A short note on multivariate dependence modeling (Q2871608)

Copulas are a tool proposed by Sklar to capture the structure of stochastic dependence of random vectors. Though there are several attempts to construct more-dimensional copulas when knowing some of marginal copulas (note that this is not always possible, see the problem of compatibility of marginal copulas), still this important area is not sufficiently covered. The reviewed paper brings one interesting attempt to fill this gap. As their main tool, the authors are applying the Iterative Proportional Fitting Procedure (IPFP, in short) proposed by \textit{W. E. Deming} and \textit{F. E. Stephan} [Ann. Math. Stat. 11, 427--444 (1940; Zbl 0024.05502, JFM 66.0652.02)]. They have focused on the particular case of constructing ternary copulas when the three binary marginal copulas are known. Similarly as not all triples of such copulas are copulas (i.e., the corresponding ternary copula does not exist), also their iterative procedure does not converges in some cases. In the later case, the IPFP tends to cycle, leading to three convergent subsequences, and based on maximal entropy , authors have proposed a method how to choose the best approximation of the desired ternary copula. Note that the presented approach is not an exact mathematical solution of the studied problem, but it should be seen as a proposal of a simple and lucid tool applicable in case that one has subjective rough estimates of the dependence relations of marginal copulas. More, it shows also a way how to employ such a partial knowledge in methods like Monte Carlo.