*2.3. Asymptotic Standard Errors*

The standard errors of the ML estimators can be estimated using the expected information matrix. For a multivariate elliptically symmetric distribution, Lange et al. (1989) indicated how to compute the expected information matrix. See also Mitchell (1989). In our case, by using score function (8), the Fisher information matrix for *θ* in the log-likelihood function defined in (7) assumes the form

$$J = E\{\mathcal{U}(\boldsymbol{\theta})\mathcal{U}^T(\boldsymbol{\theta})\} = \begin{pmatrix} J\_{11} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & J\_{22} & J\_{23} \\ \mathbf{0} & J\_{23}^T & J\_{33} \end{pmatrix} \tag{10}$$

 and

where *J*11, *J*22, *J*23 and *J*33 denote information concerning (*<sup>α</sup>*, *β*), *σ*, (*<sup>σ</sup>*, *η*) and *η*, respectively, and are given by

$$\begin{aligned} J\_{11} &=& c\_{\mathfrak{a}}(\eta)(X^{T}X)\otimes \Sigma^{-1}, \\ J\_{22} &=& \frac{n}{4} \mathsf{D}\_{p}^{T} \{2c\_{\sigma}(\eta)(\Sigma^{-1}\otimes \Sigma^{-1})N\_{p} + (c\_{\sigma}(\eta) - 1)(\mathrm{vec}\,\Sigma^{-1})(\mathrm{vec}\,\Sigma^{-1})^{T}\} D\_{p}, \\ J\_{23} &=& -\frac{nc(\eta)c\_{\sigma}(\eta)(p+2)}{(1+p\eta)^{2}} \mathsf{D}\_{p}^{T} \mathrm{vec}\,\Sigma^{-1}, \\ J\_{33} &=& -\frac{n}{2\eta^{2}} \left\{ \left(\frac{p}{(1-2\eta)^{2}}\right) \left(\frac{1+\eta\,p(1-4\eta)-8\eta^{2}}{(1+\eta\,p)(1+\eta(p+2))}\right) - \beta'(\eta) \right\}. \end{aligned}$$

with *<sup>N</sup>p* = 12 (*I p*2 + *<sup>K</sup>p*) where *<sup>K</sup>p* is the commutation matrix of order *p*2 × *p*2 (Magnus and Neudecker 2007); *<sup>c</sup>α*(*η*) = *<sup>c</sup>σ*(*η*)/(<sup>1</sup> − <sup>2</sup>*η*), *<sup>c</sup>σ*(*η*)=(<sup>1</sup> + *pη*)/(<sup>1</sup> + (*p* + <sup>2</sup>)*η*); and

$$\beta'(\eta) = -\frac{1}{2\eta^2} \left\{ \psi'\left(\frac{1+p\eta}{2\eta}\right) - \psi'\left(\frac{1}{2\eta}\right) \right\}.$$

where *ψ* (*z*) denotes the trigamma function. Note that *<sup>c</sup>α*(*η*) = *<sup>c</sup>σ*(*η*) = 1 when *η* = 0 and *<sup>N</sup>pDp* = *<sup>D</sup>p* (see for instance Magnus and Neudecker 2007), we have to recover the expressions corresponding to the normal case. Here, *X* is an *n* × 2 matrix such *X<sup>T</sup>* = 1 ··· 1 *x*1 ··· *xn* and ⊗ denotes the Kronecker product. The asymptotic sampling distribution of the ML estimator *θ* ˆ is given by

$$
\sqrt{n}(\hat{\boldsymbol{\theta}} - \boldsymbol{\theta}) \stackrel{\mathcal{D}}{\mapsto} \mathcal{N}\_r(0, \boldsymbol{V}^{-1})\_\* 
$$

where *V* = lim*n*→∞(1/*n*)*J* and *r* = {*p*(*p* + 5) + 2}/2 is the dimension of *θ*. To estimate *V*, we use *V* ˆ = *J*(*θ*<sup>ˆ</sup>)/*<sup>n</sup>*.
