**A.2 The Burr distribution**

The three parameter Burr type XII distribution function

$$B(\mathbf{x}; \boldsymbol{\eta}, \boldsymbol{\tau}, a) = \mathbf{1} - \left(\mathbf{1} + (\mathbf{x}/\boldsymbol{\eta})^{\mathsf{r}}\right)^{-a}, \text{for } \boldsymbol{\kappa} > \mathbf{0} \tag{11}$$

with parameters *η*, *τ*, *α* >0 (see e.g. [10]). Here *η* is a scale parameter and *τ* and *α* shape parameters. Note the EVI of the Burr distribution is given by *EVI* ¼ *ζ* ¼ 1*=τα* and that heavy-tailed distributions have a positive EVI and larger EVI implies heavier tails. This follows (also) from the fact that for positive EVI the Burr distribution belongs to the Pareto-type class of distributions, having a distribution function of the form 1 � *F x*ð Þ¼ *<sup>x</sup>*�1*=<sup>ζ</sup>ℓF*ð Þ *<sup>x</sup>* , with *<sup>ℓ</sup>F*ð Þ *<sup>x</sup>* a slowly varying function at infinity (see e.g. [9]). For Pareto-type, when the EVI > 1, the expected value does

not exist, and when EVI > 0.5, the variance is infinite. Note also that the Burr distribution is regularly varying with index �*τα*and therefore belongs to the class of sub-exponential distributions. Note that the *γ*-th quantile of the Burr distribution is

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$$q(\boldsymbol{\gamma}) = B^{-1}(\boldsymbol{\gamma}; \boldsymbol{\eta}, \boldsymbol{\tau}, \boldsymbol{a}) = \eta \left( (\mathbf{1} - \boldsymbol{\gamma})^{-1/a} - \mathbf{1} \right)^{1/\tau}.$$
