@article{oai:nagoya.repo.nii.ac.jp:02001360,
 author = {Takeuchi, Tsutomu T and Kono, Kai T},
 issue = {3},
 journal = {Monthly Notices of the Royal Astronomical Society},
 month = {Nov},
 note = {The need for a method to construct multidimensional distribution function is increasing recently, in the era of huge multiwavelength surveys. We have proposed a systematic method to build a bivariate luminosity or mass function of galaxies by using a copula. It allows us to construct a distribution function when only its marginal distributions are known, and we have to estimate the dependence structure from data. A typical example is the situation that we have univariate luminosity functions at some wavelengths for a survey, but the joint distribution is unknown. Main limitation of the copula method is that it is not easy to extend a joint function to higher dimensions (d > 2), except some special cases like multidimensional Gaussian. Even if we find such a multivariate analytic function in some fortunate case, it would often be inflexible and impractical. In this work, we show a systematic method to extend the copula method to unlimitedly higher dimensions by a vine copula. This is based on the pair-copula decomposition of a general multivariate distribution. We show how the vine copula construction is flexible and extendable. We also present an example of the construction of a stellar mass–atomic gas–molecular gas three-dimensional mass function. We demonstrate the maximum likelihood estimation of the best functional form for this function, as well as a proper model selection via vine copula.},
 pages = {4365--4378},
 title = {Constructing a multivariate distribution function with a vine copula: towards multivariate luminosity and mass functions},
 volume = {498},
 year = {2020}
}