A Graphical Goodness-of-Fit Test for Dependence Models in Higher Dimensions

Abstract

This article introduces a graphical goodness-of-fit test for copulas in more than two dimensions. The test is based on pairs of variables and can thus be interpreted as a first-order approximation of the underlying dependence structure. The idea is to first transform pairs of data columns with the Rosenblatt transform to bivariate standard uniform distributions under the null hypothesis. This hypothesis can be graphically tested with a matrix of bivariate scatterplots, Q-Q plots, or other transformations. Furthermore, additional information can be encoded as background color, such as measures of association or (approximate) p-values of tests of independence. The proposed goodness-of-fit test is designed as a basic graphical tool for detecting deviations from a postulated, possibly high-dimensional, dependence model. Various examples are given and the methodology is applied to a financial dataset. An implementation is provided by the R package copula. Supplementary material for this article is available online, which provides the R package copula and reproduces all the graphical results of this article.