*3.1. ML Estimation*

Consider *y*1, *y*2, ... , *yn* as a size *n* random sample from the pdf *GSC*(*φ*, *λ*, *μ*, *<sup>σ</sup>*). Hence, the log-likelihood function is given by

$$\begin{split} l(\boldsymbol{\phi}, \boldsymbol{\lambda}, \boldsymbol{\mu}, \sigma) &= n \log(\boldsymbol{\lambda}) + \sum\_{i=1}^{n} \log \{ \cosh(z\_i) \} - \sum\_{i=1}^{n} \log \left\{ 1 + \boldsymbol{\lambda}^2 \sinh^2(z\_i) \right\} \\ &+ (\boldsymbol{\phi} - 1) \sum\_{i=1}^{n} \log \left\{ -\log \left[ \frac{1}{2} - \frac{1}{\pi} \arctan \{ \boldsymbol{\lambda} \sinh(z\_i) \} \right] \right\} \\ &- n \log(\boldsymbol{\sigma}) - n \log(\boldsymbol{\pi}) - n \log(\Gamma(\boldsymbol{\phi})), \end{split} \tag{8}$$

where *zi* = *yi*−*μ σ* . To compute the ML estimation for Θ = (*φ*, *λ*, *μ*, *<sup>σ</sup>*), (8) must be maximized. That is, we have to solve the following system of equations: *δl δφ* , *δl δλ* , *δl δμ* , and *δl δσ* . More precisely, we have to solve

$$\begin{aligned} \sum\_{i=1}^{n} \log(-\log(t\_i)) - n\Psi(\phi) &= 0, \\ \sum\_{i=1}^{n} \frac{2\lambda \sinh^2(z\_i)}{1 + \lambda^2 \sinh^2(z\_i)} + (\phi - 1) \sum\_{i=1}^{n} \frac{\sinh(z\_i)}{\pi t\_i (1 + \lambda^2 \sinh^2(z\_i)) \log(t\_i)} &= \frac{n}{\lambda'}, \\ \sum\_{i=1}^{n} \frac{\lambda^2 \sinh(2z\_i)}{1 + \lambda^2 \sinh^2(z\_i)} + \frac{\lambda(\phi - 1)}{\pi} \sum\_{i=1}^{n} \frac{\cosh(z\_i)}{t\_i (1 + \lambda^2 \sinh^2(z\_i)) \log(t\_i)} &= \sum\_{i=1}^{n} \tanh(z\_i), \\ \sum\_{i=1}^{n} \frac{\lambda^2 z\_i \sinh(2z\_i)}{1 + \lambda^2 \sinh^2(z\_i)} + \frac{\lambda(\phi - 1)}{\pi} \sum\_{i=1}^{n} \frac{z\_i \cosh(z\_i)}{t\_i (1 + \lambda^2 \sinh^2(z\_i)) \log(t\_i)} - n &= \sum\_{i=1}^{n} z\_i \tanh(z\_i), \end{aligned}$$

where *ti* = 12 − 1*π* arctan{*<sup>λ</sup>* sinh(*zi*)}, and <sup>Ψ</sup>(·) is the digamma function. The system of equations given above can be solved using numerical procedures such as the Newton–Raphson procedure. An alternative is to use the NumDeriv routine with the R software (R Core Team [22]).
