This rather simple Julia package of code provides an implementation of the Williamson d-transform and of its inverse.
Do not hesitate to open an issue and talk if you are interested.
See the following references and the documentation of the package for the definition and motivations of this (rather unknown) transformation:
@book{williamson1955multiply,
title={On multiply monotone functions and their laplace transforms},
author={Williamson, Richard Edmund},
year={1955},
publisher={Mathematics Division, Office of Scientific Research, US Air Force}
}
@article{mcneil2009multivariate,
title={Multivariate Archimedean copulas, d-monotone functions and ℓ 1-norm symmetric distributions},
author={McNeil, Alexander J and Ne{\v{s}}lehov{\'a}, Johanna},
year={2009}
}
@article{mcneil2010archimedean,
title={From archimedean to liouville copulas},
author={McNeil, Alexander J and Ne{\v{s}}lehov{\'a}, Johanna},
journal={Journal of Multivariate Analysis},
volume={101},
number={8},
pages={1772--1790},
year={2010},
publisher={Elsevier}
}
@book{durante2015principles,
title={Principles of copula theory},
author={Durante, Fabrizio and Sempi, Carlo},
year={2015},
publisher={CRC press}
}