Marshall olkin extended exponentiated Gamma distribution and its applications

Abstract

Different methods for obtaining new probability distributions have been introduced in the literature in recent years, for example, (Gupta et al., 1998) proposed an interesting uni-parametric lifetime distribution, Exponentiated Gamma (EG), which hazard function has increasing and bathtub shapes. In this paper, we build a new two-parameters distribution, the Marshall Olkin Extended Exponentiated Gamma (MOEEG) distribution, which is derived from the Marshall-Olkin method and the EG distribution. The hazard function of this new distribution can accommodate monotonic, non-monotonic and unimodal shapes, allowing a better fit to greater data variability. In addition to the great flexibility of fitting the data, it contains only two parameters providing a simple parameter estimation procedure, unlike other distributions proposed in the literature that have three or more parameters. Some properties of the new distribution considered in this paper are presented such as $n$ -th time, $r$ -th moment of residual life, $r$ -thmoment of residual life inverted, stochastic ordering, entropy, mean deviation, Bonferroni and Lorenz curve, skewness, kurtosis, order statistics, and stress-strength parameter. We also apply two different estimation methods, maximum likelihood and Bayesian approach. Real data applications are presented to illustrate the usefulness of this new distribution.

Keywords

Marshall-Olkin Extended Exponentiated Gamma Marshall Olkin maximum likelihood Bayesian estimators failure rate function

1. Introduction

Survival analysis is an area of Statistics which statistical methods are used in data analysis where the variable of interest is the lifetime or failure of a particular item under study. This branch, in general, deals with the usual parametric and non-parametric methods for analyzing time-in-event data that are useful when some observations are censored, that is, the event of interest was not observed in all items during the follow-up period. The most used procedures include life tables, the Kaplan-Meier estimator for the survival function, the Cox proportional hazards model and parametric survival models.

In general, parametric models are more flexible than Cox proportional hazards model, especially when there is no proportionality of risks between groups and are based mainly on two important functions, the survival and hazard functions. These techniques are described in several textbooks as (Klein & Moeschberger, 2006; Kleinbaum & Klein, 2010; Kalbfleisch & Prentice, 2011). However, some parametric models as the exponential distribution, that is widely used for data analysis, is preferable for situations in which the hazard rate is constant. In case of monotonous failure rate, Weibull and Gamma distributions are often used (Singh et al., 2015). Different hazard function are necessary to model the various studies case, that is, according to (Hjorth, 1980), one or two parameters distributions have some important limitations such as the inability to model data that presents a bathtub risk function. On the other hand, more flexible distributions with a large number of parameters may have inaccurate estimates when sample size is small.

The construction of new distributions providing different hazard functions that can be applied to various types of data is an important research topic. Thus, our main goal in this paper is to create a new lifetime distribution that allows fit different datasets with few parameters and non-monotonous failure rates taking various shapes.

We propose a new two-parameters lifetime distribution, as a direct extension of the one-parameter Exponentiated Gamma distribution (EG) introduced by (Gupta et al., 1998). The distribution is obtained by compounding the EG distribution with a Geometric distribution based on competing and complementary risks method. It will be referred as the Marshall Olkin Extended Exponentiated Gamma (MOEEG) distribution. The hazard function of the MOEEG distribution can assume various shapes for different values of the parameters, such as increasing, bathtub and unimodal. This distribution can be very well applied to reliability/survival problems and its properties are investigated in this paper. We also explore two important inference approaches to estimate the parameters of MOEEG distribution as maximum likelihood and Bayesian methods. Based on simulation studies we compare their performance.

The paper is organized as follows: in Section 2, we present a brief description of Marshall-Olkin, competing and additional risks, and composition methods for the generation of a probability distribution. Section 3 presents the proposed distribution and some mathematical properties. The estimation of parameters are based on Bayesian and classical approaches provided in Section 4. A simulation study is carried out along with its results and presented in Section 5. An application involving survival data related to exposition of rats to poisons is shown in Section 6 to illustrate the usefulness of the proposed distribution. Finally, Section 7 closes the paper with some concluding remarks.

2. Methods for generating probability models

In this section some methods are presented based on the literature to generate continuous probability models. The first and most commonly used for building a new distribution is the method of composition between models. The second and third methods are based on competitive and additional risks. They are not very common but can derive good flexible models for survival data. In addition to these, we have the Marshall-Olkin method also widely used to build a new continuous probability model, even in the multivariate case (de Oliveira & Achcar, 2020).

2.1 Composition between distributions

From the Exponentiated Gamma distribution and by the composition of the distributions process, we obtained a new distribution called Geometric Exponentiated Gamma. For this, consider a random sample $Y_{1},Y_{2},\cdots,Y_{Z}$ from the Exponentiated Gamma distribution. Furthermore, let $Z$ be a random variable with Geometric distribution such that $Z$ represents the number of Bernoulli’s trials realized until the first failure, which the probability mass function is given by:

$\displaystyle{P_{Z}}(z;\lambda)=(1-\lambda){\lambda^{z-1}},z=1,2,\cdots,0<% \lambda<1.$ (1)

Considering the theory of competing risks, the shortest lifetime is observed among all failure factors, while with the theory of additional risks, the longest time is observed among the same factors, assuming $Y$ and $Z$ are independent variables. This way, we can define a new random variable $X$ as the minimum (competing risks) or the maximum (additional risks) of $Y_{1},Y_{2},\cdots,Y_{Z}$ . The distribution of $X$ will be given the distribution name Geometric Exponentiated Gamma. The distribution of the random variable $X$ will be called the Geometric Exponentiated Gamma distribution.

2.2 Competing risks

Let $X=\min(Y_{1},Y_{2},\cdots,Y_{Z})$ and assuming that $Y$ and $Z$ are independent random variables, we can write the probability density function (pdf) $X$ given $Z=z$ as:

$\displaystyle f(x|Z=z,\theta)=z{[1-{F_{Y}}(x|\theta)]^{z-1}}{f_{Y}}(x|\theta).$ (2)

Thus, the marginal pdf of $X$ is obtained as follows:

$\displaystyle f(x|\theta,\lambda)=\sum\limits_{z=1}^{\infty}{f(x|Z=z,\theta)}P% (Z=z)=\sum\limits_{z=1}^{\infty}{z{{[1-{F_{Y}}(x|\theta)]}^{z-1}}{f_{Y}}(x|% \theta)(1-\lambda){\lambda^{z-1}}}=\frac{{(1-\lambda){f_{Y}}(x|\theta)}}{{% \lambda(1-{F_{Y}}(x|\theta))}}\sum\limits_{z=1}^{\infty}{z{{[\lambda(1-{F_{Y}}% (x|\theta))]}^{z}}}.$ (3)

In the sum obtained in the Eq. (3), we have $|\lambda(1-{F_{Y}}(x|\theta))|<1$ , then we can see it as an infinite sum of geometric series ratio $\lambda(1-{}F_{Y}(x|\theta))$ , which converges to:

$\displaystyle\sum\limits_{z=1}^{\infty}{z{{[\lambda(1-{F_{Y}}(x|\theta))]}^{z}% }=\frac{{(1-\lambda){f_{Y}}(x|\theta)}}{{[1-\lambda{{(1-{F_{Y}}(x|\theta)]}^{2% }}}}}$ (4)

and consequently, the marginal pdf of $X=\min({Y_{1}},{Y_{2}},\ldots,{Y_{Z}})$ is given by:

$\displaystyle f(x|\theta,\lambda)=\frac{{(1-\lambda)\theta x{e^{-x}}{{(1-{e^{-% x}}(x+1))}^{\theta-1}}}}{{{{[1-\lambda(1-{{(1-{e^{-x}}(x+1))}^{\theta}})]}^{2}% }}},0<\theta,0<\lambda<1,0<x.$ (5)

The random variable $X$ with pdf Eq. (5) is called Geometric Exponentiated Gamma distribution $(\text{GEG}_{\min})$ .

2.3 Additional risks

Let $X=\max({Y_{1}},{Y_{2}},\ldots,{Y_{Z}})$ and assuming $Y$ and $Z$ as independent random variables, we can write the conditional probability density function of $X$ given $Z=z$ as:

$\displaystyle f(x|Z=z,\theta)=z{[{F_{Y}}(x|\theta)]^{z-1}}{f_{Y}}(x|\theta).$ (6)

In the same way competing risks, we have the marginal pdf of $X=\max({Y_{1}},{Y_{2}},\ldots,{Y_{Z}})$ given by:

$\displaystyle f(x|\theta,\lambda)=\frac{{(1-\lambda)\theta x{e^{-x}}{{(1-{e^{-% x}}(x+1))}^{\theta-1}}}}{{{{[1-\lambda{{(1-{e^{-x}}(x+1))}^{\theta}}]}^{2}}}},% 0<\theta,0\leqslant\lambda\leqslant 1,0<x.$ (7)

The distribution of $X=\max({Y_{1}},{Y_{2}},\ldots,{Y_{Z}})$ with pdf Eq. (7) is called Geometric Exponentiated Gamma distribution $(\text{GEG}_{\max})$ .

