We investigate some properties of the EMOG in this section.

### 3.1 Asymptotic and shapes

###
**Proposition 1**

Let *a*= inf{*x*| *G*(*x*)>0}. The asymptotics of Eqs. (2), (3) and (4) when *x*→*a* are given by

$$\begin{array}{@{}rcl@{}} &&F(x)\sim \left[\lambda\,G(x)\right]^{\alpha}, \\ &&f(x)\sim \alpha\,\lambda^{\alpha}\,g(x) \,G(x)^{\alpha-1},\\ &&h(x)\sim \alpha\,\lambda^{\alpha}\,g(x) \,G(x)^{\alpha-1}.\\ \end{array} $$

###
**Proposition 2**

The asymptotics of Eqs. (2), (3) and (4) when *x*→*∞* are given by

$$\begin{array}{@{}rcl@{}} && 1-F(x)\sim \alpha\,\bar{G}(x)^{\lambda}, \\ &&f(x)\sim \alpha\,\lambda\,g(x)\,\bar{G}(x)^{\lambda-1},\\ &&h(x)\sim \frac{\lambda\,g(x)}{\bar{G}(x)}. \end{array} $$

The shapes of the density and hazard rate functions can be described analytically. The critical points of the EMO-G density function are the roots of the equation

$$\begin{array}{@{}rcl@{}} \frac{d\, \log[f(x)]}{d x}&=&\frac{g'(x)}{g(x)}+(1-\lambda)\frac{g(x)}{1-G(x)}\\ &-&\lambda g(x)[1-G(x)]^{\lambda-1}\left\{\frac{1-\alpha}{1-[1-G(x)]^{\lambda}}+\frac{p(\alpha+1)}{1-p[1-G(x)]^{\lambda}}\right\}=0 \end{array} $$

(6)

that corresponds to points where *f*
^{′}(*x*)=0. There may be more than one root to (6). Let *λ*(*x*)=*d*
^{2} log[*f*(*x*)]/*d*
*x*
^{2}. We have

$$ \begin{aligned} {}\lambda(x)&=\frac{g^{\prime\prime}(x)g(x)-g'(x)^{2}}{g^{2}(x)}+(1-\lambda)\frac{g'(x)[1-G(x)]+g^{2}(x)}{[1-G(x)]^{2}}+\lambda(\alpha-1) \\ &\quad\times\left\{\!g'(x)\!\frac{[\!1-\!G(x)]^{\lambda-1}}{1-[\!1\,-\,G(x)]^{\lambda}}\right. \,-\,\left.(\lambda\,-\,1)g^{2}(x)\!\frac{[\!1-\!G(x)]^{\lambda-2}}{1\,-\,[\!1-G(x)]^{\lambda}} -\lambda g^{2}(x)\frac{[1-G(x)]^{2\lambda-2}}{\left\{1-[1-G(x)]^{\lambda}\right\}^{2}}\right\} \\ &\quad- p\lambda(\alpha+1) \left\{g'(x)\frac{[1-G(x)]^{\lambda-1}}{1-p[1-G(x)]^{\lambda}} -(\lambda-1)g^{2}(x)\frac{[1-G(x)]^{\lambda-2}}{1-p[1-G(x)]^{\lambda}} \right.\\ &\quad\left.-p\lambda g^{2}(x)\frac{[1-G(x)]^{2\lambda-2}}{\left\{1-p[1-G(x)]^{\lambda}\right\}^{2}}\right\}. \end{aligned} $$

(7)

If *x*=*x*
_{0} is a root of (6) then it corresponds to a local maximum if *λ*(*x*)>0 for all *x*<*x*
_{0} and *λ*(*x*)<0 for all *x*>*x*
_{0}. It corresponds to a local minimum if *λ*(*x*)<0 for all *x*<*x*
_{0} and *λ*(*x*)>0 for all *x*>*x*
_{0}. It gives a point of inflexion if either *λ*(*x*)>0 for all *x*≠*x*
_{0} or *λ*(*x*)<0 for all *x*≠*x*
_{0}.

