Volume 12, Issue 1

We derive lower and upper bounds for the survival function of an exchangeable sequence of random variables, for which the scaled minimum of each finite subgroup has a univariate exponential distribution. These bounds are sharp in the sense that both bounds themselves are attained by exchangeable sequences of the same kind, for which the (non-scaled) minimum of each subgroup has the same univariate exponential distribution as the original sequence. This result is equivalent to inequalities between infinite-dimensional stable tail dependence functions, which leads to inequalities between multivariate extreme-value copulas. In addition, it is explained how an infinite-dimensional symmetric stable tail dependence function can be obtained from its upper bound by censoring certain distributional information. This technique is applied to derive new parametric families.

Motivated by examples from extreme value theory, but without using the theory of regularly varying time series or any assumptions about the marginal distribution, we introduce the general notion of a cluster process as a limiting point process of returns of a certain event in a time series. We explore general invariance properties of cluster processes that are implied by stationarity of the underlying time series. Of particular interest in applications are the cluster size distributions, and we derive general properties and interconnections between the size of an inspected and a typical cluster. While the extremal index commonly used in extreme value theory is often interpreted as the inverse of a “mean cluster size”, we point out that this only holds true for the expected value of the typical cluster size, caused by an effect very similar to the inspection paradox in renewal theory.

Vine pair-copula constructions exist for a mix of continuous and ordinal variables. In some steps, this can involve estimating a bivariate copula for a pair of mixed continuous-ordinal variables. To assess the adequacy of copula fits for such a pair, diagnostic and visualization methods based on normal score plots and conditional Q–Q plots are proposed. The former uses a latent continuous variable for the ordinal variable. The methods are applied to data generated from some existing probability models for a mixed continuous-ordinal variable pair, and for such models, Kullback-Leibler divergence is used to assess whether simple parametric copula families can provide adequate fits. The effectiveness of the proposed visualization and diagnostic methods is illustrated on a dataset.

The long-standing Gaussian product inequality (GPI) conjecture states that E [ ∏ j = 1 n ∣ X j ∣ y j ] ≥ ∏ j = 1 n E [ ∣ X j ∣ y j ] E\left[{\prod }_{j=1}^{n}{| {X}_{j}| }^{{y}_{j}}]\ge {\prod }_{j=1}^{n}E\left[{| {X}_{j}| }^{{y}_{j}}] for any centered Gaussian random vector ( X 1 , … , X n ) \left({X}_{1},\ldots ,{X}_{n}) and any non-negative real numbers y j {y}_{j} , j = 1 , … , n j=1,\ldots ,n . In this study, we describe a computational algorithm involving sums-of-squares representations of multivariate polynomials that can be used to resolve the GPI conjecture. To exhibit the power of the novel method, we apply it to prove new four- and five-dimensional GPIs: E [ X 1 2 m X 2 2 X 3 2 X 4 2 ] ≥ E [ X 1 2 m ] E [ X 2 2 ] E [ X 3 2 ] E [ X 4 2 ] E\left[{X}_{1}^{2m}{X}_{2}^{2}{X}_{3}^{2}{X}_{4}^{2}]\ge E\left[{X}_{1}^{2m}]E\left[{X}_{2}^{2}]E\left[{X}_{3}^{2}]E\left[{X}_{4}^{2}] for any m ∈ N m\in {\mathbb{N}} , and E [ ∣ X 1 ∣ y X 2 2 X 3 2 X 4 2 X 5 2 ] ≥ E [ ∣ X 1 ∣ y ] E [ X 2 2 ] E [ X 3 2 ] E [ X 4 2 ] E [ X 5 2 ] E\left[{| {X}_{1}| }^{y}{X}_{2}^{2}{X}_{3}^{2}{X}_{4}^{2}{X}_{5}^{2}]\ge E\left[{| {X}_{1}| }^{y}]E\left[{X}_{2}^{2}]E\left[{X}_{3}^{2}]E\left[{X}_{4}^{2}]E\left[{X}_{5}^{2}] for any y ≥ 1 10 y\ge \frac{1}{10} .