2.4 General case (Marshall-Olkin)

Marshall and Olkin (1997) presented a method for generalizations probability distributions assuming a new parameter $\alpha>0$ in a family of distributions given by the equation:

$\displaystyle S*(y)=\frac{{\alpha S(y)}}{{1-\bar{\alpha}S(y)}}=\frac{{\alpha S% (y)}}{{F(y)+\alpha S(y)}},\alpha>0,$ (8)

where $F(y)$ and $S(y)$ are the cumulative and survival functions, respectively. The survival function of this new family of distributions is $S*(y)$ with the additional parameter $\alpha$ , obtained by the Eq. 8. Observe that if $\alpha=1$ then $S*(y)=S(y)$ . In addition, if the base distribution has pdf $f(y)$ and hazard function $h(y)$ , then the new probability density function corresponding to $S*(y)$ is given by:

$\displaystyle f*(y)=\frac{{\alpha f(y)}}{{{{[1-(1-\alpha)S(y)]}^{2}}}}=\frac{{% \alpha f(y)}}{{{{[F(y)+\alpha S(y)]}^{2}}}},$ (9)

and the hazard function is given by:

$\displaystyle h*(y)=\frac{{h(y)}}{{1-(1-\alpha)S(y)}}=\frac{{h(y)}}{{F(y)+% \alpha S(y)}}.$ (10)

Marshall and Olkin (1997) modified the exponential distribution based on the Eq. 9 denominated two-parameters Exponential Distribution, which can be an alternative of two parameters distributions introduced in the literature as the Weibull, Gamma and Log-normal distributions. The same modification was applied for the Weibull distribution, resulting in the Weibull distribution with three Parameters. Marshall and Olkin also developed bivariate versions of these distributions.

The three parameters Weibull distribution, obtained by Marshall Olkin extension, was studied by (Zhang & Xie, 2007). One of the important properties of this distribution is the functional form of hazard function which could be increasing, decreasing or bathtub shaped, and its particular case is the Weibull distribution for $\alpha=1$ . The characterization of this model was studied based on the Weibull probability plot (WPP). The authors presented an estimation procedure of parameters based on WPP and further developed the method of maximum likelihood estimation.

On other hand, using the method proposed by (Marshall & Olkin, 1997), (Thomas & Jose, 2004) introduced the Marshall-Olkin bivariate semi-Pareto distribution (MO-BSP) and Marshall-Olkin bivariate Pareto distribution (MO-BP) and studied several characteristics of these distributions. (Ghitany et al., 2007) investigated the properties of the inclusion of a new parameter by the method of Marshall and Olkin, based on Lomax model, also known as Pareto distribution second type. The authors showed that the proposed distribution can be expressed as a mixture model of exponential distribution using the maximum likelihood method to estimate the parameters of the distribution.

Considering now that $M$ has geometric distribution with probability function given by $P(M=m)={(1-\lambda)^{m-1}}\lambda,m=1,2,\ldots$ and $0<\lambda<1$ . The survival function of the new distribution obtained by composing process to latent structures minimum and maximum activation can be rewritten respectively as:

$\displaystyle{S_{\min}}(y|\theta,\lambda)=\frac{{\lambda S(y|\theta)}}{{1-\bar% {\lambda}S(y|\theta)}}$ (11)

and

$\displaystyle{S_{\max}}(y|\theta,\lambda)=\frac{{S(y|\theta)}}{{1-\bar{\lambda% }F(y|\theta)}},$ (12)

where ${\bar{\lambda}}=1-\lambda$ .

Both survival functions can match if we made a new re-parametrization $\alpha=\frac{1}{\lambda}\geqslant 1,(0<\lambda<1)$ . (Marshall & Olkin, 1997) arrived on this result when they proposed to extend distribution through the addition of a new shape parameter which was known as Extended Marshall-Olkin distribution. This result was possible through using the concept of extreme stability the geometric distribution.

In order to understand the geometric-extreme stable property it should be remembered that the extreme value distributions are limiting distributions for extreme, hence they are sometimes useful approximations. In practice, a random variable of interest may be the extreme of only a finite, possibly random, number $N$ of random variables. When $N$ has a geometric distribution, the random variable has a particularly nice stability property, not unlike that of extreme value distributions. In this way, the extended Marshall Olkin distribution is obtained as follows:

$\displaystyle f(x|\theta,\alpha)=\frac{{\alpha f(x|\theta)}}{{{{[1-(1-\alpha)S% (x|\theta)]}^{2}}}}=\frac{{\alpha\theta x{e^{-x}}{{(1-{e^{-x}}(x+1))}^{\theta-% 1}}}}{{{{[1-(1-\alpha)(1-{{(1-{e^{-x}}(x+1))}^{\theta}})]}^{2}}}},0<x,0<\alpha% ,0<\theta.$ (13)

The function in Eq. (13) is the pdf of the new proposed distribution called Marshall Olkin Extended Exponentiated Gamma (MOEEG). The behavior of the pdf Eq. (13) is illustrated in Figs 1 and 2, for different values of $\theta$ and $\alpha$ , respectively.

Figure 1.

Density function for different values of $\theta$ .

Figure 2.

Density function for different values of $\alpha$ .

The MOEEG hazard function is given by:

$\displaystyle h(x|\theta,\alpha)=\frac{{h(x|\theta)}}{{1-\bar{\alpha}S(x|% \theta)}}={\frac{\theta x{{\rm e}^{-x}}(1-{{\rm e}^{-x}}(x+1))^{\theta-1}}{(1-% (1-{{\rm e}^{-x}}(x+1))^{\theta})(1-\bar{\alpha}(1-(1-{{\rm e}^{-x}}(x+1))^{% \theta}))}},$ (14)

where $\bar{\alpha}=(1-\alpha)$ . Note from the function Eq. (14) that $h(x|\theta,\alpha)/h(x|\theta)$ is increasing on $x$ for $1\leqslant\alpha$ and decreasing in $x$ for $0<\alpha\leqslant$ 1 as shown in the Figs 3 and 4.

Figure 3.

Hazard function for different values of $\theta$ .

Figure 4.

Hazard function for different values of $\alpha$ .

The MOEEG cumulative and survival functions are given, respectively, by:

$\displaystyle F(x|\theta,\alpha)=\frac{{F(x|\theta)}}{{1-\bar{\alpha}S(x|% \theta)}}=\frac{{{{(1-{e^{-x}}(x+1))}^{\theta}}}}{{1-\bar{\alpha}(1-{{(1-{e^{-% x}}(x+1))}^{\theta}})}},$ (15) $\displaystyle S(x|\theta,\alpha)=\frac{{\alpha S(x|\theta)}}{{1-\bar{\alpha}S(% x|\theta)}}=\frac{{\alpha(1-{{(1-{e^{-x}}(x+1))}^{\theta}})}}{{1-\bar{\alpha}(% 1-{{(1-{e^{-x}}(x+1))}^{\theta}})}},$ (16)

where $\bar{\alpha}=(1-\alpha)$ .

3. Mathematical properties of MOEEG

.

Let $X$ be a random variable following a MOEEG distribution with parameters $\alpha$ e $\theta$ , then the pdf Eq. (13) can be rewritten in the following way:

$\displaystyle f(x|\theta,\alpha)=\sum\limits_{j=0}^{\infty}{\sum\limits_{i=0}^% {\infty}{\sum\limits_{k=0}^{i}{\left({\begin{array}[]{*{20}{c}}{\theta-1+% \theta j}\\ i\end{array}}\right)}}}\left({\begin{array}[]{*{20}{c}}i\\ k\end{array}}\right){(-1)^{i+j}}(j+1)\frac{1}{\alpha}{\left({\frac{{1-\alpha}}% {\alpha}}\right)^{j}}\theta{x^{1+k}}{e^{-x(1+i)}}.$

Proof..

Based on the expansion in Taylor series for $|z|<1$ ,

$\displaystyle{(1-p+pz)^{-a}}={(1-p)^{-a}}\sum\limits_{k=0}^{\infty}{\left({% \begin{array}[]{*{20}{c}}{-a}\\ k\end{array}}\right)\frac{{{{(pz)}^{k}}}}{{{{(1-p)}^{k}}}}}$

is obtained and the resulting density is given by

$\displaystyle f(x|\theta,\alpha)={\alpha^{-2}}\sum\limits_{j=0}^{\infty}{\left% ({\begin{array}[]{*{20}{c}}{-2}\\ j\end{array}}\right)}{\left({\frac{{1-\alpha}}{\alpha}}\right)^{j}}\alpha% \theta x{e^{-x}}{(1-{e^{-x}}(x+1))^{\theta-1+\theta j}}.$

Now, using the series expansion with $|z|<1$ ,

$\displaystyle{(1-z)^{k-1}}=\sum\limits_{m=0}^{\infty}{{{(-1)}^{m}}\left({% \begin{array}[]{*{20}{c}}{k-1}\\ m\end{array}}\right){z^{m}}},$

the pdf could be rewritten as

$\displaystyle f(x|\theta,\alpha)={\alpha^{-2}}\sum\limits_{j=0}^{\infty}{\sum% \limits_{i=0}^{\infty}\left({\begin{array}[]{*{20}{c}}{-2}\\ j\end{array}}\right)}{\left({\frac{{1-\alpha}}{\alpha}}\right)^{j}}\left({% \begin{array}[]{*{20}{c}}{\theta-1+\theta j}\\ i\end{array}}\right){(-1)^{i}}\alpha\theta x{e^{-x-xi}}{(x+1)^{i}}.$

Therefore, using the binomial expansion and also the following transformation:

$\displaystyle\left({\begin{array}[]{*{20}{c}}{-2}\\ j\end{array}}\right)={(-1)^{j}}\left({\begin{array}[]{*{20}{c}}{2+j-1}\\ j\end{array}}\right)={(-1)^{j}}(j+1),$

the pdf could be described as follows

and the proof is complete. ∎

This form of writing the density function facilitates obtaining its integral for calculations of the $n$ - th moment, the moment generating function among others.