The critical points of the (hrf) of *X* are obtained from the equation

$$ {{\begin{aligned} {}\frac{d\, \log[h(x)]}{d x}&=\frac{g'(x)}{g(x)}+(1-\lambda)\frac{g(x)}{1-G(x)}+\lambda(\alpha-1)\frac{g(x)[1-G(x)]^{\lambda-1}}{1-[1-G(x)]^{\lambda}}-\lambda p\frac{g(x)[1-G(x)]^{\lambda-1}}{1-p[1-G(x)]^{\lambda}}\\ &-\!\frac{p\alpha\lambda g(x)[\!1-\!G(x)]^{\lambda-1}\!\left\{1-[\!1\,-\,G(x)]^{\lambda}\!\right\}^{\alpha-1}\,-\,\lambda\alpha g(x)[\!1\,-\,G(x)]^{\lambda-1}\left\{1\!-p[1\!-G(x)]^{\lambda}\!\right\}^{\alpha-1}}{\left\{1-p[\!1-G(x)]^{\lambda}\right\}^{\alpha}-\left\{1-[\!1-G(x)]^{\lambda}\right\}^{\alpha}}\,=\,0. \end{aligned}}} $$

(8)

There may be more than one root to (8). Let *τ*(*x*)=*d*
^{2} log[*h*(*x*)]/*d*
*x*
^{2}. If *x*=*x*
_{0} is a root of (8) then it refers to a local maximum if *τ*(*x*)>0 for all *x*<*x*
_{0} and *τ*(*x*)<0 for all *x*>*x*
_{0}. It corresponds to a local minimum if *τ*(*x*)<0 for all *x*<*x*
_{0} and *τ*(*x*)>0 for all *x*>*x*
_{0}. It gives an inflexion point if either *τ*(*x*)>0 for all *x*≠*x*
_{0} or *τ*(*x*)<0 for all *x*≠*x*
_{0}.

### 3.2 Linear mixtures

We can demonstrate that the cdf (2) of *X* admits the expansion

$$\begin{array}{@{}rcl@{}} F(x)=\sum\limits_{k=0}^{\infty}b_{k}\,H_{k}(x; \boldsymbol{\xi}), \end{array} $$

(9)

where \(b_{k}=\sum \limits _{i,j=0}^{\infty }w_{i,j,k}\),

$$ w_{i,j,k}=w_{i,j,k}(\alpha,\lambda, p)=(-1)^{i+j+k}\,\dbinom{-\alpha}{i}\,\dbinom{\alpha}{j}\,\dbinom{(i+j)\lambda}{k}\,p^{i}, $$

and *H*
_{
k
}(*x*;*ξ*)=*G*(*x*;*ξ*)^{k} denotes the exponentiated-G (“exp-G”) cdf with power parameter *k*.

The density function of *X* can be expressed as an infinite linear mixture of exp-G density functions

$$ f(x)=\sum\limits_{k=0}^{\infty}\,b_{k+1}\,h_{k+1}(x;\boldsymbol{\xi}), $$

(10)

where (for *k*≥0) *h*
_{
k+1}(*x*;*ξ*)=(*k*+1) *g*(*x*;*ξ*) *G*(*x*;*ξ*)^{k} denotes the density function of the random variable *Y*
_{
k+1}∼exp-G(*k*+1). Equation (10) reveals that the EMO-G density function is a linear mixture of exp-G density functions. Thus, some of its mathematical properties can be derived directly from those properties of the exp-G distribution. For example, the ordinary and incomplete moments and (mgf) of *X* can be obtained from those quantities of the exp-G distribution. Some structural properties of the exp-G distributions are well-defined by Mudholkar and Hutson (1996), Gupta and Kundu (2001) and Nadarajah and Kotz (2006), among others.

The formulae derived throughout the paper can be easily handled in most symbolic computation software platforms such as Maple, Mathematica and Matlab. These platforms have currently the ability to deal with analytic expressions of formidable size and complexity. Established explicit expressions to calculate statistical measures can be more efficient than computing them directly by numerical integration. The infinity limit in these sums can be substituted by a large positive integer such as 20 or 30 for most practical purposes.

