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The formation history of binary black hole systems is imprinted on the distribution of their masses, spins, and eccentricity. While much has been learned studying these parameters in turn, recent studies have explored the joint distribution of binary black hole parameters in two or more dimensions. Most notably, it has recently been argued that binary black hole mass ratio and effective inspiral spin χeff are anticorrelated. We point out a previously overlooked subtlety in such 2D population studies: in order to conduct a controlled test for correlation, one ought to fix the two marginal distributions – lest the purported correlation be driven by improved fit in just one dimension. We address this subtlety using a tool from applied statistics: the copula density function. We use the previous work correlating mass ratio and χeff as a case study to demonstrate the power of copulas in gravitational-wave astronomy while scrutinizing their astrophysical inferences. Our findings, however, affirm their conclusions that binary black holes with unequal component masses exhibit larger χeff (98.7 per cent credibility). We conclude by discussing potential astrophysical implications of these findings as well as prospects for future studies using copulas. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A common approach towards modelling bivariate scatterplots is decomposition into Gaussian components, i.e. Gaussian mixture modelling. This implicitly assumes linear relationships between the variables within each of the components in the mixture. An alternative, namely dependence modelling by mixtures of copulas, is advocated in this paper. This approach allows separate modelling of the univariate marginal distributions and the dependence which can possibly be non-linear and/or asymmetric. It also accommodates the use of a variety of parametric families for modelling each component and for each variable. The variety of dependence structures can be extended by introducing rotated versions of the copulas. Gaussian mixture modelling on the one hand, and separate modelling of univariate marginal distributions and dependence on the other hand, are illustrated by application to pulsar period – period-derivative observations. Parameter estimation for mixtures of copulas is performed using the method of maximum likelihood and selected copula models are subjected to non-parametric goodness-of-fit testing. [ABSTRACT FROM AUTHOR]