3.1 Moments of MOEEG

The general expression of expectation, the $n-th$ time or time of order $n$ of a random variable $X$ is given by $E({X^{n}})$ . The moments are very important to characterize probability distributions. The first, second, third and fourth moments characterize central tendency, dispersion, skewness and kurtosis, respectively, of a probability distribution.

The $n-th$ moment is defined as:

$\displaystyle E({X^{n}})=\int\limits_{0}^{\infty}{{x^{n}}f(x)dx},$

and now, using the theorem 1 the $n-th$ moment for MOEEG is given by:

$\displaystyle E({X^{n}})=\sum\limits_{j=0}^{\infty}{\sum\limits_{i=0}^{\infty}% {\sum\limits_{k=0}^{i}{\left({\begin{array}[]{*{20}{c}}{\theta-1+\theta j}\\ i\end{array}}\right)}}}\left({\begin{array}[]{*{20}{c}}i\\ k\end{array}}\right){(-1)^{i+j}}(j+1)\frac{1}{\alpha}{\left({\frac{{1-\alpha}}% {\alpha}}\right)^{j}}$

(17) $\displaystyle\theta{(1+i)^{-2-k-n}}\Gamma(2+k+n).$
3.2 Moment generating function of MOEEG

The moment generating function (mgf) of a random variable is an alternative specification of a probability distribution.

Let $X$ be a random variable, then the moment generating function of the variable $X$ is defined as:

$\displaystyle M_{X}(t)={E}(e^{tX}),$

since ${E}(e^{tX})$ exists in some interval $(-h,h)$ for some real number $h>0$ . The mgf only needs to be defined in a neighborhood of zero, because the moments will be obtained through successive differentiation applied at zero (Curtiss, 1942). Through the series expansions, the mgf of MOEEG is written as:

$\displaystyle{M_{X}}(t)=E({e^{tX}})=E\left({\sum\limits_{n=0}^{\infty}{\frac{{% {t^{n}}{X^{n}}}}{{n!}}}}\right)=\sum\limits_{n=0}^{\infty}{\frac{{{t^{n}}E({X^% {n}})}}{{n!}}}=\sum\limits_{n=0}^{\infty}\sum\limits_{j=0}^{\infty}\sum\limits% _{i=0}^{\infty}{\sum\limits_{k=0}^{i}{\left({\begin{array}[]{*{20}{c}}{\theta-% 1+\theta j}\\ i\end{array}}\right)}}\left({\begin{array}[]{*{20}{c}}i\\ k\end{array}}\right){(-1)^{i+j}}(j+1)\frac{{{t^{n}}}}{{\alpha n!}}{\left({% \frac{{1-\alpha}}{\alpha}}\right)^{j}}{}\times\theta{(1+i)^{-2-k-n}}\Gamma(2+k% +n).$ (18)

3.3 Skewness and kurtosis measures

Skewness is a measure of symmetry, or more accurately, the lack of symmetry. A distribution is symmetric if the tails, both the right and the left of the center point are equal. The value of the asymmetry can be positive, negative, or even undefined. The asymmetry formula below is known as the skewness coefficient Fisher-Pearson.

$\displaystyle\gamma=E\left(\frac{(x-\mu)^{3}}{\sigma^{3}}\right)=\frac{{E({X^{% 3}})-3E(X)E({X^{2}})+2{E^{3}}(X)}}{{Va{r^{3/2}}(X)}}.$ (19)

The kurtosis is a measure of the “tailedness” of the distribution that characterizes the peak or “flattening” of the curve of the pdf. Data sets with high kurtosis tend to have heavy tails, or outliers. Low kurtosis tends to have slight tails, or lack of outliers. The standard kurtosis measure is based on a reduced scale version of the fourth moment of the data or population (Pearson, 1905). It is defined as:

$\displaystyle\kappa={\frac{{\mu}^{4}}{{\sigma}^{4}}}={\frac{E({X}^{4})-4E(X)E(% {X}^{3})+6E({X}^{2})(E(X))^{2}-3(E(X))^{4}}{(\textit{Var}(X))^{2}}}.$ (20)

Table 1 shows that the variance of MOEGG distribution is an increasing function when the skewness and kurtosis are decreasing functions of the parameters.

Table 1

Skewness and Kurtosis Table

(a) $\theta=$ 0.7
$\alpha$	Skewness	Kurtosis	Variance
0.01	9.10	138.48	0.08
0.1	3.73	24.26	0.49
0.5	2.05	9.22	1.33
1	1.58	6.63	1.84
1.5	1.35	5.63	2.15
2	1.20	5.09	2.38
3	1.01	4.51	2.69
5	0.81	4.02	3.05
10	0.57	3.65	3.46
(b) $\alpha=$ 0.5
$\theta$	Skewness	Kurtosis	Variance
0.01	16.26	367.58	0.04
0.1	5.08	38.82	0.37
0.5	2.34	10.96	1.15
1	1.82	8.00	1.49
1.5	1.62	7.13	1.63
2	1.52	6.74	1.71
3	1.42	6.38	1.77
5	1.35	6.14	1.81
10	1.30	6.02	1.80

3.4

r

-th order moment for residual life and reversed residual life

Suppose a component survives up to time $t>0$ , the residual life is the period beyond $t$ until the time of failure and defined by the conditional random variable $X|X>t$ . Therefore the $r$ th-order moment of the residual life is defined as:

$\displaystyle{\mu_{r}}(t)=E({{{(X-t)}^{r}}|X>t})=\frac{{\int\limits_{t}^{% \infty}{{{(x-t)}^{r}}f(x)}dx}}{{S(t)}}.$ (21)

Again, using the Theorem 1, the $r$ th-order moment of the residual life is given by:

$\displaystyle{\mu_{r}}(t)=\frac{1}{{S(t)}}\sum\limits_{j=0}^{\infty}{\sum% \limits_{i=0}^{\infty}{\sum\limits_{k=0}^{i}{\sum\limits_{m=0}^{r}{\left({% \begin{array}[]{*{20}{c}}r\\ m\end{array}}\right)}\left({\begin{array}[]{*{20}{c}}{\theta-1+\theta j}\\ i\end{array}}\right)}}}\left({\begin{array}[]{*{20}{c}}i\\ k\end{array}}\right){(-1)^{i+j+m}}(j+1)\frac{1}{\alpha}{\left({\frac{{1-\alpha% }}{\alpha}}\right)^{j}}{}\times\theta{t^{m}}\int\limits_{t}^{\infty}{{x^{1+k+r% -m}}{e^{-x(1+i)}}dx}.$ (22)

Solving the integral in Eq. (22), the $r$ th-order moment of the residual life could be rewritten as:

$\displaystyle{\mu_{r}}(t)=\frac{1}{{S(t)}}\sum\limits_{j=0}^{\infty}{\sum% \limits_{i=0}^{\infty}{\sum\limits_{k=0}^{i}{\sum\limits_{m=0}^{r}{\left({% \begin{array}[]{*{20}{c}}r\\ m\end{array}}\right)}\left({\begin{array}[]{*{20}{c}}{\theta-1+\theta j}\\ i\end{array}}\right)}}}\left({\begin{array}[]{*{20}{c}}i\\ k\end{array}}\right){(-1)^{i+j+m}}(j+1)\frac{1}{\alpha}{\left({\frac{{1-\alpha% }}{\alpha}}\right)^{j}}{}\times\theta{t^{m}}{(1+i)^{-2-k-r+m}}\Gamma(2+k+r-m,t% (i+1)),r\geqslant 1,$ (23)

where $\Gamma(s;t)=\int\limits_{t}^{\infty}{{x^{s-1}}{e^{-x}}dx}$ is the upper incomplete gamma function.