### 3.3 Quantile power series

We obtain explicit expressions for the moments and generating function of the EMO family using a power series for the qf *x*=*Q*(*u*)=*F*
^{−1}(*u*) of *X* by expanding (5). If the G qf, say *Q*
_{
G
}(*u*), does not have a closed-form expression, this function can usually be expressed as a power series

$$\begin{array}{@{}rcl@{}} Q_{G}(u)=\sum\limits_{i=0}^{\infty} a_{i}\,u^{i}, \end{array} $$

(11)

where the coefficients *a*
_{
i
}’s are suitably chosen real numbers depending on the parameters of the parent distribution. For several important distributions such as the normal, Student t, gamma and beta distributions, *Q*
_{
G
}(*u*) does not have explicit expressions but it can be expanded as in Eq. (11).

We use throughout the paper a result of Gradshteyn and Ryzhik (2000) for a power series raised to a positive integer *n* (for *n*≥1)

$$\begin{array}{@{}rcl@{}} Q_{G}(u)^{n}=\left(\sum\limits_{i=0}^{\infty} a_{i}\,u^{i}\right)^{n}=\sum\limits_{i=0}^{\infty} c_{n,i}\,u^{i}, \end{array} $$

(12)

where the coefficients *c*
_{
n,i
} (for *i*=1,2,…) are easily determined from the recurrence equation, with \(c_{n,0}={a_{0}^{n}}\),

$$ c_{n,i}=(i\,a_{0})^{-1}\,\sum\limits_{m=1}^{i}\,[m(n+1)-i]\,a_{m}\,c_{n,i-m}. $$

(13)

Clearly, *c*
_{
n,i
} can be easily evaluated numerically from *c*
_{
n,0},…,*c*
_{
n,i−1} and then from the quantities *a*
_{0},…,*a*
_{
i
}.

Next, we derive an expansion for the argument of *Q*
_{
G
}(·) in Eq. (5)

$$A=1-\frac{(1-u^{1/\alpha})^{1/\lambda}}{(1-p\,u^{1/\alpha})^{1/\lambda}}. $$

Using the the generalized binomial expansion four times since *u*∈(0,1), we can write

$$ A=\sum\limits_{r,s,t=0}^{\infty}\sum\limits_{m=0}^{t}(-1)^{r+s+t+m}\,\,p^{r}\,\dbinom{-\lambda^{-1}}{r}\,\dbinom{\lambda^{-1}}{s}\, \dbinom{(r+s)\alpha^{-1}}{t}\dbinom{t}{m}\,u^{m} $$

Then, the qf of *X* can be expressed from (5) as

$$ Q(u)=Q_{G}\left(\sum\limits_{m=0}^{\infty}\delta_{m}\,u^{m}\right), $$

(14)

where

$$ \delta_{m} = \left\{ \begin{array}{ccccccc} 1-{\sum\nolimits}_{r,s,t=0}^{\infty}(-1)^{r+s+t}\,\,p^{r}\,\dbinom{-\lambda^{-1}}{r}\,\dbinom{-\lambda^{-1}}{s}\,\dbinom{(r+s)\alpha^{-1}}{t},\,\,m = 0,\\ {\sum\nolimits}_{r,s,t=0}^{\infty}(-1)^{r+s+t+m}\,\,p^{r}\,\dbinom{-\lambda^{-1}}{r}\,\dbinom{\lambda^{-1}}{s}\,\dbinom{(r+s)\alpha^{-1}}{t}\dbinom{t}{m},\,\,m > 0.\\ \end{array} \right. $$

By combining (11) and (14), we have

$$ Q(u)=\sum\limits_{i=0}^{\infty} a_{i}\,\left(\sum\limits_{m=0}^{\infty}\delta_{m}\,u^{m}\right)^{i}, $$

and then using (12) and (13),

$$ Q(u)=\sum\limits_{m=0}^{\infty} e_{m}\,u^{m}, $$

(15)

where \(e_{m}={\sum \nolimits }_{i=0}^{\infty } a_{i}\,d_{i,m}\), \(d_{i,0}={\delta _{0}^{i}}\) and, for *m*>1,

$$d_{i,m}=(m\,\delta_{0})^{-1}\,\sum\limits_{n=1}^{m}[n(i+1)-m]\,\delta_{n}\,d_{i,m-n}. $$

