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Let Λ be the collection of all probability distributions for (X , X ˜) , where X is a fixed random vector and X ˜ ranges over all possible knockoff copies of X (in the sense of Candes et al. (2018)). Three topics are developed in this paper: (i) A new characterization of Λ is proved; (ii) A certain subclass of Λ , defined in terms of copulas, is introduced; (iii) The (meaningful) special case where the components of X are conditionally independent is treated in depth. In real problems, after observing X = x , each of points (i)–(ii)–(iii) may be useful to generate a value x ˜ for X ˜ conditionally on X = x. • The paper provides some new theoretical insights on the knockoffs procedure introduced by Barber and Candes (2015). Let Λ be the set of all possible knockofffs. In Section 2, a new characterization of Λ is proved. • In Section 3, a certain (proper) subclass Λ 0 ⊂ Λ is introduced. The elements of Λ 0 admit a simple and explicit representation in terms of copulas. • In Section 4, we focus on the case where the variables X 1 , ... , X p are conditionally independent. Under this assumption, we show how to build a knockoff X ˜. [ABSTRACT FROM AUTHOR]

Assume that X = ( X 1 , … , X d ) , d ⩾ 2 is a random vector having joint cumulative distribution function H with continuous marginal cumulative distribution functions F 1 , … , F d respectively. Sklar’s decomposition yields a unique copula C such that H ( x 1 , … , x d ) = C ( F 1 ( x 1 ) , … , F d ( x d ) ) for all ( x 1 , … , x d ) ∈ R d . Here F 1 , … , F d and C are the unknown parameters, the one of interest being the copula C . We assume C to belong to the Archimedean family, that is C = C ψ , for some Archimedean generator ψ . We exploit the well known fact that such a generator is in one-to-one correspondence with the distribution function of a nonnegative random variable R with no atom at zero. In order to adopt a Bayesian approach for inference, a prior on the Archimedean family may be selected via a prior on the cumulative distribution function F of R . A mixture of Pólya trees is proposed for F , making the model very flexible, yet still manageable. The induced prior is concentrated on the space of absolutely continuous d -dimensional Archimedean copulas and explicit forms for the generator and its derivatives are available. To the best of our knowledge, others in the literature have not yet considered such an approach. An extensive simulation study is carried out to compare our estimator with a popular frequentist nonparametric estimator. The results clearly indicate that if intensive computing is available, our estimator is worth considering, especially for small samples. [ABSTRACT FROM AUTHOR]

Abstract: The most popular multivariate control chart for monitoring the mean of a distribution is probably the Hotelling T 2 rule. Unfortunately, this rule relies on the assumption that the distribution under control is Gaussian, which is rarely true in practice. The objective of this paper is to propose a new approach for the non-normal multivariate case. It consists in the construction of a tolerance region obtained from a density level set estimation. The method follows a “plug-in” approach in which the density of the observations is previously estimated. This estimation is conducted using copulas modeling, an increasingly popular tool in multivariate modeling. [Copyright &y& Elsevier]

Abstract: Inspired by the notion of lower semilinear copulas, we introduce a new class of copulas. These copulas, called lower semiquadratic copulas, are constructed by quadratic interpolation on segments connecting the diagonal of the unit square to the lower and left boundary of the unit square. Moreover, we unveil the necessary and sufficient conditions on a diagonal function and two auxiliary real functions u and v to obtain a copula that has this diagonal function as diagonal section. Under some mild assumptions, we characterize the smallest and the greatest lower semiquadratic copulas with a given diagonal section. [Copyright &y& Elsevier]

Abstract: In this paper we study a class of M-estimators in a regression model under bivariate random censoring and provide a set of sufficient conditions that ensure asymptotic . The cornerstone of our approach is a new estimator of the joint distribution function of the censored lifetimes. A copula approach is used to modelize the dependence structure between the bivariate censoring times. The resulting estimators present the advantage of being easily computable. A simulation study enlighten the finite sample behavior of this technique. [Copyright &y& Elsevier]

