*2.2. Series Representation*

The normal cdf is related to the error function erf as follows:

$$\Phi(z) = \frac{1}{2} \left[ 1 + \text{erf} \left( \frac{z}{\sqrt{2}} \right) \right] \,\text{\,\,\,}\tag{11}$$

where erf(*z*) = √2*π z*0 *<sup>e</sup>*<sup>−</sup>*<sup>t</sup>*2d*<sup>t</sup>*. Provided that erf(*z*/√2) can be linearly represented by:

$$\begin{split} \text{erf}\left(\frac{z}{\sqrt{2}}\right) &= \frac{2}{\sqrt{\pi}} \sum\_{n=0}^{\infty} \frac{(-1)^n \cdot (z/\sqrt{2})^{2n+1}}{n!(2n+1)} \\ &= \sqrt{\frac{2}{\pi}} \cdot \sum\_{n=0}^{\infty} \left(-\frac{1}{2}\right)^n \frac{z^{2n+1}}{n!(2n+1)} \end{split} \tag{12}$$

replacing Equation (12) in Equation (11), we obtain:

$$\Phi(z) = \frac{1}{2} + \frac{1}{\sqrt{2\pi}} \sum\_{n=0}^{\infty} \left( -\frac{1}{2} \right)^{n} \frac{z^{2n+1}}{n!(2n+1)} \,. \tag{13}$$

Now, considering |*G*(*x*)| < 1, we can write:

$$\frac{2G(\mathbf{x}) - 1}{G(\mathbf{x})[1 - G(\mathbf{x})]} = \frac{2G(\mathbf{x}) - 1}{G(\mathbf{x})} \cdot \frac{1}{1 - G(\mathbf{x})} = \left(2 - \frac{1}{G(\mathbf{x})}\right) \sum\_{k=0}^{\infty} G(\mathbf{x})^k \tag{14}$$

and replacing *z* of the right member of Equation (13) by the expression in Equation (14), we have:

$$\Phi\left(\frac{2G(\mathbf{x})-1}{G(\mathbf{x})[1-G(\mathbf{x})]}\right) = \frac{1}{2} + \frac{1}{\sqrt{2\pi}} \sum\_{n=0}^{\infty} \frac{(-1/2)^n}{n!(2n+1)} \left[ \left(2 - \frac{1}{G(\mathbf{x})}\right) \sum\_{k=0}^{\infty} G(\mathbf{x})^k \right]^{2n+1}$$

$$= \frac{1}{2} + \frac{1}{\sqrt{2\pi}} \sum\_{n=0}^{\infty} \frac{(-1/2)^n}{n!(2n+1)} \underbrace{\left(2 - \frac{1}{G(\mathbf{x})}\right)^{2n+1}}\_{\mathbf{A}1} \underbrace{\left[\sum\_{k=0}^{\infty} G(\mathbf{x})^k\right]^{2n+1}}\_{\mathbf{A}2}.\tag{15}$$

The right member of Equation (15) has two factors, namely, A1 and A2, that can be rewritten as power series. Concerning to A1, the binomial theorem allows us to write:

$$\begin{aligned} \left(2 - \frac{1}{G(\mathbf{x})}\right)^{2n+1} &= \sum\_{j=0}^{2n+1} \binom{2n+1}{j} 2^{2n+1-j} \left(-\frac{1}{G(\mathbf{x})}\right)^j \\ &= \sum\_{j=0}^{2n+1} \binom{2n+1}{j} (-1)^j \cdot 2^{2n+1-j} \cdot G(\mathbf{x})^{-j} \\ &= \sum\_{j=0}^{2n+1} \delta\_j \cdot G(\mathbf{x})^{-j} \end{aligned} \tag{16}$$

It is a known result related to power series raised to powers that:

$$\left[\sum\_{k=0}^{\infty} a\_k G(\mathbf{x})^k\right]^N = \sum\_{k=0}^{\infty} c\_k G(\mathbf{x})^k \,\mathrm{.}\tag{17}$$

where *c*0 = *aN*0 , *ck* = 1*ka*0 ∑*ks*=<sup>1</sup>(*sN* − *k* + *<sup>s</sup>*)*asck*−*<sup>s</sup>* for *k* ≥ 1 and *N* ∈ N. Setting *N* = 2*n* + 1 and *ak* = 1 for all *k* ≥ 0, we ge<sup>t</sup> to the expression A2 in Equation (15) and we can use the result in Equation (17) to write as follows:

$$\left[\sum\_{k=0}^{\infty} G(\mathbf{x})^k\right]^{2n+1} = \sum\_{k=0}^{\infty} \mathbf{c}\_k \cdot G(\mathbf{x})^k \,. \tag{18}$$

such that *c*0 = 1, *ck* = 1 *k* ∑*k <sup>s</sup>*=<sup>1</sup>(*s*[<sup>2</sup>*<sup>n</sup>* + 1] − *k* + *<sup>s</sup>*)*ck*−*<sup>s</sup>* for *k* ≥ 1 and 2*n* + 1 ∈ N. Now replacing A1 and A2 of the Equation (15) by the right members of the Equations (16) and (18) respectively, we obtain the result below:

$$\Phi\left(\frac{2G(x)-1}{G(x)[1-G(x)]}\right) = \frac{1}{2} + \frac{1}{\sqrt{2\pi}} \sum\_{n=0}^{\infty} \frac{(-1/2)^n}{n!(2n+1)} \cdot \sum\_{j=0}^{2n+1} \delta\_j \cdot G(x)^{-j} \cdot \sum\_{k=0}^{\infty} c\_k \cdot G(x)^k$$

$$= \frac{1}{2} + \sum\_{n=0}^{\infty} \sum\_{j=0}^{2n+1} \sum\_{k=0}^{\infty} \underbrace{\binom{2n+1}{j} \frac{(-1)^{n+j} \cdot 2^{n+1-j}}{n!(2n+1)\sqrt{2\pi}} c\_k}\_{\eta\_{j,n,k}} \cdot G(x)^{k-j}$$

$$= \frac{1}{2} + \sum\_{n,k=0}^{\infty} \sum\_{j=0}^{2n+1} \eta\_{j,n,k} \cdot G(x)^{k-j} \cdot \tag{19}$$

The Fubini's theorem on differentiation allows us to write the derivative of Equation (19) with respect to *x* as follows:

$$f\_{\mathbb{G}}(\mathbf{x}) = \sum\_{n,k=0}^{\infty} \sum\_{j=0}^{2n+1} \eta\_{j,n,k} \cdot \underbrace{(k-j)g(\mathbf{x})G(\mathbf{x})^{k-j-1}}\_{\mathbb{S}^{k-j}} \,. \tag{20}$$

Since *gk*−*j*(*x*) is the pdf of a random variable of the exponentiated family, as described in [15,16], one can say that (20) is the Normal-*G* pdf (5) expressed as a linear combination of pdfs of exponentiated distributions. Such useful property is typically found and detailed in works on new classes of distributions; see for instance: [17–20].
