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lbelzile committed Jul 9, 2024
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Expand Up @@ -43,6 +43,7 @@ The different models implemented are described in @Dombry:2016, but some other m


## Logistic

The logistic model (`log`) of @Gumbel:1960 has distribution function
\begin{align*}
\Pr(\boldsymbol{X} \leq \boldsymbol{x})= \exp \left[ - \left(\sum_{i=1}^{n} {x_i}^{-\alpha}\right)^{\frac{1}{\alpha}}\right]
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@Joe:1990 mentions generalizations of the distribution as given above but the constraints were not enforced elsewhere in the literature. The proof that the distribution is valid follows from Theorem~1 of @Stephenson:2003 as it is a max-mixture. Note that the parametrization of the asymmetric negative logistic distribution does not match the bivariate implementation of `rbvevd`.


## Multilogistic distribution

This multivariate extension of the logistic, termed multilogistic (`bilog`) proposed by @Boldi:2009, places mass on the interior of the simplex. Let $\boldsymbol{W} \in \mathbb{S}_d$ be the solution of
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## Coles and Tawn Dirichlet distribution

The Dirichlet (`ct`) model of @Coles:1991

The Dirichlet (`ct`) model of @Coles:1991 has spectral density
\begin{align*}
h_{\boldsymbol{W}}(\boldsymbol{w}) = \frac{1}{d} \frac{\Gamma \left(1+\sum_{j=1}^d \alpha_j\right)}{\prod_{j=1}^d \alpha_jw_j}
\left(\sum_{j=1}^d \alpha_jw_j\right)^{-(d+1)}\prod_{j=1}^d \alpha_j \prod_{j=1}^d \left(\frac{\alpha_jw_j}{\sum_{k=1}^d
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