In this section, we derive an explicit expression for the EGG moments. Let \(\beta _{s}(\alpha,{\boldsymbol {\tau }})=E\left (Y_{\alpha }^{s}\right)\) be the *s*th moment of the EG class (2) with parameters *α* and *τ*. The *s*th moment of *X*, say \(\mu _{s}^{\prime }\), can be expressed in terms of the corresponding moments of EG distributions. In fact, we obtain a very simple expression from (8)

$$\begin{array}{@{}rcl@{}} \mu_{s}^{\prime}=E(X^{s})=\sum^{\infty}_{j=0}\,w_{j}\,\beta_{s}((j+1)\alpha,{\boldsymbol{\tau}}). \end{array} $$

((9))

Equation (9) reveals that the moments of *X* are linear combinations of the corresponding moments of *Y*
_{(j+1)α
} for *j*≥0. So, the moments of the EGG family are expressed as infinite linear combinations of those EG moments.

Next, we provide three alternative expressions for *β*
_{
s
}(*α*,*τ*). First, by substituting *u*=*Q*
_{
α,τ
}(*y*) given by (3) in the *s*th moment of *Y*
_{
α
} determined from (2), we obtain

$$\begin{array}{@{}rcl@{}} \beta_{s}(\alpha,{\boldsymbol{\tau}})={\int_{0}^{1}} Q_{\alpha,{\boldsymbol{\tau}}}(u)^{s} du. \end{array} $$

((10))

A second formula follows from (2) by changing variable *v*=*H*(*y*)

$$\begin{array}{@{}rcl@{}} \beta_{s}(\alpha,{\boldsymbol{\tau}})=s\,\int_{0}^{\infty} p_{s,{\boldsymbol{\tau}}}(v)\,\mathrm{e}^{-\alpha v}dv=P_{s,{\boldsymbol{\tau}}}(-\alpha), \end{array} $$

((11))

where \(p_{s,{\boldsymbol {\tau }}}(v)=\frac {H^{-1}(v)^{s-1}}{h\left (H^{-1}(v)\right)}\) and \(P_{s,{\boldsymbol {\tau }}}(-\alpha)=\mathfrak {L}(p_{s,{\boldsymbol {\tau }}}(v);\alpha)\) is the Laplace transform of *p*
_{
s,τ
}(*v*). A third formula for *β*
_{
s
}(*α*,*τ*) comes from the Mellin transform of *h*(*y*) e^{−αH(y)}, namely

$$\beta_{s}(\alpha,{\boldsymbol{\tau}})= \alpha\,\mathrm{M}\left(h(y)\,\mathrm{e}^{-\alpha H(y)};s+1\right). $$

Several tables provide Mellin transforms for common functions. The Laplace and Mellin transforms are defined in Prudnikov et al. (1986). Equations (9)–(11) are the main results of this section. They can be used to obtain analytically or numerically the moments of several EGG special models.

Further, we provide an application of (9) in conjunction with (10) and (11). Substituting *H*(*y*) of the XTG distribution given in the “Appendix: new special EGG models” into Eq. (11) and assuming that *m*=*s*/*β* is an integer, and after some algebra, we have

$$\begin{array}{@{}rcl@{}} \beta_{s}(\alpha,\beta,\lambda)&=&s\,\int_{0}^{\infty} v^{s-1}\,\exp(\lambda \alpha)\,\exp\left\{-\lambda \alpha \exp\left[(v/\lambda)^{\beta}\right]\right\}dv \\ &=&m\,\lambda^{s}\,\exp(\lambda\alpha)\,\frac{\partial^{m-1}(\lambda\alpha)^{-t}\Gamma(t,\lambda\alpha)} {\partial t^{m-1}}\Big{|}_{t=0}, \end{array} $$

((12))

where \(\Gamma (t,\alpha)=\int _{t}^{\infty }\,w^{\alpha -1}\,\mathrm {e}^{-w}dw\) is the upper incomplete gamma function. Equation (12) gives the moments of the XTG distribution. Hence, we can express the XTGG moments by combining Eqs. (9) and (12).

The central moments (*μ*
_{
s
}) and cumulants (*κ*
_{
s
}) of *X* can be obtained from (9) as

$$\mu_{s}=\sum_{j=0}^{s}(-1)^{j}\,\binom{s}{j}\,\mu_{1}^{\prime s}\,\mu_{s-j}^{\prime} \quad \text{and} \quad \kappa_{s}=\mu^{\prime}_{s}-\sum_{j=1}^{s-1}\,\binom{s-1}{j-1}\,\kappa_{j}\,\mu^{\prime}_{s-j}, $$

respectively, where \(\kappa _{1}=\mu ^{\prime }_{1}\). The skewness and kurtosis measures can be determined from the central moments using well-known relationships. Plots of the skewness and kurtosis for some parameter values as functions of *α* for the XTGG and MWG distributions are displayed in Figs. 3 and 4, respectively.

Finally, for empirical purposes, the shape of many distributions can be usefully described by the incomplete moments. These types of moments play an important role for measuring inequality, for example, income quantiles and Lorenz and Bonferroni curves. The *s*th incomplete moment of *X* is determined from (8) as

$$\begin{array}{@{}rcl@{}} m_{s}(y)={\int_{0}^{y}}x^{s}\,f(x;p,\alpha,{\boldsymbol{\tau}}) dx=\sum^{\infty}_{j=0}w_{j}\,\int_{0}^{Q_{(j+1)\alpha,{\boldsymbol{\tau}}}(y)}\,Q_{(j+1)\alpha,{\boldsymbol{\tau}}}(u)^{s} du. \end{array} $$

((13))

Both integrals in (13) can be evaluated numerically for most EGG distributions.