On other hand, the mean residual life (mrl) function is a particular case of the $r$ th-order moment of the residual life this occurs when $r=1$ , is the remaining life ( $X-t$ ) expected for the unit, given that time $t$ the unit was operating. Theories and applications using mrl extend for a lot of fields. Accelerated testing, a set of fuzzy modeling, mixtures, evaluation insurance of human life expectancy, maintenance and replacement of bridges are just a few examples of applications function mrl, (Steele et al., 2011). The mrl function is defined as:

$\displaystyle\mu(t)=E(X-t|X>t)=\frac{{\int\limits_{t}^{\infty}{S(x)}dx}}{{S(t)% }}.$ (24)

Using the Eq. (23) and $r=1$ , we have:

$\displaystyle\mu(t)=\sum\limits_{j=0}^{\infty}{\sum\limits_{i=0}^{\infty}{\sum% \limits_{k=0}^{i}{\frac{1}{{S(t)}}\left({\begin{array}[]{*{20}{c}}{\theta-1+% \theta j}\\ i\end{array}}\right)}}}\left({\begin{array}[]{*{20}{c}}i\\ k\end{array}}\right)(j+1)\frac{1}{\alpha}{\left({\frac{{1-\alpha}}{\alpha}}% \right)^{j}}\theta{}\times({(-1)^{i+j+1}}{(i+1)^{-2-k}}t\Gamma(2+k,t(i+1))+{(-% 1)^{i+j}}{(i+1)^{-3-k}}\Gamma(3+k,t(i+1))).$ (25)

Note that, taking $t=0$ , the average is obtained as:

$\displaystyle\mu(0)=E(X)=\sum\limits_{j=0}^{\infty}{\sum\limits_{i=0}^{\infty}% {\sum\limits_{k=0}^{i}{\left({\begin{array}[]{*{20}{c}}{\theta-1+\theta j}\\ i\end{array}}\right)}}}\left({\begin{array}[]{*{20}{c}}i\\ k\end{array}}\right){(-1)^{i+j}}(j+1)\frac{1}{\alpha}{\left({\frac{{1-\alpha}}% {\alpha}}\right)^{j}}\theta{}\times{(1+i)^{-3-k}}\Gamma(3+k).$ (26)

In terms of reliability, it is known that the mean residual life function and the ratio of two consecutive moments of residual life determine the distribution exclusively (Lata Gupta & Gupta, 1983). On the other hand, the time elapsed since the failure of an item on condition that this failure has occurred in $[0,t]$ is known in the literature as the mean past residual life. The $r$ -th order reversed residual time is defined as:

$\displaystyle{m_{r}}(t)=E({{{(t-X)}^{r}}|X\leqslant t})=\frac{{\int\limits_{0}% ^{t}{{{(t-x)}^{r}}f(x)}dx}}{{F(t)}}.$ (27)

Using similar arguments as used for Eq. (23), the $r$ -th order reversed residual time is given by:

$\displaystyle{m_{r}}(t)=\frac{1}{{F(t)}}\sum\limits_{j=0}^{\infty}{\sum\limits% _{i=0}^{\infty}{\sum\limits_{k=0}^{i}{\sum\limits_{m=0}^{r}{\left({\begin{% array}[]{*{20}{c}}r\\ m\end{array}}\right)}\left({\begin{array}[]{*{20}{c}}{\theta-1+\theta j}\\ i\end{array}}\right)}}}\left({\begin{array}[]{*{20}{c}}i\\ k\end{array}}\right){(-1)^{i+j-m+r}}(j+1)\frac{1}{\alpha}\theta{t^{m}}{}\times% {\left({\frac{{1-\alpha}}{\alpha}}\right)^{j}}{(i+1)^{-2-k-r+m}}\gamma(2+k+r-m% ,t(i+1)),r\geqslant 1,$ (28)

where $\gamma(s;t)=\int\limits_{0}^{t}{{x^{s-1}}{e^{-x}}dx}$ is the lower incomplete gamma function.

Suppose now that a component with lifetime $X$ that has failed at or before the time $t,t\geqslant 0$ . Consider the conditional random variable $(t-X|X\leqslant t)$ . This conditional random variable shows, in fact, the time elapsed since the component failure given that its lifetime is less than or equal to $t$ . This random variable can also be called downtime (or time since failure), for more details you can see (Nanda et al., 2003; Kundu & Nanda, 2010).

The mean past lifetime (mpl) is a particular case of $r$ th-order moment of the reversed residual life when $r=1$ , and it is given by:

$\displaystyle m(t)=\sum\limits_{j=0}^{\infty}{\sum\limits_{i=0}^{\infty}{\sum% \limits_{k=0}^{i}{\frac{1}{{F(t)}}\left({\begin{array}[]{*{20}{c}}{\theta-1+% \theta j}\\ i\end{array}}\right)}}}\left({\begin{array}[]{*{20}{c}}i\\ k\end{array}}\right)(j+1)\frac{1}{\alpha}{\left({\frac{{1-\alpha}}{\alpha}}% \right)^{j}}\theta{}\times\left({{{(-1)}^{i+j+1}}{{(i+1)}^{-3-k}}\gamma(3+k,t(% i+1))+{{(-1)}^{i+j}}t{{(i+1)}^{-2-k}}\gamma(2+k,t(i+1))}\right).$ (29)

3.5 Stochastic ordering

The simplest way to compare two random variables is comparing their expected values. However, this comparison is not very informative since it is based only on two numbers. Furthermore, in most situations information about the behavior of the random variable have become much more detailed, such as its distribution functions, Laplace transforms, generating moments functions, hazard functions and other functionals. The comparison of these characteristics of random variables results in the establishment of different relationships stochastic order among these same random variables, much more informative than the mere comparison of their expected values (Szekli, 2012).

The simplest and most popular method of comparing the magnitudes of two random variables is through their means and medians. It may happen that in some cases, the mean of $X$ is greater than the $Y$ , while the median $X$ is lower than the mean of $Y$ . However, this confusion will not arise if the random variables are stochastically ordered. Similarly, the same may occur if compared the $X$ and $Y$ variability based only on the numeric measures such as standard deviation, and so on. Moreover, these characteristics distributions may not exist in some cases. These methods are much more informative than based on just some numerical characteristics of distributions. Comparisons of random variables based on such functions, generally establish partial orders between them and are called stochastic ordering (Kochar, 2012).

Distributions ordering, particularly among distributions that model lifetime play an important role in the statistical literature. Six different stochastic orders were considered, the usual, likelihood ratio ordering, mean residual life order, reverse hazard rate order, hazard rate ordering, and expectancy for two MOEEG independent random variables. In this way, if $X$ and $Y$ are independent random variables with cumulative distribution functions ${F_{X}}$ e ${F_{Y}}$ respectively, then $X$ is said to be smaller than $Y$ in:

•
stochastic order $(X{\leqslant_{st}}Y)$ if ${F_{X}}(x)\geqslant{F_{Y}}(x)\forall x$ ;
•
hazard rate ordering $(X{\leqslant_{hr}}Y)$ if ${h_{X}}(x)\geqslant{h_{Y}}(x)\forall x$ ;
•
reverse hazard rate order $({X\leqslant_{rh}}Y)$ if $P(t-X>x|X\leqslant t)\geqslant P(t-Y>y|Y\leqslant t),\forall x\geqslant 0,\forall t$ ;
•
mean residual life order $(X{\leqslant_{\textit{mrl}}}Y)$ if ${\mu_{X}}(x)\leqslant{\mu_{Y}}(x)\forall x$ ;
•
likelihood ratio ordering $(X{\leqslant_{lr}}Y)$ if $\frac{{{f_{X}}(x)}}{{{f_{Y}}(x)}}$ is increasing in $x$ .

Following (Kochar, 2012), we have the following implications chain between the stochastic orders:

$\displaystyle\begin{array}[]{c}X{\leqslant_{lr}}Y\Rightarrow X{\leqslant_{hr}}% Y\Rightarrow X{\leqslant_{\textit{mrl}}}Y\\ \Downarrow\\ X{\leqslant_{hr}}Y\\ \end{array}$ (30)

.

Let $X\sim\textit{MOEEG}({\alpha_{1}},{\theta_{1}})$ and $Y\sim\textit{MOEEG}({\alpha_{2}},{\theta_{2}})$ . If ${\theta_{1}}={\theta_{2}}=\theta$ e ${\alpha_{2}}>{\alpha_{1}}$ , then $(X\leqslant_{st}Y)$ , $(X\leqslant_{hr}Y)$ , $(X\leqslant_{\textit{mrl}}Y)$ and $(X\leqslant_{lr}Y)$ .

Proof..

Given the ratio of the likelihoods $\frac{{{f_{X}}(x)}}{{{f_{Y}}(x)}}$ then $\frac{d}{{dx}}\log\frac{{{f_{X}}(x)}}{{{f_{Y}}(x)}}\leqslant 0$ . Now, if ${\theta_{1}}={\theta_{2}}=\theta$ and ${\alpha_{2}}>{\alpha_{1}}$ , implying $(X\leqslant_{st}Y)$ , therefore $(X\leqslant_{hr}Y)$ , $(X\leqslant_{\textit{mrl}}Y)$ , $(X\leqslant_{lr}Y)$ , ∎
3.6 Entropies

The entropy of a random variable measures the variation of the uncertainty. A large value of entropy indicates the greater uncertainty in the data. Some popular entropy measures are Rényi entropy (Rényi et al., 1961) and Shannon entropy (Shannon, 1951). In this section, the expressions for these measures are determined for MOEEG distribution.

