**Appendix A**

The first derivatives of -(*θ*) are given by

$$\begin{array}{rcl}\frac{\partial\ell(\theta)}{\partial\mu} &=& \frac{1}{\sigma} \left[\frac{\mathbf{x}-\boldsymbol{\mu}}{\sigma} - (\boldsymbol{\mu}-1)\boldsymbol{\kappa}(\mathbf{x})\right] \\ \frac{\partial\ell(\theta)}{\partial\sigma} &=& -\frac{1}{\sigma} \left[1 - \left(\frac{\boldsymbol{x}-\boldsymbol{\mu}}{\sigma}\right)^2 + (\boldsymbol{\mu}-1)\left(\frac{\boldsymbol{x}-\boldsymbol{\mu}}{\sigma}\right)\boldsymbol{\kappa}(\mathbf{x})\right] \\ \frac{\partial\ell(\theta)}{\partial\boldsymbol{\alpha}} &=& \frac{1}{\boldsymbol{a}} - \frac{2^{\boldsymbol{a}}\log(2)}{2^{\boldsymbol{a}}-1} + \log\left[1 + \Phi\left(\frac{\boldsymbol{x}-\boldsymbol{\mu}}{\sigma}\right)\right] \end{array}$$

The second derivatives of *l*(*θ*) are

$$\begin{array}{ll} \frac{\partial^2 \ell(\theta)}{\partial \theta^2} &=& -\frac{1}{\sigma^2} - \left(\frac{\mathfrak{a} - 1}{\sigma^2}\right) \left[ \left(\frac{\mathfrak{x} - \mu}{\sigma}\right) \kappa(\mathfrak{x}) + \kappa^2(\mathfrak{x}) \right] \\\\ \frac{\partial^2 \ell(\theta)}{\partial \mu \partial \sigma} &=& -\frac{2}{\sigma^2} \left(\frac{\mathfrak{x} - \mu}{\sigma}\right) + \left(\frac{\mathfrak{a} - 1}{\sigma^2}\right) \kappa(\mathfrak{x}) \left[1 - \left(\frac{\mathfrak{x} - \mu}{\sigma}\right)^2 - \left(\frac{\mathfrak{x} - \mu}{\sigma}\right) \kappa(\mathfrak{x})\right] \\\\ \frac{\partial^2 \ell(\theta)}{\partial \mu \partial \alpha} &=& -\frac{k(\mathfrak{x})}{\sigma} \\\\ \frac{\partial^2 \ell(\theta)}{\partial \sigma^2} &=& \frac{1}{\sigma^2} - \frac{3}{\sigma^2} \left(\frac{\mathfrak{x} - \mu}{\sigma}\right)^2 + \left(\frac{\mathfrak{x} - 1}{\sigma^2}\right) \left(\frac{\mathfrak{x} - \mu}{\sigma}\right) \kappa(\mathfrak{x}) \left[2 - \left(\frac{\mathfrak{x} - \mu}{\sigma}\right)^2 - \left(\frac{\mathfrak{x} - \mu}{\sigma}\right) \kappa(\mathfrak{x})\right] \\\\ \frac{\partial^2 \ell(\theta)}{\partial \sigma \partial \alpha} &=& -\left(\frac{\mathfrak{x} - \mu}{\sigma^2}\right) k(\mathfrak{x}) \\\\ \frac{\partial^2 \ell(\theta)}{\partial \mathbf{a}^2} &=& -\frac{1}{\mathfrak{a}^2} + \frac{2^\mu \log^2(2)}{(2^\mu - 1)^2} \end{array}$$
