*3.4. Fisher Information Matrix*

The Fisher information (FI) matrix for the GTPN distribution is given by *IF*(*θ*) = *Iab<sup>a</sup>*,*<sup>b</sup>* ∈ {*<sup>σ</sup>*,*λ*,*<sup>α</sup>*}. Consider the notation

$$T\_{jk} = T\_{jk}(\lambda) = \int\_0^\infty [\log(w)]^k w^j \phi(w - \lambda) dw, \quad j \in \{1, 2\}, \quad k \in \{0, 1, 2\}.$$

Therefore,

$$\begin{aligned} I\_{\sigma\nu} &= -\frac{\alpha}{\sigma^2} + \frac{\alpha(2\alpha+1)}{\sigma^2 \Phi(\lambda)} T\_{20} - \frac{\lambda\alpha(\alpha+1)}{\sigma^2 \Phi(\lambda)} T\_{10\prime} \\ I\_{\sigma\lambda} &= \frac{\alpha}{\sigma\Phi(\lambda)} T\_{10\prime} \\ I\_{\sigma a} &= \frac{1}{\sigma} + \frac{1}{\sigma\Phi(\lambda)} [\lambda(T\_{10} + T\_{11}) - T\_{20} - 2T\_{21}], \\ I\_{\lambda\lambda} &= 1 - \lambda\_s^2(\lambda) - \xi^2(\lambda), \\ I\_{\lambda a} &= -\frac{1}{\alpha\Phi(\lambda)} T\_{11\prime} \\ I\_{\mu a} &= \frac{1}{\alpha^2} + \frac{2}{\alpha^2\Phi(\lambda)} T\_{22} - \frac{\lambda}{\alpha^2\Phi(\lambda)} T\_{12} \end{aligned}$$

We observe that *Tjk*(*λ*) is a continuous function and lim|*λ*|→<sup>+</sup>∞ *Tjk*(*λ*) = 0, then *Tjk*(*λ*) < +<sup>∞</sup>. Note that for *λ* = 0 and *α* = 1 this matrix is reduced to

$$FI(\sigma,0,1) = \begin{pmatrix} \frac{2}{\sigma^2} & \frac{1}{\sigma} \sqrt{\frac{2}{\pi}} & -\frac{4}{\sigma} \hat{T}\_{21} \\ \cdot & 1 - \frac{2}{\pi} & -2\hat{T}\_{11} \\ \cdot & \cdot & 1 + 4\hat{T}\_{22} \end{pmatrix} / \sigma$$

where *T jk* = *Tjk*(0). Additionally, we note that

$$\det(FI(\sigma,0,1)) = \frac{2}{\sigma^2} \left[ \frac{(\pi - 3)}{\pi} (1 + 4\hat{\Gamma}\_{22} + 4\hat{\Gamma}\_{21}) - 4(\hat{\Gamma}\_{11} - \sqrt{\frac{2}{\pi}}\hat{\Gamma}\_{21})^2 \right] \approx \frac{0.1021506}{\sigma^2} > 0, \quad \forall \sigma > 0, \mu > 0$$

where det(·) denotes the determinant operator. Therefore, the FI matrix for the reduced model (HN) is invertible.