Rényi Entropy is a measure of variation of uncertainty that has been used in many applications and characterizations of probability distributions. Rényi entropy is defined as:

$\displaystyle{H_{I}}(\delta)=\frac{1}{{1-\delta}}\ln\left({\int\limits_{0}^{% \infty}{{f^{\delta}}(x)dx}}\right),\delta>0,\delta\neq 1.$ (31)

Using the same series expansions given in Theorem 1 was obtained the following expression:

$\displaystyle{H_{I}}(\delta)=\frac{1}{{1-\delta}}\ln\left(\sum\limits_{j=0}^{% \infty}{\sum\limits_{i=0}^{\infty}{\sum\limits_{k=0}^{i}{\left({\begin{array}[% ]{*{20}{c}}i\\ k\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{\delta(\theta-1)+\theta j% }\\ i\end{array}}\right)}}}\left({\begin{array}[]{*{20}{c}}{-2\delta}\\ j\end{array}}\right){(-1)^{i}}{\theta^{\delta}}\right.{}\times\left.{\left({% \frac{{1-\alpha}}{\alpha}}\right)^{j}}\frac{1}{{{\alpha^{\delta}}}}{(\delta+i)% ^{-1-k-\delta}}\Gamma(\delta+k+i)\right),\delta>0,\delta\neq 1.$ (32)

The Shannon entropy concept refers to the uncertainty of a probability distribution and the measure that has been proposed is intended to quantify this uncertainty. The Shannon entropy draws attention to the fact that the entropy $H$ is not a function of the random variable $X$ , however from the probability distribution of this variable. In other words, not dependent on the values that $X$ takes, but their odds (Artuso, 2011). If $X$ is a non-negative random variable with continuous pdf $f(x)$ then the Shannon entropy is defined as:

$\displaystyle H(f)=E[-\ln f(x)]=-\int\limits_{0}^{\infty}{f(x)\ln(f(x))dx}.$ (33)

According to (Fattah et al., 2017), the Shannon entropy is:

$\displaystyle\underset{\delta\rightarrow 1}{\lim}{H_{\delta}}(x)=H(f).$

Since the proposed limit has indeterminate form $\frac{0}{{0}}$ , the L’Hôpital’s rule was used and the following result was obtained:

$\displaystyle H(f)=-\int\limits_{0}^{\infty}{f(x)\ln(f(x))dx}=-\sum\limits_{j=% 0}^{\infty}{\sum\limits_{i=0}^{\infty}{\sum\limits_{k=0}^{i}{\left({\begin{% array}[]{*{20}{c}}{\theta-1+\theta j}\\ i\end{array}}\right)}}}\left({\begin{array}[]{*{20}{c}}i\\ k\end{array}}\right){(-1)^{i+j}}(j+1)\frac{1}{\alpha}{\left({\frac{{1-\alpha}}% {\alpha}}\right)^{j}}\theta{}\times\left(\ln\left({\sum\limits_{j=0}^{\infty}{% \sum\limits_{i=0}^{\infty}{\sum\limits_{k=0}^{i}{\left({\begin{array}[]{*{20}{% c}}{\theta-1+\theta j}\\ i\end{array}}\right)}}}\left({\begin{array}[]{*{20}{c}}i\\ k\end{array}}\right){{(-1)}^{i+j}}(j+1)\frac{1}{\alpha}{{\left({\frac{{1-% \alpha}}{\alpha}}\right)}^{j}}\theta}\right){(1+i)^{-2-k}}\Gamma(2+k)\right){}% -{({1+i})^{-3-k}}\Gamma({3+k})({1+i}){}+{\displaystyle\frac{{({({-{k^{2}}-k})% \ln({1+i})+({{k^{2}}+k})\Psi(k)+2k+1})\Gamma(k)({1+k})}}{{{{({1+i})}^{k}}{{({1% +i})}^{2}}}}}),$ (34)

where $\Psi(z)=\frac{{{\Gamma^{\prime}}(z)}}{{\Gamma}(z)}$ .

3.7 Mean deviation

The amount of scatter in a population is evidently measured to some extent by the totality of deviations from the mean (in the case of a symmetrical distribution) and the median (in the case of an asymmetrical distribution). If $X$ is a random variable with MOEEG distribution with mean $\mu=E(X)$ and median $m$ , then the mean deviation from the mean and the mean deviation from the median are defined respectively by:

$\displaystyle{\vartheta_{1}}=\int\limits_{0}^{\infty}{|x-\mu|f(x)dx}\quad e% \quad{\vartheta_{2}}=\int\limits_{0}^{\infty}{|x-m|f(x)dx}.$ (35)

The mean deviation can be simplified as (more details on this simplification can be found in (Nadarajah & Kotz, 2006):

$\displaystyle{\vartheta_{1}}=2\mu F(\mu)-2I(\mu)\quad\text{and}\quad{\vartheta% _{2}}=\mu+2mF(m)-m-2I(m),$ (36)

where $F(\mu)$ and $F(m)$ are obtained by the Eq. (15), and $I(m)$ is obtained by

$\displaystyle I(m)=\int\limits_{0}^{m}{xf(x)dx}.$

Considering the pdf defined in Eq. (13), we have

$\displaystyle I(m)=\sum\limits_{j=0}^{\infty}{\sum\limits_{i=0}^{\infty}{\sum% \limits_{k=0}^{i}{\left({\begin{array}[]{*{20}{c}}{\theta-1+\theta j}\\ i\end{array}}\right)}}}\left({\begin{array}[]{*{20}{c}}i\\ k\end{array}}\right){(-1)^{i+j}}(j+1)\frac{1}{\alpha}{\left({\frac{{1-\alpha}}% {\alpha}}\right)^{j}}\theta{}\times{(i+1)^{-3-k}}\gamma(3+k,m(1+i)).$ (37)

The result is analogous for $I(\mu)$ .

3.8 Bonferroni and Lorenz curves

Bonferroni and Lorenz curves were proposed by (Bonferroni,1941). These curves have applications not only in economics to study income and poverty, but also in other fields like reliability, demography, insurance and medicine.

Bonferroni curve is defined as:

$\displaystyle B(p)=\frac{1}{{p\mu}}\int\limits_{0}^{q}{xf(x)dx}$ (38)

Now, solving this integral for pdf of the MOEEG distribution, we have

$\displaystyle B(p)=\frac{1}{{p\mu}}\sum\limits_{j=0}^{\infty}{\sum\limits_{i=0% }^{\infty}{\sum\limits_{k=0}^{i}{\left({\begin{array}[]{*{20}{c}}{\theta-1+% \theta j}\\ i\end{array}}\right)}}}\left({\begin{array}[]{*{20}{c}}i\\ k\end{array}}\right){(-1)^{i+j}}(j+1)\frac{1}{\alpha}{\left({\frac{{1-\alpha}}% {\alpha}}\right)^{j}}\theta{}\times{(i+1)^{-3-k}}\gamma(3+k,q(1+i)),$ (39)

where $\mu=E(X)$ e $q={F^{-1}}(p)$ .

The Lorenz curve is a graphic representation of the empirical cumulative distribution function of the probability distribution of wealth. In such use, many economists consider it a social inequality measure to represent the unequal distribution of wealth. Lorenz curve is calculated as:

$\displaystyle L(p)=\frac{1}{\mu}\int\limits_{0}^{q}{xf(x)dx}.$ (40)

Applying in the MOEEG distribution, we have

$\displaystyle L(p)=\frac{1}{{\mu}}\sum\limits_{j=0}^{\infty}{\sum\limits_{i=0}% ^{\infty}{\sum\limits_{k=0}^{i}{\left({\begin{array}[]{*{20}{c}}{\theta-1+% \theta j}\\ i\end{array}}\right)}}}\left({\begin{array}[]{*{20}{c}}i\\ k\end{array}}\right){(-1)^{i+j}}(j+1)\frac{1}{\alpha}{\left({\frac{{1-\alpha}}% {\alpha}}\right)^{j}}\theta{}\times{(i+1)^{-3-k}}\gamma(3+k,q(1+i))$ (41)

3.9 Order statistics

The order statistics play an important role in Statistical Inference, and are for the population quantiles as the sample moments are for the population moments. It is known that ${X_{(1)}}\leqslant\dots\leqslant{X_{(n)}}$ denotes the statistical order of the random sample ${X_{1}}\leqslant\dots\leqslant{X_{n}}$ of continuous random variable with cdf $F(x)$ and pdf $f(x)$ , then the pdf of ${X_{(j)}}$ is given by

$\displaystyle{f_{{X_{(j)}}}}(x)=\frac{{n!}}{{(j-1)!(n-j)!}}f(x){(F(x))^{j-1}}{% (1-F(x))^{n-j}},$

for $j=1,\dots,n$ . This way, the pdf of the $j$ -th order statistics for the MOEEG distribution is given by:

