A New Graphical Tool for Copula Selection

Abstract

The selection of copulas is an important aspect of dependence modeling. In many practical applications, only a limited number of copulas is tested, and the modeling applications usually are restricted to the bivariate case. One explanation is the fact that no graphical copula tool exist which allows to assess the goodness-of-fit of a large set of (possible higher dimensional) copula functions at once. This paper pursues to overcome this problem by developing a new graphical tool for the copula selection, based on a statistical analysis technique called 'principal coordinate analysis'. The advantage is threefold. In the first place, when projecting the empirical copula of a modeling application on a two-dimensional copula space, it allows us to visualize the fit of a whole collection of multivariate copulas at once. Secondly, the visual tool allows to identify 'search' directions for potential fit improvements (e.g. through the use of copula transforms). Finally, in the bivariate case the tool makes it also possible to give a two-dimensional visual overview of a large number of known copula families, for a common concordance value, leading to a better understanding and a more efficient use of the different copula families. The practical use of the new graphical tool is illustrated for two two-dimensional and two three-dimensional fitting applications.

Copula model selection using image recognition

“…Furthermore, when sample size is small, it is natural to focus on the simplest models, given the likely lack of statistical power to detect more complex model features. We note that bivariate copulas can be used to hierarchically build complex multivariate dependence structures (Aas et al, 2009); for a selection tool projecting multivariate copulas to two dimensions see Michiels and De Schepper (2013). In the case where datasets are richer, the problem the modeller faces is not so much one of selecting between different models, but more one of designing a model that is flexible enough to reflect idiosyncratic features of the data, see e.g.…”

$$D_s$$ D s -optimality in copula models

“…( 2009 ), or Fermanian ( 2013 ). Other more specific avenues consist in applying graphical tools (Michiels and Schepper 2013 ) or information based criteria (Grønneberg and Hjort 2014 ). In fully parametric models, as considered in this paper, the latter can be formulated in terms of functions of the Fisher information matrix, which will allow us to generate optimal designs for copula model discrimination.…”

Optimal discrimination design for copula models

“…For that purpose a rather general approach is to use omnibus goodness-of-fit tests that require minimum assumptions, for recent reviews see, e.g., [2], [19], or [15]. Other more specific avenues consist in applying graphical tools ( [27]) or information based criteria ( [20]). In fully parametric models, as considered in this paper, the latter can be formulated in terms of functions of the Fisher information matrices, which will allow us to generate optimal designs for copula model discrimination.…”