Equation (15) is the main result of this section. It allows to obtain various mathematical quantities for the EMO-G family as can be seen in the next sections. Note that

$$ Q(u)^{r} = \left(\sum\limits_{m=0}^{\infty} e_{m} u^{m}\right)^{r} = \sum\limits_{m=0}^{\infty} f_{r,m}\,u^{m}, $$

(16)

where *f*
_{
r,m
} is obtained from the *e*
_{
m
}’s using (13).

The effects of the shape parameters on the skewness and kurtosis can be determined from quantile measures. The shortcomings of the classical kurtosis measure are well-known. The Bowley skewness (Kenney and Keeping 1962) is one of the earliest skewness measures defined by the average of the quartiles minus the median divided by half the interquartile range, namely

$$ B=\frac{Q\left(\frac{3}{4}\right)+Q\left(\frac{1}{4}\right)-2Q\left(\frac{1}{2}\right)}{Q\left(\frac{3}{4}\right)-Q\left(\frac{1}{4}\right)}. $$

Since only the middle two quartiles are considered and the outer two quartiles are ignored, this adds robustness to the measure. The Moors kurtosis (Moors 1998) is based on octiles

$$ M=\frac{Q\left(\frac{3}{8}\right)-Q\left(\frac{1}{8}\right)+Q\left(\frac{7}{8}\right)-Q\left(\frac{5}{8}\right)}{Q\left(\frac{6}{8}\right)-Q\left(\frac{2}{8}\right)}. $$

These measures are less sensitive to outliers and they exist even for distributions without moments. In Figs. 6 and 7, we plot the measures *B* and *M* for the EMOFr and EMON distributions (discussed in Section 2), respectively. These plots reveal how both measures *B* and *M* vary on the shape parameters.

### 3.4 Moments

Hereafter, we shall assume that *G* is the cdf of a random variable *Z* and that *F* is the cdf of a random variable *X* having density function (3). The *r*th ordinary moment of *X* can be obtained from the (*r*,*k*)th Probability Weighted Moment (PWM) of *Z* defined by

$$\begin{array}{@{}rcl@{}} \tau_{r,k}=\mathrm{E}[Z^{r}\,G(Z)^{k}]={\int\nolimits}_{-\infty}^{\infty} z^{r}\,G(z)^{k}\,g(z)dz. \end{array} $$

(17)

In fact, we have

$$\begin{array}{@{}rcl@{}} \mathrm{E}(X^{r})=\sum\limits_{k=0}^{\infty} (k+1)\,b_{k+1}\,\tau_{r,k}. \end{array} $$

(18)

Thus, the moments of any EMO-G distribution can be expressed as an infinite linear combination of the PWMs of G. A second formula for *τ*
_{
r,k
} can be based on the parent qf *Q*
_{
G
}(*u*)=*G*
^{−1}(*u*). Setting *G*(*x*)=*u*, we obtain

$$\begin{array}{@{}rcl@{}} \tau_{r,k}={{\int\nolimits}_{0}^{1}} Q_{G}(u)^{r}\,u^{k} d u, \end{array} $$

(19)

where the integral follows from (16) as

$$\begin{array}{@{}rcl@{}} \mathrm{E}(X^{r}) = {{\int\nolimits}_{0}^{1}}\,Q(u)^{r} du = \sum\limits_{m=0}^{\infty}\frac{f_{r,m}}{m+1}. \end{array} $$

(20)

The PWMs for some well-known distributions will be determined in the following sections using alternatively Eqs. (17) and (19).