Abstract: A meta-elliptical model is a distribution function whose copula is that of an elliptical distribution. The tail dependence function in such a bivariate model has a parametric representation with two parameters: a tail parameter and a correlation parameter. The correlation parameter can be estimated by robust methods based on the whole sample. Using the estimated correlation parameter as plug-in estimator, we then estimate the tail parameter applying a modification of the method of moments approach proposed in the paper by . We show that such an estimator is consistent and asymptotically normal. Further, we derive the joint limit distribution of the estimators of the two parameters. We illustrate the small sample behavior of the estimator of the tail parameter by a simulation study and on real data, and we compare its performance to that of the competitive estimators. [Copyright &y& Elsevier]

Abstract: Fully nonparametric tests for the independence between random vectors are studied in this paper. The test statistics are functionals of an empirical process defined as the difference between the joint empirical copula and the product of the empirical copulas associated to the vectors that are suspected to be independent. The validity of a weighted bootstrap procedure is established, which allows for a quick computation of p-values. A special attention is given to the asymptotic behavior of the tests under contiguous sequences of distributions. Finally, a characteristic of the copulas in the Archimedean class in terms of independence of vectors is exploited in order to propose a new goodness-of-fit procedure. [Copyright &y& Elsevier]

Abstract: We review the distributional transform of a random variable, some of its applications, and some related multivariate distributional transformations. The distributional transform is a useful tool, which allows in many respects to deal with general distributions in the same way as with continuous distributions. In particular it allows to give a simple proof of Sklar''s theorem in the general case. It has been used in the literature for stochastic ordering results. It is also useful for an adequate definition of the conditional value at risk measure and for many further purposes. We also discuss the multivariate quantile transform as well as the multivariate extension of the distributional transform and some of their applications. In the final section we consider an application to an extension of a limit theorem for the empirical copula process, also called empirical dependence function, to general not necessarily continuous distributions. This is useful for constructing and analyzing tests of dependence properties for general distributions. [Copyright &y& Elsevier]

Abstract: A new family of copulas is introduced that provides flexible dependence structure while being tractable and simple to use for multivariate discrete data modeling. The construction exploits finite mixtures of uncorrelated normal distributions. Accordingly, the cumulative distribution function is simply the product of univariate normal distributions. At the same time, however, the mixing operation introduces association. The properties of the new family of copulas are examined and a concrete application is used to show its applicability. [Copyright &y& Elsevier]

Abstract: In this paper the concept of copulas is implemented into the methodology for solving the imputation problem in correlated incomplete data. We use the Gaussian copula as alternative to the joint distribution for modeling the conditional distribution, conditioned by the observed values of measurements. The general formula for imputation and its application for compound symmetry correlation structure are given. [Copyright &y& Elsevier]

Abstract: The weighted arithmetic mean of two copulas is a copula. In some cases, geometric and harmonic means also provide copulas. There are copulas specially appropriate to be combined by using weighted geometric means. With this method of construction we combine Farlie–Gumbel–Morgentern and Ali–Mikhail–Haq copulas to obtain families of copulas which can be expressed in terms of double power series. The Gumbel–Barnett copula is also considered and a new copula is proposed, which arises as the first order approximation of the weighted geometric mean of two copulas. Invariance of two multivariate distributions (Cuadras–Augé and Johnson–Kotz) by weighted geometric and arithmetic means is also studied. [Copyright &y& Elsevier]

Abstract: Conditional probability distributions have been commonly used in modeling Markov chains. In this paper we consider an alternative approach based on copulas to investigate Markov-type dependence structures. Based on the realization of a single Markov chain, we estimate the parameters using one- and two-stage estimation procedures. We derive asymptotic properties of the marginal and copula parameter estimators and compare performance of the estimation procedures based on Monte Carlo simulations. At low and moderate dependence structures the two-stage estimation has comparable performance as the maximum likelihood estimation. In addition we propose a parametric pseudo-likelihood ratio test for copula model selection under the two-stage procedure. We apply the proposed methods to an environmental data set. [Copyright &y& Elsevier]

In this paper we address the dependence structure of the minimum and maximum of n iid random variables X1,…,Xn by determining their copula. It is then easy to give an alternative proof for their asymptotic independence and to calculate Kendall''s τ and Spearman''s ρ for (X(1),X(n)). This will show that the dependence between the variables is already small for small sample sizes. Finally, it can be shown that 3τn⩾ρn⩾τn>0. Although closed-form expressions are available for τn and ρn, we cannot compare them directly but have to use the concept of positive likelihood ratio dependence to establish this result. [Copyright &y& Elsevier]