  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 an stellar mass--atomic
gas--molecular gas 3-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.
