Abstract:
We propose notions of calibration for probabilistic forecasts of general multivariate quantities. Probabilistic copula calibration is a natural analogue of probabilistic calibration in the univariate setting. It can be assessed empirically by checking for the uniformity of the copula probability integral transform (CopPIT), which is invariant under coordinate permutations and coordinatewise strictly monotone transformations of the predictive distribution and the outcome. The CopPIT histogram can be interpreted as a generalization and variant of the multivariate rank histogram, which has been used to check the calibration of ensemble forecasts. Climatological copula calibration is an analogue of marginal calibration in the univariate setting. Methods and tools are illustrated in a simulation study and applied to compare raw numerical model and statistically postprocessed ensemble forecasts of bivariate wind vectors.

Abstract:
We define a copula process which describes the dependencies between arbitrarily many random variables independently of their marginal distributions. As an example, we develop a stochastic volatility model, Gaussian Copula Process Volatility (GCPV), to predict the latent standard deviations of a sequence of random variables. To make predictions we use Bayesian inference, with the Laplace approximation, and with Markov chain Monte Carlo as an alternative. We find both methods comparable. We also find our model can outperform GARCH on simulated and financial data. And unlike GARCH, GCPV can easily handle missing data, incorporate covariates other than time, and model a rich class of covariance structures.

Abstract:
The class of probability distributions possessing the almost-lack-of-memory property appeared about 20 years ago. It reasonably took place in research and modeling, due to its suitability to represent uncertainty in periodic random environment. Multivariate version of the almost-lack-of-memory property is less known, but it is not less interesting. In this paper we give the copula of the bivariate almost-lack-of-memory (BALM) distributions and discuss some of its properties and applications. An example shows how the Marshal-Olkin distribution can be turned into BALM and what is its copula. 1. Introduction The class of probability distributions called “almost-lack-of-memory (ALM) distributions” was introduced in Chukova and Dimitrov [1] as a counterexample of a characterization problem. Dimitrov and Khalil [2] found a constructive approach considering the waiting time up to the first success for extended in time Bernoulli trials. Similar approach was used in Dimitrov and Kolev [3] in sequences of extended in time and correlated Bernoulli trials. The fact that nonhomogeneous in time Poisson processes with periodic failure rates are uniquely related to the ALM distributions was established in Chukova et al. [4]. It gave impetus to several additional statistical studies on estimations of process parameters (see, e.g., [5, 6] to name a few) of these properties. Best collection of properties of the ALM distributions and related processes can be found in Dimitrov et al. [7]. Meanwhile, Dimitrov et al. [8] extended the ALM property to bivariate case and called the obtained class BALM distributions. For the BALM distributions, a characterization via a specific hyperbolic partial differential equation of order 2 was obtained in Dimitrov et al. [9]. Roy [10] found another interpretation of bivariate lack-of-memory (LM) property and gave a characterization of class of bivariate distributions via survival functions possessing that LM property for all choices of the participating in it four nonnegative arguments. One curious part of the BALM distributions is that the two components of the 2-dimensional vector satisfy the properties characterizing the bivariate exponential distributions with independent components only in the nodes of a rectangular grid in the first quadrant. However, inside the rectangles of that grid any kind of dependence between the two components may hold. In addition, the marginal distributions have periodic failure rates. This picture makes the BALM class attractive for modeling dependences in investment portfolios, financial mathematics, risk

Abstract:
We collect well known and less known facts about the bivariate normal distribution and translate them into copula language. In addition, we prove a very general formula for the bivariate normal copula, we compute Gini's gamma, and we provide improved bounds and approximations on the diagonal.

Abstract:
Copula modelling has in the past decade become a standard tool in many areas of applied statistics. However, a largely neglected aspect concerns the design of related experiments. Particularly the issue of whether the estimation of copula parameters can be enhanced by optimizing experimental conditions and how robust all the parameter estimates for the model are with respect to the type of copula employed. In this paper an equivalence theorem for (bivariate) copula models is provided that allows formulation of efficient design algorithms and quick checks of whether designs are optimal or at least efficient. Some examples illustrate that in practical situations considerable gains in design efficiency can be achieved. A natural comparison between different copula models with respect to design efficiency is provided as well.

Abstract:
We propose a new framework for dependence structure learning via copula. Copula is a statistical theory on dependence and measurement of association. Graphical models are considered as a type of special case of copula families, named product copula. In this paper, a nonparametric algorithm for copula estimation is presented. Then a Chow-Liu like method based on dependence measure via copula is proposed to estimate maximum spanning product copula with only bivariate dependence relations. The advantage of the framework is that learning with empirical copula focuses only on dependence relations among random variables, without knowing the properties of individual variables. Another advantage is that copula is a universal model of dependence and therefore the framework based on it can be generalized to deal with a wide range of complex dependence relations. Experiments on both simulated data and real application data show the effectiveness of the proposed method.

Abstract:
We show that all multivariate Extreme Value distributions, which are the possible weak limits of the $K$ largest order statistics of iid sequences, have the same copula, the so called K-extremal copula. This copula is described through exact expressions for its density and distribution functions. We also study measures of dependence, we obtain a weak convergence result and we propose a simulation algorithm for the K-extremal copula.

Abstract:
A framework named Copula Component Analysis (CCA) for blind source separation is proposed as a generalization of Independent Component Analysis (ICA). It differs from ICA which assumes independence of sources that the underlying components may be dependent with certain structure which is represented by Copula. By incorporating dependency structure, much accurate estimation can be made in principle in the case that the assumption of independence is invalidated. A two phrase inference method is introduced for CCA which is based on the notion of multidimensional ICA.