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fix typo
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lrnv committed Nov 8, 2023
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2 changes: 1 addition & 1 deletion src/ArchimedeanCopulas/WilliamsonCopula.jl
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Expand Up @@ -11,7 +11,7 @@ Constructors
WilliamsonCopula(ϕ::Function, d)
WilliamsonCopula(ϕ::Function, X::Distributions.UnivariateDistribution, d)
The WilliamsonCopula is the barebone Archimedean Copula that directly leverages the Williamson transform and inverse transform (in their greatest generalities), that are implemented in [WilliamsonTransforms.jl](https://www.github.com/lrnv/WilliamsonTransforms.jl). You can construct it by providing the Williamson-d-tranform as a (non-negative) random variable `X::Distributions.UnivariateDistribution`, or by providing the $d$-monotone generator `ϕ::Function` as a function from ``\\mathbb R_+`` to ``[0,1]``, decreasing and d-monotone. The other component will be computed automatically. You also have the option to provide both, which will probably be faster if you have an analytical expression for both.
The WilliamsonCopula is the barebone Archimedean Copula that directly leverages the Williamson transform and inverse transform (in their greatest generalities), that are implemented in [WilliamsonTransforms.jl](https://www.github.com/lrnv/WilliamsonTransforms.jl). You can construct it by providing the Williamson-d-tranform as a (non-negative) random variable `X::Distributions.UnivariateDistribution`, or by providing the ``d``-monotone generator `ϕ::Function` as a function from ``\\mathbb R_+`` to ``[0,1]``, decreasing and d-monotone. The other component will be computed automatically. You also have the option to provide both, which will probably be faster if you have an analytical expression for both.
About `WilliamsonCopula(X::Distributions.UnivariateDistribution, d)`: For a univariate non-negative random variable ``X``, with cumulative distribution function ``F`` and an integer ``d\\ge 2``, the Williamson-d-transform of ``X`` is the real function supported on ``[0,\\infty[`` given by:
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