*2.3. Mode*

**Proposition 3.** *The mode of the GTPN*(*<sup>σ</sup>*, *λ*, *α*) *model is attained:*

$$\begin{aligned} 1. \quad \text{at } z &= \frac{\sigma}{2^{1/a}} \left( \lambda + \sqrt{\lambda^2 + 4 \left( 1 - \frac{1}{a} \right)} \right)^{1/a} \text{ whenever } a \ge 1 \text{ or } 0 < a < 1 \text{ and } \lambda > 0, \\ 2. \quad \text{at } z &= 0 \text{ in otherwise.} \end{aligned}$$

**Proof.** Let *l* = log(*f*), where *f* is the density function defined in (2), of a direct computation we have

$$\frac{\partial l}{\partial z} = l\_z = -\frac{1}{z} \left[ \alpha \left( \frac{z}{\sigma} \right)^{2\alpha} - \alpha \lambda \left( \frac{z}{\sigma} \right)^{\alpha} - (\alpha - 1) \right].$$

Note that *lz* vanishes when *α zσ* 2*α* − *αλ zσ α* − (*α* − 1) = 0. Using the auxiliary variable *w* = *zσ α* the last equation is rewritten as follows:

$$
\hbar w^2 - \hbar \lambda w - (\mathfrak{a} - 1) = 0.\tag{3}
$$

In the rest of the proof we use the discriminant of the quadratic equation in Equation (3), which is given by Δ = *α*2*λ*<sup>2</sup> + <sup>4</sup>*α*(*α* − 1) = *α*<sup>2</sup>[*λ*<sup>2</sup> + 4(1 − 1*α* )] and its zeros are given by: *w* = (*λ* ± &*λ*<sup>2</sup> + 4(1 − 1*α* ))/2.

If *α* ≥ 1 then Δ ≥ *λ*2. In consequence, the mode is attained at *z*1 = *σ* 21/*α λ* + *<sup>λ</sup>*<sup>2</sup> + 4 1 − 1*α*1/*<sup>α</sup>*. If 0 < *α* < 1 then Δ < *λ*2. Here, two cases may occur. The first when 0 < Δ < *λ*2, in which

case if *λ* > 0 its mode is attained at *z* = *σ*21/*α λ* + *<sup>λ</sup>*<sup>2</sup> + 4 1 − 1*α*1/*<sup>α</sup>*, since *lzz* < 0, and if *λ* < 0, then the zeros of Equation (3) are negative, implying that function *l* is strictly decreasing. Its mode is therefore attained at zero. The other case is when Δ < 0, then we have that *αw*<sup>2</sup> − *αλw* − (*α* − 1) > 0 for all *w* ≥ 0, implying that *lz* < 0 for all *z* ≥ 0. Therefore, *l* is strictly decreasing and thus its mode is zero.

**Remark 1.** *Note that α* ≥ 1 *or λ* > 0 *implies that the mode of the GTPN model is attached in a positive value.*
