.046, 1.436, 2.592, 0.140, 1.492, 2.600, 0.150, 1.580, 2.670, 0.248, 1.719, 2.717, 0.280, 1.794, 2.819, 0.313, 1.915, 2.820, 0.389, 1.920, 2.878, 0.487, 1.963, 2.950, 0.622, 1.978, 3.003, 0.900, 2.053, 3.102, 0.952, 2.065, 3.304, 0.996, 2.117, 3.483, 1.003, 2.137, 3.500, 1.010, 2.141, 3.622, 1.085, 2.163, 3.665, 1.092, 2.183, 3.695, 1.152, 2.240, 4.015, 1.183, 2.341, 4.628, 1.244, 2.435, 4.806, 1.249, 2.464, 4.881, 1.262, 2.543, 5.140. These data sets were recently studied by [21]. The unit for measurement is 1000 h for both data sets.

Table 4. MLEs, standard errors of the estimates (in parentheses), $\widehat{\ell }$ and goodness-of-fits statistics for the second data set.

Model	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\theta }}$	$\widehat{\mathit{c}}$	$\widehat{\mathit{k}}$	$-\widehat{\mathit{\ell }}$	$\mathit{AIC}$	$\mathit{KS}$	${\mathit{A}}^{\ast }$	${\mathit{W}}^{\ast }$
EBXII-G	0.3573	−69.4175	1.3364	2.6885	102.0361	212.0721	0.0935	0.6693	0.0803
	(0.4481)	(7.6873)	(0.7013)	(1.7470)
BXII-G		−47.7605	1.0344	3.6365	102.3057	210.6114	0.0967	0.7748	0.0861
		(4.0336)	(0.1569)	(0.3727)
EBXII-L	0.2364	$-3.5\times {10}^6$	2.9710	3.0922	99.5671	207.1343	0.1067	0.4639	0.0860
	(0.0461)	(0.4027)	(0.1331)	(0.1867)
EBXII-P	1.0879	−3.6258	1.4626	1.5307	108.0832	224.1664	0.1498	1.7402	0.2545
	(0.6565)	(1.1587)	(0.4915)	(0.6687)
BurrXII			2.1335	0.6151	116.1148	236.2296	0.2545	10.4160	1.3521
			(0.2804)	(0.0982)

lists the MLEs, their standard errors of the parameters, $\widehat{\ell }$ and goodness-of-fits statistics from the fitted models. Table 4 shows that the EBXII-L model fits this data set better than the other models according to the statistics $\widehat{\ell }$, AIC and $A^{\ast }$. At the same time, we can say that the data set is better fitted by the EBXII-G according to the statistics K and $W^{\ast }$.

We draw the fitted densities and cdfs, and the P–P plots in Figure 8 and Figure 9, respectively, for the data set. Clearly, the EBXII-L distribution provides a closer fit to the empirical cdf. From all these results, we may choose the EBXII-L model as the best model for this data set.

The plots of the fitted densities and cdfs are displayed in Figure 8. We also draw P–P plot of all models in Figure 9. These plots shows that the EBXII-L provides the good fit to these data compared to the other models.

7.3. Failure Data

The third real data consists of the number of successive failures for the air conditioning systems reported for each member in a fleet of 13 Boeing 720 jet airplanes. The pooled data with 213 observations was considered by [22,23].

Table 5. MLEs, standard errors of the estimates (in parentheses), $\widehat{\ell }$ and goodness-of-fits statistics for the third data set.

Model	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\theta }}$	$\widehat{\mathit{c}}$	$\widehat{\mathit{k}}$	$-\widehat{\mathit{\ell }}$	$\mathit{AIC}$	$\mathit{KS}$	${\mathit{A}}^{\ast }$	${\mathit{W}}^{\ast }$
EBXII-G	324.1786	−59.7180	0.2562	7.4612	1178.5170	2365.0330	0.0410	0.5110	0.0592
	(0.5597)	(0.1066)	(0.0085)	(0.1257)
BXII-G		−349.9338	0.8401	1.7449	1180.261	2366.5210	0.0413	0.5832	0.0678
		(1.2261)	(0.2915)	(0.5884)
EBXII-L	377.6428	−267.0753	0.3544	5.6072	1189.0280	2386.0560	0.1026	3.9962	0.7387
	(0.1188)	(0.1809)	(0.0226)	(0.2711)
EBXII-P	68.2377	−4.1156	0.3268	3.9270	1186.3710	2380.7430	0.0645	1.4650	0.1768
	(2.2356)	(0.9638)	(0.0319)	(0.3690)
BurrXII			53.7460	0.0047	1335.4150	2674.8300	0.3697	44.8020	9.2016
			(1.6200)	(0.0004)

lists the MLEs, their standard errors of the parameters, $\widehat{\ell }$ and goodness-of-fits statistics from the fitted models. Table 5 shows that the EBXII-G model fits this data set better than the other models according to the statistics $\widehat{\ell }$, AIC and $A^{\ast }$.

We draw the fitted densities and cdfs, and the P–P plots in Figure 10 and Figure 11, respectively, for the data set. Clearly, the EBXII-G distribution provides a closer fit to the empirical cdf. From all these results, we may choose the EBXII-G model as the best model for this data set.

8. Conclusions

In this work, we introduce a new Burr XII power series class of distributions, which is obtained by compounding Burr XII and power series distributions and has a strong physical motivation. The new distribution contains several important lifetime models. We derive explicit expressions for the ordinary and incomplete moments and generating functions. We discuss the maximum likelihood estimation of the model parameters. The maximum likelihood estimation procedure is presented. We assess the performance of the maximum likelihood estimators in terms of biases, variances, and mean square of errors by means of two simulation studies. The usefulness of the new models is illustrated by means of three real data sets. The new models provide consistently better fits than other competitive models for these data sets.