The central moments (*μ*
_{
s
}) and cumulants (*κ*
_{
s
}) of *X* can be obtained from Eqs. (18) and (20) as

$$ \mu_{s}=\sum\limits_{j=0}^{s}(-1)^{j} \binom{s}{j} \mu_{1}^{\prime s} \mu_{s-j}^{\prime}, \qquad \kappa_{s}=\mu^{\prime}_{s}-\sum\limits_{j=1}^{s-1} \binom{s-1}{j-1} \kappa_{j} \mu^{\prime}_{s-j}, $$

where \(\kappa _{1}=\mu ^{\prime }_{1}\). The skewness and kurtosis measures can be calculated from the ordinary moments using well-known relationships. The *p*th descending factorial moment of *X* is

$$ \mu_{(p)}^{\prime}=\mathrm{E}[X^{(p)}]=\mathrm{E}[X(X-1)\times\cdots \times(X-p+1)]=\sum\limits_{k=0}^{p} s(p,k) \mu^{\prime}_{k}, $$

where *s*(*r*,*k*)=(*k*!)^{−1}[*d*
^{k}
*x*
^{(r)}/*d*
*x*
^{k}]_{
x=0} is the Stirling number of the first kind. So, we can obtain the factorial moments from the ordinary moments given before.

### 3.5 Incomplete moments

The *n*th incomplete moment of *X* is defined as \(m_{r}(y)= \int _{-\infty }^{y}x^{r}\,f(x)dx\). For empirical purposes, the shape of many distributions can be usefully described by the incomplete moments. Here, we propose two methods to determine the incomplete moments of the new family. First, we can express *m*
_{
r
}(*y*) as

$$\begin{array}{@{}rcl@{}} m_{r}(y)= \sum\limits_{k=0}^{\infty}\ (k+1)\,b_{k+1}\,{\int\nolimits}_{0}^{G(y;\,\boldsymbol{\xi})}\,Q_{G}(u)^{r}\, u^{k}\,du. \end{array} $$

(21)

The integral in (21) can be evaluated at least numerically for most baseline distributions.

A second method for the incomplete moments of *X* follows from (21) using Eqs. (12) and (13). We obtain

$$\begin{array}{@{}rcl@{}} m_{r}(y)= \sum\limits_{k,m=0}^{\infty} \frac{(k+1)\,b_{k+1}\,c_{r,m}}{m+k+1}\,G(y;\boldsymbol{\xi})^{m+k+1}. \end{array} $$

(22)

Equations (21) and (22) are the main results of this section.

### 3.6 Generating function

Here, we provide three formulae for the mgf *M*(*s*)=*E*(e^{sX}) of *X*. A first formula for *M*(*s*) comes from Eq. (10) as

$$\begin{array}{@{}rcl@{}} M(s)=\sum\limits_{k=0}^{\infty} b_{k+1}\,M_{k+1}(s), \end{array} $$

(23)

where *M*
_{
k+1}(*s*) is the generating function of the exp-G(*k*+1) distribution. Hence, *M*(*s*) can be determined from an infinite linear combination of the exp-G generating functions.

A second formula for *M*(*s*) can be derived from Eq. (10) as

$$\begin{array}{@{}rcl@{}} M(s)=\sum\limits_{k=0}^{\infty} (k+1)\,b_{k+1}\,\rho_{k}(s), \end{array} $$

(24)

where

$$\begin{array}{@{}rcl@{}} \rho_{k}(s)={{\int\nolimits}_{0}^{1}}\exp\left[s\,Q_{G}(u)\right] u^{k} d u. \end{array} $$

(25)

We can derive the mgfs of several EMO distributions directly from Eqs. (24) and (25). For example, the mgfs of the exponentiated Marshall-Olkin exponential (EMOE) (such that *λ*
*s*<1) and EMO-standard logistic (for *s*<1) distributions are given by

$$\begin{array}{@{}rcl@{}} M(s)\,=\,\sum\limits_{k=0}^{\infty} (k\,+\,1\,)\,b_{k+1}\,\,B(k\,+\,1,1\,-\,\lambda s)\quad \text{and}\quad M(s)=\sum\limits_{k=0}^{\infty} (k+1\,)\,b_{k+1}\,B(s+k\,+\,1,1-s), \end{array} $$

respectively.