$\displaystyle{f_{{X_{(j)}}}}(x)=\sum\limits_{i,k,m=0}^{\infty}{\sum\limits_{h=% 0}^{m}{{{(-1)}^{i+m}}}}\left({\begin{array}[]{*{20}{c}}{n-j}\\ i\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{i+j+1}\\ k\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{\theta(i+j+k)-1}\\ m\end{array}}\right)\left({\begin{array}[]{*{20}{c}}m\\ h\end{array}}\right){}\times\frac{{n!}}{{(j-1)!(n-j)!}}\times\frac{\theta}{{{% \alpha^{i+j}}}}{\left({\frac{{1-\alpha}}{\alpha}}\right)^{k}}{x^{1+h}}{e^{-x(1% +m)}}.$ (42)

Moments of order statistics play an important role in quality control testing reliability to predict the failure of future items based on the times of few early failures. The $n$ -th time of $X_{j{n}}$ can be expressed as:

$\displaystyle E[X_{j:n}^{t}]=\sum\limits_{i,k,m=0}^{\infty}{\sum\limits_{h=0}^% {m}{{{(-1)}^{i+m}}}}\left({\begin{array}[]{*{20}{c}}{n-j}\\ i\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{i+j+1}\\ k\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{\theta(i+j+k)-1}\\ m\end{array}}\right)\left({\begin{array}[]{*{20}{c}}m\\ h\end{array}}\right){}\times\frac{{n!}}{{(j-1)!(n-j)!}}\times\frac{\theta}{{{% \alpha^{i+j}}}}{\left({\frac{{1-\alpha}}{\alpha}}\right)^{k}}{(1+m)^{-2-h-t}}% \Gamma(2+h+t).$ (43)

3.10 Stress strength parameter

Some researches that involve a stress-strength model have an interest in the estimation of reliability denoted by $R=P({X_{2}}<{X_{1}})$ , where ${X_{1}}$ and ${X_{2}}$ are independent random variables that belong to the same family of univariate distributions. The algebraic form for stress strength parameter has been established for most of the known standard distributions. Considering ${X_{1}}$ and ${X_{2}}$ MOEEG independent random variables distribution with parameters $({\alpha_{1}},{\theta_{1}})$ e $({\alpha_{2}},{\theta_{2}})$ respectively, then, we have:

$\displaystyle R=P({X_{2}}<{X_{1}})=\int\limits_{0}^{\infty}{{f_{1}}(x)}{F_{2}}% (x)dx=\int\limits_{0}^{\infty}{}\frac{{{\alpha_{1}}{\theta_{1}}x{e^{-x}}{{(1-{% e^{-x}}(x+1))}^{{\theta_{1}}-1}}}}{{{{[1-(1-{\alpha_{1}})(1-{{(1-{e^{-x}}(x+1)% )}^{\theta_{1}}})]}^{2}}}}\times\frac{{{{(1-{e^{-x}}(x+1))}^{\theta_{2}}}}}{{1% -{\bar{\alpha}_{2}}(1-{{(1-{e^{-x}}(x+1))}^{\theta_{2}}})}}dx.$

In the stress-strength modelling, $R=P(Y<X)$ is a measure of component reliability when it is subjected to random stress $Y$ and has strength $X$ . For a particular situation, consider $Y$ as the pressure of a chamber generated by ignition of a solid propellant and $X$ as the strength of the chamber, then $R$ represents the probability of successful firing of the engine. In this context, $R$ can be considered as a measure of system performance and it is naturally arise in electrical and electronic systems. It may be mentioned that $R$ is of greater interest than just reliability since it provides a general measure of the difference between two populations and has applications in many area. For example, if $X$ is the response for a control group, and $Y$ refers to a treatment group, $R$ is a measure of the effect of the treatment (Asgharzadeh et al., 2011). Therefore, setting ${\theta_{1}}={\theta_{2}}=\theta$ , we have:

$\displaystyle R={\frac{\alpha_{{1}}(\alpha_{{2}}\ln(\alpha_{{2}})-\alpha_{{2}}% \ln(\alpha_{{1}})+\alpha_{{1}}-\alpha_{{2}})}{(-\alpha_{{2}}+\alpha_{{1}})^{2}% }}.$ (44)

For $({\alpha_{1}},{\theta_{1}})\neq({\alpha_{2}},{\theta_{2}})$ we have

$\displaystyle R=\int\limits_{0}^{\infty}{}\frac{{{\alpha_{1}}{\theta_{1}}x{e^{% -x}}{{(1-{e^{-x}}(x+1))}^{{\theta_{1}}-1}}}}{{{{[1-(1-{\alpha_{1}})(1-{{(1-{e^% {-x}}(x+1))}^{\theta_{1}}})]}^{2}}}}\times\frac{{{{(1-{e^{-x}}(x+1))}^{\theta_% {2}}}}}{{1-{\bar{\alpha}_{2}}(1-{{(1-{e^{-x}}(x+1))}^{\theta_{2}}})}}dx=\sum% \limits_{j=0}^{\infty}{\sum\limits_{k=0}^{\infty}{}\sum\limits_{i=0}^{\infty}{% \sum\limits_{h=0}^{i}{\left({\begin{array}[]{*{20}{c}}{-1}\\ j\end{array}}\right)}}}\left({\begin{array}[]{*{20}{c}}i\\ h\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{-2}\\ k\end{array}}\right)\left({\begin{array}[]{*{20}{c}}{{\theta_{1}}(1+j)+{\theta% _{2}}(1+k)-1}\\ i\end{array}}\right){(-1)^{i}}{}\times\frac{1}{{{\alpha_{1}}{\alpha_{2}}}}{% \left({\frac{{1-{\alpha_{2}}}}{{{\alpha_{2}}}}}\right)^{j}}{\left({\frac{{1-{% \alpha_{1}}}}{{{\alpha_{1}}}}}\right)^{k}}{\alpha_{1}}{\theta_{1}}{(1+i)^{-2-h% }}\Gamma(2+h).$ (45)

4. Inference methods

This section presents the Maximum Likelihood and Bayesian estimation methods to obtain estimates of the parameters $\alpha$ and $\theta$ .

4.1 Maximum Likelihood method

Suppose we have a complete random sample $T_{1},\ldots,T_{n}$ from MOEEG $(\alpha,\theta)$ distribution where the parameters $\alpha$ and $\theta$ are unknown, then the likelihood function is given by:

$\displaystyle L(\alpha,\theta|\bm{x})\propto\frac{\alpha^{n}\theta^{n}\prod_{i% =1}^{n}\bigl{(}1-(x_{i}+1)e^{-x_{i}}\bigr{)}^{\theta-1}}{\prod_{i=1}^{n}\Bigl{% [}1-(1-\alpha)(1-(1-(x_{i}+1)e^{-x_{i}})^{\theta})\Bigr{]}^{2}}.$ (46)

The Maximum Likelihood Estimators (MLE) of the parameters $\alpha$ and $\theta$ are values, based on $\bm{T}=(T_{1},\ldots,T_{n})$ , that maximize the likelihood function $L(\alpha,\theta|\bm{x})$ . Now, to compute the MLE of $\alpha$ and $\theta$ , first we obtain the log-likelihood function based on the observed sample and given by:

$\displaystyle l(\alpha,\theta|\bm{x})=n\log(\alpha)+n\log(\theta)+(\theta-1)% \sum_{i=1}^{n}\log\bigl{(}1-(x_{i}+1)e^{-x_{i}}\bigr{)}$ (47) $\displaystyle=-2\sum_{i=1}^{n}\log\Bigl{(}1-(1-\alpha)(1-(1-(x_{i}+1)e^{-x_{i}% })^{\theta})\Bigr{)}.$ (48)

Deriving Eq. (48) for the parameters $\alpha$ and $\theta$ is obtained the score vector given by:

$\displaystyle{U_{\alpha}}(\alpha,\theta|x)={\frac{\partial}{\partial{\alpha}}}% l\left({\alpha},{\theta|x}\right)={\frac{n}{{\alpha}}}-2\sum_{i=1}^{n}{\frac{1% -\left(1-{{e}^{-x_{{i}}}}\left(x_{{i}}+1\right)\right)^{{\theta}}}{1-\left(1-{% \alpha}\right)\left(1-\left(1-{{e}^{-x_{{i}}}}\left(x_{{i}}+1\right)\right)^{{% \theta}}\right)}},$ (49) $\displaystyle{U_{\theta}}(\alpha,\theta|x)={\frac{\partial}{\partial{\theta}}}% l\left({\alpha},{\theta}|x\right)={\frac{n}{{\theta}}}+\sum_{i=1}^{n}\ln{\left% (1-{{e}^{-x_{{i}}}}\left(x_{{i}}+1\right)\right)}{}-2\sum_{i=1}^{n}{\frac{% \left(1-{\alpha}\right)\left(1-{{e}^{-x_{{i}}}}\left(x_{{i}}+1\right)\right)^{% {\theta}}\ln{\left(1-{{e}^{-x_{{i}}}}\left(x_{{i}}+1\right)\right)}}{1-\left(1% -{\alpha}\right)\left(1-\left(1-{{e}^{-x_{{i}}}}\left(x_{{i}}+1\right)\right)^% {{\theta}}\right)}}.$ (50)

