*2.4. Quantiles*

**Proposition 4.** *The quantile function for the GTPN*(*<sup>σ</sup>*, *λ*, *α*) *is given by*

$$Q(p) = \sigma \left[ \Phi^{-1}(1 - (1 - p)\Phi(\lambda)) + \lambda \right]^{\frac{1}{x}}.$$

**Proof.** Follows from a direct computation, applying the definition of quantile function.

**Corollary 1.** *The quartiles of the GTPN distribution are*


*2.5. Central Moments*

**Proposition 5.** *Let Z* ∼ *GTPN*(*<sup>σ</sup>*, *λ*, *α*) *and r* = 1, 2, . . .*. The r-th non-central moment is given by*

$$\mu\_r = \mathbb{E}(Z^r) = \frac{\left(\sigma\lambda^{1/\alpha}\right)^r b\_r(\lambda, \alpha)}{2\Phi(\lambda)\sqrt{\pi}}.$$

*where br*(*<sup>λ</sup>*, *α*) = ∑∞*<sup>k</sup>*=<sup>0</sup> (*<sup>r</sup>*/*<sup>α</sup>k* ) *λ*2 −*k* Γ *<sup>k</sup>*+<sup>1</sup> 2 , *λ*22 *and* (*<sup>r</sup>*/*<sup>α</sup>k* ) = 1*k*! *k*−1 ∏ *<sup>n</sup>*=1 (*r*/*α* − *n*) *is the generalized binomial coefficient. When r*/*α* ∈ N*, the sum in br*(*<sup>λ</sup>*, *α*) *stops at <sup>r</sup>*/*<sup>α</sup>.*

**Proof.** Considering the stochastic representation of the GTPN model in Section 2.1, it is immediate that E(*Zr*) = *σ<sup>r</sup>*E(*<sup>Y</sup> rα* ), where *Y* ∼ *TPN*(1, *<sup>λ</sup>*). This expected value can be computed using Proposition 2.2 in Gómez et al. [13].

**Remark 2.** *When r*/*α* ∈ N*, Closed Forms Can Be Obtained for μr*

Figure 3 illustrates the mean, variance, skewness, and kurtosis coefficients for the GTPN(*σ* = 1, *λ*, *α*) model for some combinations of its parameters.

**Figure 3.** Plots of the (**a**) expectation, (**b**) variance, (**c**) skewness and (**d**) kurtosis for GTPN(*σ* = 1, *λ*, *α*) for *α* ∈ {0.75, 1, 1.5} as a function of *λ*. In (**d**), the dashed line represents the kurtosis of the normal distribution.

### *2.6. Bonferroni and Lorenz Curves*

In this subsection we present the Bonferroni and Lorenz curves (see Bonferroni [14]). These curves have applications not only in economics to study income and poverty, but also in medicine, reliability, etc. The Bonferroni curve is defined as

$$B(p) = \frac{1}{p\mu} \int\_0^q zf(z)dz, \quad 0 \le p < 1.$$

where *μ* = *<sup>E</sup>*(*Z*), *q* = *<sup>F</sup>*−<sup>1</sup>(*p*). The Lorenz curve is obtained by the relation *<sup>L</sup>*(*p*) = *p<sup>B</sup>*(*p*). Particularly, it can be checked that for the GTPN model the Bonferroni curve is given by

$$B(p) = \frac{1}{p\mu} \left[ E(Z) - \frac{\sigma}{\Phi(\lambda)\sqrt{2\pi}} \sum\_{k=0}^{\infty} \binom{\frac{1}{a}}{k} \lambda^{\frac{1}{a}-k} 2^{\frac{k-1}{2}} \Gamma(\frac{k+1}{2}, \frac{\left(\left(\frac{q}{\sigma}\right)^a - \lambda\right)^2}{2}) \right].$$

These curves serve as graphic methods for analysis and comparison, e.g., the inequality of non-negative distributions. See, for example, for a more detailed discussion [15].

Figure 4 shows the Bonferroni curve for the GTPN(*σ* = 1, *λ*, *α*) model, considering different values for *λ* and *α*.

**Figure 4.** Bonferroni curve for the generalized truncation positive normal (GTPN)(*<sup>σ</sup>*, *λ*, *α*) model.