We propose a new method to construct a stationary process and random field with a given decreasing covariance function and any one-dimensional marginal distribution. The result is a new class of stationary processes and random fields. The construction method utilizes a correlated binary sequence, and it allows a simple and practical way to model dependence structures in a stationary process and random field as its dependence structure is induced by the correlation structure of a few disjoint sets in the support set of the marginal distribution. Simulation results of the proposed models are provided, which show the empirical behavior of a sample path.

We analyze optimal low-rank approximations and correspondence analysis of the dependence structure given by arbitrary bivariate checkerboard copulas. Methodologically, we make use of the truncation of singular value decompositions of doubly stochastic matrices representing the copulas. The resulting (truncated) representations of the dependence structures are sparse, in particular, compared to the number of squares on the checkerboard. The additive structure of the decomposition carries through to statistical functionals of the copula, such as Kendall’s τ \tau or Spearman’s ρ \rho , and also motivates similarity measures for checkerboard copulas. We link our analysis to continuous decompositions of copula densities and copula-generating algorithms and discuss further general properties of the decomposition and its truncation. For example, truncated series might lack nonnegativity, and approximation errors increase for monotonicity-like copulas. We provide algorithms and extensions that account for and counteract these properties. The low-rank representation is illustrated for various copula examples, and some analytical results are derived. The resulting correspondence analysis profile plots are analyzed, providing graphical insights into the dependence structure implied by the copula. An illustration is provided with an empirical data set on fuel injector spray characteristics in jet engines.

We study the relations between the shapes of strict triangular norms and their generators. The generators are not unique; to avoid this ambiguity, we introduce the class of balanced generators. We find conditions for their existence and a procedure of their computation. We formulate their influence on local properties of the corresponding triangular norms, in particular at the extremes of their domain. Many strict triangular norms are also copulas; thus, our results can also be applied to bivariate extreme value distributions described by some families of copulas.

Motivated by recently investigated results on dependence measures and robust risk models, this article provides an overview of dependence properties of many well known bivariate copula families, where the focus is on the Schur order for conditional distributions, which has the fundamental property that minimal elements characterize independence and maximal elements characterize perfect directed dependence. We give conditions on copulas that imply the Schur ordering of the associated conditional distribution functions. For extreme-value copulas, we prove the equivalence of the lower orthant order, the Schur order for conditional distributions, and the pointwise order of the associated Pickands dependence functions. Furthermore, we provide several tables and figures that list and illustrate various positive dependence and monotonicity properties of copula families, in particular, from classes of Archimedean, extreme-value, and elliptical copulas. Finally, for Chatterjee’s rank correlation, which is consistent with the Schur order for conditional distributions, we give some new closed-form formulas in terms of the parameter of the underlying copula family.

This article studies the conditional dependency between random variables, conditionally upon a covariate (vector). The conditional copula fully characterizes this conditional dependency. A way to summarize this dependence structure taking into account the impact of the covariate is via the average conditional copula, which under fairly general conditions coincides with the partial copula. A mean is just one way to summarize this conditional dependence behaviour. In this article, we introduce the notions of median conditional copula and more generally quantile conditional copula. We investigate the existence of these concepts and establish explicit expressions for calculating them. Examples are given to illustrate the concepts, and the practical use of them is demonstrated in real data examples.

Comprehensive families of copulas including the three basic copulas (at least as limit cases) are useful tools to model countermonotonicity, independence, and comonotonicity of pairs of random variables on the same probability space. In this contribution, we study how the transition from a (basic) copula to a copula modeling a different dependence behavior can be realized by means of ordinal sums based on one of the three basic copulas, perturbing one of the three basic copulas (considering some appropriate parameterized transformations) and truncating the results using the Fréchet-Hoeffding bounds. We provide results and examples showing the flexibility and the restrictions for obtaining new copulas or comprehensive families and illustrate the development of their dependence parameters.