**2. Model Properties**

In this section we introduce the main properties of the GTPN model such as density, quantile and hazard functions, moments, among others.

### *2.1. Stochastic Representation and Particular Cases*

As mentioned previously, we say that a r.v. *Z* has GTPN(*<sup>σ</sup>*, *λ*, *α*) distribution if *Z* = *σ <sup>Y</sup>*1/*<sup>α</sup>*, where *Y* ∼ *TPN*(1, *<sup>λ</sup>*). By construction, the following models are particular cases for the GTPN distribution:


Figure 1 summarizes the relationships among the GTPN and its particular cases. We highlight that *λ* = 0 and *α* = 1 are within the parametric space (not on the boundary). Therefore, to decide between the GTPN versus the TPN, GHN, or HN distributions we can use classical hypothesis tests such as the likelihood ratio test (LRT), score test (ST), or gradient test (GT).

**Figure 1.** Particular cases for the GTPN distribution.

### *2.2. Density, Cdf and Hazard Functions*

**Proposition 1.** *For Z* ∼ *GTPN*(*<sup>σ</sup>*, *λ*, *<sup>α</sup>*)*, the density function is given by*

$$f(z; \sigma, \lambda, a) = \frac{a}{\sigma^a \Phi(\lambda)} z^{a-1} \phi \left( \left( \frac{z}{\sigma} \right)^a - \lambda \right), \quad z \ge 0. \tag{2}$$

*where σ*, *α* > 0 *and λ* ∈ R

**Proof.** Considering the stochastic representation discussed in Section 2.1, we have that *z* = *g*(*y*) = *σy*1/*<sup>α</sup>* and

$$f\_Z(z) = f\_Y(\mathcal{g}^{-1}(z)) \left| \frac{d\mathcal{g}^{-1}(z)}{dz} \right| = \frac{1}{\Phi(\lambda)} \phi(\mathcal{g}^{-1}(z) - \lambda) \frac{az^{a-1}}{\mathcal{o}^a}.$$

Replacing *g*<sup>−</sup><sup>1</sup>(*z*)=(*z*/*σ*)*<sup>α</sup>* the result is obtained.

**Proposition 2.** *For Z* ∼ *GTPN*(*<sup>σ</sup>*, *λ*, *<sup>α</sup>*)*, the cdf and hazard function are given by*

$$F(z; \sigma, \lambda, a) = \frac{\Phi(\left(\frac{z}{\sigma}\right)^a - \lambda) + \Phi(\lambda) - 1}{\Phi(\lambda)} \qquad \text{and} \qquad h(z) = \frac{f(z)}{1 - F(z)} = \frac{az^{a-1}\phi\left(\left(\frac{z}{\sigma}\right)^a - \lambda\right)}{\sigma^a \left[1 - \Phi\left(\left(\frac{z}{\sigma}\right)^a - \lambda\right)\right]},$$

*respectively, for all z* ≥ 0*.*

Figure 2 shows the density and hazard functions for the GTPN(*σ* = 1, *λ*, *α*) model, considering some combinations for (*<sup>λ</sup>*, *<sup>α</sup>*). Note that the GTPN model can assume decreasing and unimodal shapes for the density function and decreasing or increasing shapes for the hazard function.

**Figure 2.** Density and hazard functions for the GTPN(*σ* = 1, *λ*, *α*) model with different combinations of *λ* and *α*.