The MLE ${{\hat{\alpha}}_{\textit{MLE}}}$ and ${{\hat{\theta}}_{\textit{MLE}}}$ for the parameters $\alpha$ and $\theta$ , respectively, are obtained by solving non-linear equations ${U_{\alpha}}(\alpha,\theta|x)=0$ and ${U_{\theta}}(\alpha,\theta|x)=0$ . Given the vector of unknown parameters $\Theta=(\alpha,\theta)$ , for large samples is known that the MLE asymptotic distribution $\Theta$ is

$\displaystyle(\hat{\Theta}-\Theta)\sim{N_{2}}({0,{I^{-1}}(\Theta)}),\text{ for% }n\rightarrow\infty,$

where ${I}(\Theta)$ is the Fisher information matrix observed given by:

$\displaystyle I(\Theta)=\left({\begin{array}[]{*{20}{c}}{{I_{11}}}&{{I_{12}}}% \\ {{I_{21}}}&{{I_{22}}}\end{array}},\right)$ (51)

with elements:

$\displaystyle I_{11}=\frac{{{\partial^{2}}l(\alpha,\theta|x)}}{{\partial{% \alpha^{2}}}}=-{\frac{n}{{{\alpha}}^{2}}}-2\sum_{i=1}^{n}\left(-{\frac{\left(1% -\left(1-{{e}^{-x_{{i}}}}\left(x_{{i}}+1\right)\right)^{{\theta}}\right)^{2}}{% \left(1-\left(1-{\alpha}\right)\left(1-\left(1-{{e}^{x_{{i}}}}\left(x_{{i}}+1% \right)\right)^{{\theta}}\right)\right)^{2}}}\right),$ $\displaystyle I_{12}=I_{21}=\frac{{{\partial^{2}}l(\alpha,\theta|x)}}{{% \partial{\theta^{2}}}}=-{\frac{n}{{{\theta}}^{2}}}-2\sum_{i=1}^{n}\left(-{% \frac{\left({\alpha}-1\right)\left(1-{{e}^{-x_{{i}}}}\left(x_{{i}}+1\right)% \right)^{{\theta}}\left(\ln{\left(1-{{e}^{-x_{{i}}}}\left(x_{{i}}+1\right)% \right)}\right)^{2}{\alpha}}{\left(-\left(1-{{e}^{-x_{{i}}}}\left(x_{{i}}+1% \right)\right)^{{\theta}}-{\alpha}+{\alpha}\left(1-{{e}^{-x_{{i}}}}\left(x_{{i% }}+1\right)\right)^{{\theta}}\right)^{2}}}\right),$

and

$\displaystyle I_{22}=\frac{{{\partial^{2}}l(\alpha,\theta|x)}}{{\partial\theta% \partial\alpha}}=-2\sum_{i=1}^{n}\left(-{\frac{\left(1-{{e}^{-x_{{i}}}}\left(x% _{{i}}+1\right)\right)^{{\theta}}\ln{\left(1-{{e}^{-x_{{i}}}}\left(x_{{i}}+1% \right)\right)}}{\left(-\left(1-{{e}^{-x_{{i}}}}\left(x_{{i}}+1\right)\right)^% {{\theta}}-{\alpha}+{\alpha}\left(1-{{e}^{-x_{{i}}}}\left(x_{{i}}+1\right)% \right)^{{\theta}}\right)^{2}}}\right).$

The previous results are used to compute the $100\%(1-\tau)$ approximate confidence intervals for the parameter $\Theta=(\alpha,\theta)$ as follows

$\displaystyle{{\hat{\alpha}}_{\textit{MLE}}}\pm{z_{\frac{\tau}{2}}}\sqrt{{% \mathop{\rm var}}({{\hat{\alpha}}_{\textit{MLE}}})};{{\hat{\theta}}_{\textit{% MLE}}}\pm{z_{\frac{\tau}{2}}}\sqrt{{\mathop{\rm var}}({{\hat{\theta}}_{\textit% {MLE}}})},$

where ${z_{\frac{\tau}{2}}}$ is the $\frac{\tau}{2}$ -th percentile of the standard normal distribution.

4.2 Bayesian method

In a Bayesian analysis, the inference is based on the posterior distribution of parameters $\alpha$ and $\theta$ , denoted by $p(\alpha,\theta|\bm{x})$ , which in turn is used for inferences and decisions involving $\alpha$ and $\theta$ . The posterior distribution $p(\alpha,\theta|\bm{x})$ is obtained from the combination of the information provided by a prior distribution $\pi(\alpha,\theta)$ and the information supplied by the data through the likelihood $L(\alpha,\theta|\bm{x})$ . Thus, using Bayes’theorem, the posterior distribution is given by

$\displaystyle p(\alpha,\theta|\bm{x})\propto\pi(\alpha,\theta)L(\alpha,\theta|% \bm{x}).$

The prior distribution represents the knowledge or uncertainty state about the parameter $\alpha$ and $\theta$ before the experiment is running, and the posterior distribution describes the updated information about $(\alpha,\theta)$ after the data $\bm{x}$ is observed.

A common specification of prior considered in the literature is given by the product of independent prior distributions. Therefore, for the MOEEG distribution we consider the product of Gamma prior distributions given by

$\displaystyle\pi(\alpha)\propto{\alpha^{a-1}}{e^{\alpha b}},\alpha>0$ $\displaystyle\pi(\theta)\propto{\theta^{c-1}}{e^{\theta d}},\theta>0$

where the hyperparameters $a, b, c$ and $d$ are known assuming specific values to provide no prior information. The hyperparameters could be choosen such as 0.01 or 0.001 to provide absence of prior information.

Thus, the joint posterior distribution for the parameters $\alpha$ and $\theta$ of MOEEG distribution is proportional to the product of the likelihood function Eq. (44) and the priors $\pi(\alpha)$ and $\pi(\theta)$ , resulting in the joint posterior density given by:

$\displaystyle p(\alpha\theta|\bm{x})\propto\frac{\alpha^{n}\theta^{n}\prod_{i=% 1}^{n}\bigl{(}1-(x_{i}+1)e^{-x_{i}}\bigr{)}^{\theta-1}}{\prod_{i=1}^{n}\Bigl{[% }1-(1-\alpha)(1-(1-(x_{i}+1)e^{-x_{i}})^{\theta})\Bigr{]}^{2}}\pi(\alpha)\pi(% \theta).$ (52)

As we are unable to find an analytic expression for marginal posterior distributions and hence to extract characteristics of parameters such as Bayes estimators and credible intervals, we need use the MCMC algorithm to obtain a sample of values of $\alpha$ and $\theta$ from the joint posterior. Specifically, we run an algorithm for simulating a long chain of draws from the posterior distribution, and basic inferences on posterior summaries of the parameters or functionals of the parameters are calculated from the samples. This way, by using an auxiliary probability distribution, $q(\theta,\cdot)$ , we generated possible samples values of the posterior distribution where the generated value is accepted by satisfying this condition as follows

$\displaystyle\alpha({\theta^{(t)}},{\theta^{(*)}})=\min\left[{1;\frac{{p({% \theta^{(*)}})q({\theta^{(t)}}|{\theta^{(*)}})}}{{p({\theta^{(t)}})q({\theta^{% (*)}}|{\theta^{(t)}})}}}\right]\geqslant u(0,1)$

The following steps are considered:

Propose an auxiliary distribution that generates possible values of $\theta$ ;

Start counter $t=1$ ;

Choose an initial value for ${\theta^{(t)}}$ ;

Generate a value for ${\theta^{(t)}}$ using $q(\theta,\cdot)$ ;

Test the condition $\alpha({\theta^{(t)}},{\theta^{(*)}})\geqslant u(0,1)$ ;

If the condition is satisfied: ${\theta^{(t+1)}}={\theta^{(*)}},t=t+1$ and return to step 4;

If the condition is not satisfied: ${\theta^{(t+1)}}={\theta^{(t)}},t=t+1$ and return to step 4;

Repeat it until you reach the convergence.

