**3. Inference**

In this section we discuss the ML method for parameter estimation in the GTPN model.

### *3.1. Maximum Likelihood Estimators*

For a random sample *z*1, ... , *zn* from the GTPN(*<sup>σ</sup>*, *λ*, *α*) model, the log-likelihood function for *θ* = (*<sup>σ</sup>*, *λ*, *α*) is given by

$$\ell(\boldsymbol{\theta}) = n \log(a) + (n - 1) \sum\_{i=1}^{n} \log(z\_i) - na \log(\sigma) - n \log(\Phi(\lambda)) - \frac{n}{2} \log(2\pi) - \frac{1}{2} \sum\_{i=1}^{n} \left( \left(\frac{z\_i}{\sigma}\right)^a - \lambda \right)^2.$$

Therefore, the score assumes the form *<sup>S</sup>*(*θ*)=(*<sup>S</sup>σ*(*θ*), *<sup>S</sup>λ*(*θ*), *<sup>S</sup>α*(*θ*)), where

$$\begin{aligned} S\_{\sigma}(\theta) &= -\frac{n\alpha}{\sigma} + \sum\_{i=1}^{n} \left( \left(\frac{z\_{i}}{\sigma}\right)^{\alpha} - \lambda \right) \alpha \left(\frac{z\_{i}}{\sigma}\right)^{\alpha - 1} \frac{z\_{i}}{\sigma^{2}}, \\ S\_{\lambda}(\theta) &= -n\xi(\lambda) + \sum\_{i=1}^{n} \left( \left(\frac{z\_{i}}{\sigma}\right)^{\alpha} - \lambda \right) \qquad \text{and} \\ S\_{a}(\theta) &= \frac{n}{\alpha} + \sum\_{i=1}^{n} \log z\_{i} - n\log(\sigma) - \sum\_{i=1}^{n} \left( \left(\frac{z\_{i}}{\sigma}\right)^{\alpha} - \lambda \right) \left(\frac{z\_{i}}{\sigma}\right)^{\alpha} \log\left(\frac{z\_{i}}{\sigma}\right) \end{aligned}$$

where *ξ*(*λ*) = *φ*(*λ*) Φ(*λ*) is the negative of the inverse Mills ratio. The ML estimators are obtained by solving the equation *S*(*θ*) = **0**3, where **0***p* denotes a vector of zeros with dimension *p*. This equation has the following solution for *λ*

$$\widehat{\lambda}(\widehat{\boldsymbol{\sigma}},\widehat{\boldsymbol{\kappa}}) = \frac{\sum\_{i=1}^{n} \left(\frac{\mathbb{Z}\_{i}}{\widehat{\boldsymbol{\sigma}}}\right)^{2\widehat{\boldsymbol{\kappa}}} - n}{\sum\_{i=1}^{n} \left(\frac{\mathbb{Z}\_{i}}{\widehat{\boldsymbol{\sigma}}}\right)^{\widehat{\boldsymbol{\kappa}}}}.\tag{4}$$

,

Replacing Equation (4) in *<sup>S</sup>λ*(*θ*) = 0 and *<sup>S</sup>α*(*θ*) = 0, the problem is reduced to two equations. The solution of this problem needs to be solved by numerical methods such as Newton-Raphson. Below we discuss initial values for the vector *θ* to initialize the algorithm.

### *3.2. Initial Point to Obtain the Maximum Likelihood Estimators*

In this subsection, we discuss the initial points for the iterative methods to find the ML estimators in the GTPN distribution.

### 3.2.1. A Naive Point Based on the HN Model

In Section 2 we discuss that GTPN(*<sup>σ</sup>*, *λ* = 0, *α* = 1) <sup>≡</sup>HN(*σ*). Based on this fact, and considering that the ML estimator for *σ* in the HN distribution has a closed-form, we can consider as an initial point *θnaive* = 1*n n*∑*i*=1*z*2*i* , 0, 1.