### 3.7 Mean deviations

The mean deviations about the mean (\(\delta _{1}=E(|X-\mu ^{\prime }_{1}|)\)) and about the median (*δ*
_{2}=*E*(|*X*−*M*|)) of *X* can be expressed as

$$\begin{array}{@{}rcl@{}} \delta_{1}=2 \mu^{\prime}_{1}\,F\left(\mu^{\prime}_{1}\right)-2 m_{1}\left(\mu^{\prime}_{1}\right) \qquad\text{and}\qquad\delta_{2}=\mu^{\prime}_{1}-2 m_{1}(M), \end{array} $$

(26)

respectively, \(F(\mu ^{\prime }_{1})\) is easily evaluated from Eq. (2),

$$M=Q_{G}\left[1-\left(\frac{1-2^{-1/\alpha}}{1-p\times2^{-1/\alpha}}\right)^{1/\lambda}\right] $$

is the median of *X*, \(\mu ^{\prime }_{1}=\mathrm {E}(X)\) comes from (18) and *m*
_{1}(*y*) is the first incomplete moment of *X* determined from (22) with *r*=1.

Next, we provide three alternative ways to compute *δ*
_{1} and *δ*
_{2}. A general equation for *m*
_{1}(*z*) is given by (21). A second general formula for *m*
_{1}(*z*) can be obtained from (10) as

$$\begin{array}{@{}rcl@{}} m_{1}(z)= \sum\limits_{k=0}^{\infty} b_{k+1}\,J_{k+1}(z), \end{array} $$

(27)

where

$$\begin{array}{@{}rcl@{}} J_{k+1}(z)={\int\nolimits}_{-\infty}^{z} x\,h_{k+1}(x)dx. \end{array} $$

(28)

Equation (28) is the basic quantity to compute the mean deviations for the exp-G distributions. A simple application of (27) and (28) can be conducted to the exponentiated Marshall-Olkin Weibull (EMOW) distribution. The exponentiated Weibull density function (for *x*>0) with power parameter *k*+1, shape parameter *c* and scale parameter *β* is given by

$$\begin{array}{@{}rcl@{}} h_{k+1}(x)=c\,(k+1)\,\beta^{c}\,x^{c-1}\,\exp\left\{-(\beta x)^{c}\right\}\,\left[1-\exp\left\{-(\beta x)^{c}\right\}\right]^{k}, \end{array} $$

and then

$$\begin{array}{@{}rcl@{}} J_{k+1}(z)&=&c\,(k+1)\,\beta^{c}\,\sum\limits_{r=0}^{\infty}(-1)^{r}\,{k \choose r}\, {{\int\nolimits}_{0}^{z}} x^{c}\,\exp\left\{-(r+1)(\beta x)^{c}\right\} dx. \end{array} $$

The last integral reduces to the incomplete gamma function

$$\begin{array}{@{}rcl@{}} J_{k+1}(z)=c\,(k+1)\,\beta^{c}\,\sum\limits_{r=0}^{\infty} (-1)^{r}\,{k \choose r}\,\gamma\left(c+1,(r+1)(\beta z)^{c}\right), \end{array} $$

where \(\gamma (a,x)= {\int _{0}^{x}} w^{a-1}\,\mathrm {e}^{-w}\mathrm {d}w\).

A third general formula for *m*
_{1}(*z*) can be derived by setting *u*=*G*(*x*) in (10)

$$\begin{array}{@{}rcl@{}} m_{1}(z)=\sum\limits_{k=0}^{\infty} (k+1)\,b_{k+1}\,\,T_{k}(z), \end{array} $$

(29)

where *T*
_{
k
}(*z*) is given by

$$\begin{array}{@{}rcl@{}} T_{k}(z)={\int\nolimits}_{0}^{G(z)}Q_{G}(u)\,u^{k} du. \end{array} $$

(30)

Applications of these equations are straightforward to obtain Bonferroni and Lorenz curves. These curves are defined (for a given probability *π*) by \(B(\pi)=m_{1}(q)/(\pi \,\mu ^{\prime }_{1})\) and \(L(\pi)=m_{1}(q)/\mu ^{\prime }_{1}\), respectively, where *q*=*F*
^{−1}(*π*)=*Q*(*π*) comes from the qf of *X* for a given probability *π*.