Table 2

Simulation results

	$n$	BIAS $(\alpha)$	BIAS $(\theta)$	RMSE $(\alpha)$	RMSE $(\theta)$	CP ( $\alpha$ )	CP ( $\theta$ )
$\alpha=0.5$ and $\theta=5$
MLE	7	1.33	3.48	2.49	5.13	0.70	0.93
Bayes		0.52	3.15	1.22	4.21	0.94	0.95
MLE	10	1.06	2.66	2.08	3.78	0.74	0.95
Bayes		0.48	2.56	1.26	3.42	0.94	0.95
MLE	60	0.19	0.87	0.27	1.12	0.89	0.95
Bayes		0.17	0.87	0.24	1.10	0.94	0.95
MLE	100	0.14	0.64	0.19	0.83	0.91	0.95
Bayes		0.13	0.63	0.17	0.82	0.95	0.95
$\alpha=5$ and $\theta=5$
MLE	7	6.44	14.82	17.30	25.75	0.65	0.94
Bayes		4.96	6.74	7.01	12.89	0.99	0.99
MLE	10	6.01	9.39	17.68	15.74	0.72	0.95
Bayes		5.28	5.02	7.69	9.45	0.98	0.99
MLE	60	3.55	2.02	5.45	2.81	0.85	0.96
Bayes		2.34	1.93	3.62	2.52	0.96	0.96
MLE	100	2.36	2.56	3.65	2.16	0.88	0.95
Bayes		1.78	1.56	1.59	2.01	0.96	0.95
$\alpha=0.5$ and $\theta=0.8$
MLE	7	1.29	0.54	2.48	0.81	0.71	0.95
Bayes		0.51	0.49	1.03	0.67	0.96	0.96
MLE	10	1.05	0.41	2.26	0.58	0.78	0.97
Bayes		0.42	0.38	0.73	0.51	0.95	0.96
MLE	60	0.18	0.13	0.25	0.18	0.90	0.95
Bayes		0.16	0.13	0.22	0.17	0.95	0.95
MLE	100	0.13	0.10	0.18	0.13	0.90	0.95
Bayes		0.13	0.10	0.16	0.13	0.95	0.95
$\alpha=1.5$ and $\theta=0.8$
MLE	7	2.75	0.96	4.52	1.49	0.65	0.94
Bayes		1.71	0.68	3.84	1.06	0.97	0.97
MLE	10	2.57	0.72	4.18	1.16	0.72	0.94
Bayes		1.40	0.58	2.82	0.91	0.96	0.96
MLE	60	0.73	0.20	1.52	0.27	0.87	0.94
Bayes		0.56	0.20	0.76	0.27	0.94	0.94
MLE	100	0.49	0.15	0.69	0.19	0.90	0.95
Bayes		0.43	0.15	0.58	0.19	0.94	0.94

5. Simulation study

In this section a simulation study is carried out in order to compare the performance of the proposed inference approaches: Maximum likelihood estimation and Bayesian estimation. To accomplish this goal, we have generated $B=1000$ pseudo-random samples from the Marshall-Olkin extended Exponentiated Gamma distribution for different values of the parameters $(\alpha,\theta)$ and different sample sizes as $n=$ 7, 10, 60 and 100. Moreover, samples were obtained via inverse transformation, thus generating a vector $u$ , with size $n$ , such that $u\sim U(0,1)$ . Since the MOEEG distribution does not have an explicit inverse distribution function $x={F^{-1}}(u)$ to generate pseudo-random samples, we have to find the solution $x$ of $F(x|\alpha,\theta)=U(0,1)$ using numerical method as the Newton-Raphson.

To compare the different estimation methods for each parameter, we computed the bias, root mean-squared error, average absolute difference between the true and estimate distributions functions and the maximum absolute difference between the true and estimate distributions functions as:

$\displaystyle{\textit{BIAS}(\hat{\alpha})=\frac{1}{B}\sum\limits_{i=1}^{B}({{% \hat{\alpha}}_{i}}-\alpha)}\quad{\textit{BIAS}(\hat{\theta})=\frac{1}{B}\sum% \limits_{i=1}^{B}({{\hat{\theta}}_{i}}-\theta)}$

(53) $\displaystyle{\textit{RMSE}(\hat{\alpha})=\sqrt{\frac{1}{B}\sum\limits_{i=1}^{% B}({{\hat{\alpha}}_{i}}-\alpha{)^{2}}}}\quad{\textit{RMSE}(\hat{\theta})=\sqrt% {\frac{1}{B}\sum\limits_{i=1}^{B}({{\hat{\theta}}_{i}}-\theta{)^{2}}}},$

respectively.

Other criterion for comparison of the estimation methods consists on checking the frequentist coverage probabilities of the intervals for each parameter of the distribution. Table 2 shows the values of the parameters obtained by different estimators where it could be seen that the Bayesian estimator tends to be less biased than the maximum likelihood estimator.

From Table 2, we observe that the Bayesian approach presents, in general, the bias a little smaller than the maximum likelihood method for small sample size $n=7$ and $n=10$ , for both parameters, while for larger values of $n$ the non-biased property is satisfied. Similarly it is noted for the RMSE measurement, that is, for $n$ small the results are a little better for the Bayes estimates. As expected, when $n$ increases, the results have similar values. In terms of coverage probability, the simulation study indicates that the Bayesian posterior intervals provide more accurate coverage probabilities close to nominal level 95% for any sample size while the coverage probabilities under MLE are very poor for small $n$ , turning the preference in favor of the Bayesian method. This can be explained by the use of asymptotic confidence intervals under the MLE despite the small sample size. This fact provides an advantage of using the Bayesian estimation method since the intervals do not depend on the sample size.
6. An application to survival data

An application was carried out to test the fit of the new MOEEG distribution with others already existing in the literature. For this purpose the dataset from (Box & Cox, 1964), which provide the lifetime of 48 animals exposed to certain types of poisons and then to certain types of treatments, was used. We can observe from the TTT-Plot in Fig. (5) that the hazard function is increasing.

Figure 5.

TTT plot.

Distributions considered for comparative purposes and their respective estimated parameters by the likelihood approach consist of the MOEEG with $\alpha=0.0036$ and $\theta=2.0509$ , Exponential (GE) with $\theta=0.3681$ , Nadarajah Haghighi Exponential (NHE) with $\alpha=0.0090$ and $\beta=153,0002$ , Weibull with $\alpha=2.0606$ and $\beta=0.5445$ , Exponential Power (EP) with $\lambda=0.7603$ and $\theta=1.3798$ and also the Gamma distribution with $\beta=0.1112$ and $\alpha=4.3079$ .

Several common criteria proposed in the literature can also be used for model selection. These include Akaike’s Information Criterion (AIC; Akaike (1974)), the Bayesian Information Criterion (BIC; Schwarz (1978)), Bozdogan’s consistent AIC (CAIC; Liang and Zou (2008)). These criteria essentially take into account the complexity of the distribution (number of parameters), sample size, among other conditions. The best model will be the one that represents a lower value to AIC, BIC e CAIC.

$\displaystyle\textit{AIC}=-2l(\theta)+2d$ $\displaystyle\textit{BIC}=-2l(\theta)+\log(n)$ (54) $\displaystyle\textit{CAIC}=\textit{AIC}+\frac{{2(d+2)(d+3)}}{{n-d-3}}$

Table 3

Selection criteria for the proposed model

Modelo	AIC	BIC	CAIC
MOEEG	$-$ 9.70	$-$ 5.96	$-$ 8.77
NHE	12.76	16.50	13.69
Gama	$-$ 8.34	$-$ 4.59	$-$ 7.40
GE	61.36	63.23	61.90
EP	5.67	9.41	6.60
Weibull	$-$ 2.50	1.23	$-$ 1.57

Figure 6.

Adjusted distributions.

From Table 3 we conclude that the MOEEG distribution had a good fit for the proposed dataset, after all, its results were better than the other distributions compared. Therefore, we show the flexibility of the new distribution proposed in this paper making it a good alternative to fit many data types.

7. Conclusions

In this paper, a new probabilistic distribution MOEEG was introduced based on the composition and Marshall-Olkin methods. Some properties and results were derived for MOEEG distribution such as $n-th$ time, $r-th$ moment of residual life, $r-th$ moment of residual life inverted, stochastic ordering, entropy, mean deviation, Bonferroni and Lorenz curve, skewness, kurtosis, order statistics, and stress-strength parameter. Maximum Likelihood and Bayesian methods are applied to estimate the parameters of the MOEEG distribution and an evaluation of their performance was carried out. We also illustrate the application of this distribution to a dataset showing the best fit of the MOEEG compared to other well-known distributions in the literature.

The main advantage of the proposed distribution is the possibility of the hazard function taking different shapes as monotonic, non-monotonic or unimodal, allowing great flexibility to fit different datasets. Besides, another advantage of this distribution is the presence of only two parameters providing a straightforwardly estimation procedure, mainly in computational aspects.

In conclusion, the results presented in this paper reinforce the fact that the search of appropriate lifetime distribution could be extremely difficult, especially, depending on the shape of the empirical hazard function of the data. The proposed methodology could be very useful due the flexibility of the generated model, especially in the medical data analysis. The results could be also extended to other cross-over trials in clinical issues, reliability analysis in engineering, risk analysis in economics, among many others areas.

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Marshall olkin extended exponentiated Gamma distribution and its applications

Abstract

Keywords

1. Introduction

2. Methods for generating probability models

2.1 Composition between distributions

.

Proof..

3.1 Moments of MOEEG

(17) θ ⁢ ( 1 + i ) - 2 - k - n ⁢ Γ ⁢ ( 2 + k + n ) . 3.2 Moment generating function of MOEEG

.

Proof..

4.1 Maximum Likelihood method

References

(17) $\displaystyle\theta{(1+i)^{-2-k-n}}\Gamma(2+k+n).$
3.2 Moment generating function of MOEEG