### 3.2.2. An Initial Point Based on Centiles

Let *qt*, *t* = 1, ... , 99, the *t*-th sample centile based on *z*1, ... , *zn*. An initial point to *θ* can be obtained by matching *qu*, *q*50 and *q*100−*u*, with *u* ∈ {1, 2, ... , 48, <sup>49</sup>}, with their respective theoretical counterparts. Defining *p* = *u*/100, the equations obtained are

$$\begin{aligned} q\_{\mu} &= \sigma \left[ \Phi^{-1} (1 - (1 - p)\Phi(\lambda)) + \lambda \right]^{\frac{1}{\pi}}, \qquad q\_{50} = \sigma \left[ \Phi^{-1} (1 - 0.5\Phi(\lambda)) + \lambda \right]^{\frac{1}{\pi}} \qquad \text{and} \\\ q\_{100-\mu} &= \sigma \left[ \Phi^{-1} (1 - p\Phi(\lambda)) + \lambda \right]^{\frac{1}{\pi}}. \end{aligned}$$

The solutions for *σ* and *α* are

'

$$
\widetilde{\vartheta} = \widetilde{\vartheta}(\widetilde{\lambda}) = \frac{q\_{100-u}}{\left[\Phi^{-1}\left(1 - p\Phi(\widetilde{\lambda})\right) + \widetilde{\lambda}\right]^{1/\widetilde{\widetilde{\alpha}}(\widetilde{\lambda})}} \qquad \text{and} \qquad \widetilde{u} = \widetilde{u}(\widetilde{\lambda}) = \frac{\log\left(\frac{\Phi^{-1}\left(1 - (1-p)\Phi(\widetilde{\lambda})\right) + \widetilde{\lambda}}{\Phi^{-1}\left(1 - p\Phi(\widetilde{\lambda})\right) + \widetilde{\lambda}}\right)}{\log\left(\frac{q\_{u}}{q\_{100-u}}\right)}.
$$

where *λ* is obtained from the non-linear equation

$$
\widetilde{\sigma}(\widetilde{\lambda}) \left[ \Phi^{-1} \left( 1 - 0.5 \Phi(\widetilde{\lambda}) \right) + \widetilde{\lambda} \right]^{1/\overline{\alpha}(\overline{\lambda})} = q\_{50}.
$$

Therefore, the initial point based on this method is given by *θcent* = '*σ*, '*λ*, '*α*.

### *3.3. An Initial Point Based on the Method of Moments*

A more robust initial point can be obtained using the method of moments. The equations to solve are *μr* = *<sup>z</sup>r*, *r* = 1, 2, 3. The solution for *σ* is

$$
\widetilde{\sigma}^{\star} = \widetilde{\sigma}^{\star}(\widetilde{\lambda}^{\star}, \widetilde{\kappa}^{\star}) = \frac{2\sqrt{\pi \widetilde{\pi}} \Phi(\lambda^{\star})}{\left(\widetilde{\lambda}^{\star}\right)^{1/\widetilde{\alpha}^{\star}} b\_{1}(\widetilde{\lambda}^{\star}, \widetilde{\kappa}^{\star})}.
$$

The solution for *λ* and *α* (say ' *λ* and '*<sup>α</sup>*, respectively) are obtained from the non-linear equations

'

$$\frac{(\widetilde{\sigma}^\*(\lambda, a)\lambda^{1/a})^2 b\_2(a, \lambda)}{2\Phi(\lambda)\sqrt{\pi}} = \overline{z^2} \qquad \text{and} \qquad \frac{(\widetilde{\sigma}^\*(\lambda, a)\lambda^{1/a})^3 b\_3(a, \lambda)}{2\Phi(\lambda)\sqrt{\pi}} = \overline{z^3}.$$

Therefore, the initial point based on this method is given by *θmom* = '*σ*, '*λ*, '*α*.
