*2.3. Quantile Function*

By inverting Equation (4), the quantile function associated with the Normal-*G* class is obtained. For simplification, let us write *v* = *FG*(*x*). From Equation (4) we have:

$$\Phi^{-1}(v) = \frac{2G(\mathbf{x}) - 1}{G(\mathbf{x})[1 - G(\mathbf{x})]} \Rightarrow \Phi^{-1}(v)G(\mathbf{x})^2 + [2 - \Phi^{-1}(v)]G(\mathbf{x}) - 1 = 0, \dots$$

that is, a quadratic equation for *<sup>G</sup>*(*x*), that admits the following two solutions:

$$\frac{\Phi^{-1}(v) - 2 - \sqrt{4 + \Phi^{-1}(v)^2}}{2\Phi^{-1}(v)} \quad \text{and} \quad \frac{\Phi^{-1}(v) - 2 + \sqrt{4 + \Phi^{-1}(v)^2}}{2\Phi^{-1}(v)}$$

If the first solution above is picked, then *<sup>G</sup>*(*x*) might assume values lesser than 0 (see *v* = 0.95 for example). On the other hand, the second one allows us to verify that 0 < *<sup>G</sup>*(*x*) < 1 is valid for all *x* ∈ R. Finally, we can write the quantile function of Equation (4) as follows:

$$Q\_F(v) = Q\_G \left[ \frac{\Phi^{-1}(v) - 2 + \sqrt{4 + \Phi^{-1}(v)^2}}{2\Phi^{-1}(v)} \right],\tag{21}$$

.

such that *QG*(·) is the quantile function of the baseline *G*. A uniform random number generator and (21) make the simulation of random variables following (3) quite simple. Namely, if *Z* ∼ U(0, <sup>1</sup>), then *QF*(*Z*) follows a Normal-*G* distribution.

### *2.4. Raw Moments, Incomplete Moments and Moment Generating Function*

Provided that *X* follows a Normal-*G* distribution, the *r*th raw moment of *X* is *E*(*Xr*) = ∞ −∞ *x<sup>r</sup> fG*(*x*)d*<sup>x</sup>*, where *fG*(*x*) is given in Equation (20) and *r* ∈ Z∗ +. Using Fubini's theorem to change the order of integration and series, we have:

$$E(X^r) = \sum\_{n,k=0}^{\infty} \sum\_{j=0}^{2n+1} \eta\_{j,n,k} \int\_{-\infty}^{\infty} \mathbf{x}^r \mathbf{g}\_{k-j}(\mathbf{x}) d\mathbf{x} \tag{22}$$

$$\hat{\eta} = \sum\_{n,k=0}^{\infty} \sum\_{j=0}^{2n+1} \eta\_{j,n,k} E(Y\_{k-j}^r) \tag{23}$$

where *Yk*−*<sup>j</sup>* follows the exponentiated distribution whose pdf is *gk*−*j*(*x*) shown in Equation (20).

Despite the upper infinity limit in the sums, expressions like Equation (23) are not intractable. According to [21], one can ge<sup>t</sup> fairly accurate results truncating each infinite sum by 20; they used numerical routines to compute accurately similar expressions for the moments of some Kumaraswamy generalized distributions.

The *r*th moment can also be represented in terms of the quantile function of the baseline. Defining *u* = *Gk*−*j*(*x*) and replacing *x* in Equation (22) by *QG u*1/(*k*−*j*) , we have:

$$E(X^r) = \sum\_{n,k=0}^{\infty} \sum\_{j=0}^{2n+1} \eta\_{j,n,k} \int\_0^1 \left[ \mathcal{Q}\_G \left( u^{\frac{1}{k-j}} \right) \right]' \,\mathrm{d}u \,\dots$$

The *r*th incomplete moment of *X* is given by the following expression:

$$T\_I(z) = \int\_{-\infty}^z x^r f\_G(x) dx = \sum\_{n,k=0}^\infty \sum\_{j=0}^{2n+1} \eta\_{j,n,k} T\_r^\*(z) \tag{24}$$

where *T*∗ *r* (*z*) is the *r*th incomplete moment of *Yk*−*j*. One can also write Equation (24) in terms of the quantile function of *G*:

$$T\_{\mathcal{I}}(z) = \sum\_{n,k=0}^{\infty} \sum\_{j=0}^{2u+1} \eta\_{j,n,k} \int\_0^{[G(\alpha)]^{k-j}} \left[ Q\_G \left( u^{\frac{1}{k-j}} \right) \right]^r \mathbf{d}u \dots$$

The mgf is a function associated with a random variable, whose moments can be straightforwardly derived using it. It is also useful to check whether two functions of random variables are equal since there is a bijection between pdfs and mgfs (when they exist). The mgf <sup>M</sup>*X*(*t*) of *X* is the expected value of *<sup>e</sup>tX*, where *t* ∈ (−*ι*, *ι*), *ι* > 0. Given that <sup>M</sup>*Yk*−*<sup>j</sup>* (*t*) is the mgf of *Yk*−*j*, on the lines of Equation (23), we can write:

$$\begin{split} \mathbf{M}\_{X}(t) &= \sum\_{n,k=0}^{\infty} \sum\_{j=0}^{2n+1} \eta\_{j,n,k} \int\_{-\infty}^{\infty} \varepsilon^{tx} g\_{k-j}(x) dx \\ &= \sum\_{n,k=0}^{\infty} \sum\_{j=0}^{2n+1} \eta\_{j,n,k} \mathbf{M}\_{Y\_{k-j}}(t) \ . \end{split}$$
