Hierarchical Kendall copulas: Properties and inference

Abstract

While there is substantial need for dependence models in higher dimensions, most existing models quickly become rather restrictive and barely balance parsimony and flexibility. Hierarchical constructions may improve on that by grouping variables in different levels. In this paper, the new class of hierarchical Kendall copulas is proposed and discussed. Hierarchical Kendall copulas are built up by flexible copulas specified for groups of variables, where aggregation is facilitated by the Kendall distribution function, the multivariate analog to the probability integral transform for univariate random variables. After deriving properties of the general model formulation, particular focus is given to inference techniques of hierarchical Kendall copulas with Archimedean components, for which closed‐form analytical expressions can be derived. A substantive application to German stock returns finally shows that hierarchical Kendall copulas perform very well for real data, out‐of‐ as well as in‐sample. The Canadian Journal of Statistics 42: 78–108; 2014 © 2014 Statistical Society of Canada

A note on upper-patched generators for Archimedean copulas

“…In this sense some efforts to leave the Archimedean framework have be done in the recent literature, as for instance for the nested or hierarchical copulas (see e.g. [4,22,23,29]). Furthermore, in our strategy, the tail behaviour must be estimated, which in practice may be sensitive to the size of the considered data-set.…”

Detection of block-exchangeable structure in large-scale correlation matrices

“…Several commonly used models make an implicit or explicit use of this kind of complexity reduction; examples include latent variable models such as frailty or random effects models, Markov random fields or graphical models, hierarchical copula models [4,36], factor copulas [21,27], or nested (hierarchical) Archimedean copulas [25]. Definition 1.…”

“…, F d are the univariate marginal distributions of X and C is a copula, i.e., a joint distribution function with standard uniform marginals [48]. To achieve flexibility and parsimony when d is large, the complexity of the problem needs to be reduced through an ingenious construction of C; examples are vines [28], factor models [24,26], or hierarchical constructions [4,25,36]. The cluster algorithm proposed in this paper is particularly well suited for such approaches: Equicorrelated clusters can first be identified through it and modeled by exchangeable lower-dimensional copulas.…”

Standardized Drought Indices: A Novel Univariate and Multivariate Approach

“…For meaningful estimation of the empirical Kendall distribution in high dimensions (here multiples of 12) long data records are required. As in such high dimensions the Kendall distribution becomes almost degenerate at 0 (see for example Brechmann ()), the quantification of dry conditions may become problematic.…”

“…For meaningful estimation of the empirical Kendall distribution in high dimensions (here multiples of 12) long data records are required. As in such high dimensions the Kendall distribution becomes almost degenerate at 0 (see for example Brechmann (2014)), the quantification of dry conditions may become problematic. Hao andAghaKouchak (2013, 2014) introduced a bivariate parametric and non-parametric version of the multivariate standardized drought index (MSDI) respectively, by enhancing the SPI idea to bivariate data (in their example precipitation and soil moisture time series).…”