Implicit Copulas: An Overview Smith, Michael Stanley
Econometrics and statistics,
October 2023, 2023-10-00, Letnik:
28
Journal Article
Odprti dostop
Implicit copulas are the most common copula choice for modeling dependence in high dimensions. This broad class of copulas is introduced and surveyed, including elliptical copulas, skew t copulas, ...factor copulas, time series copulas and regression copulas. The common auxiliary representation of implicit copulas is outlined, and how this makes them both scalable and tractable for statistical modeling. Issues such as parameter identification, extended likelihoods for discrete or mixed data, parsimony in high dimensions, and simulation from the copula model are considered. Bayesian approaches to estimate the copula parameters, and predict from an implicit copula model, are outlined. Particular attention is given to implicit copula processes constructed from time series and regression models, which is at the forefront of current research. Two econometric applications—one from macroeconomic time series and the other from financial asset pricing—illustrate the advantages of implicit copula models.
The empirical beta copula Segers, Johan; Sibuya, Masaaki; Tsukahara, Hideatsu
Journal of multivariate analysis,
March 2017, 2017-03-00, Letnik:
155
Journal Article
Recenzirano
Odprti dostop
Given a sample from a continuous multivariate distribution F, the uniform random variates generated independently and rearranged in the order specified by the componentwise ranks of the original ...sample look like a sample from the copula of F. This idea can be regarded as a variant on Baker’s J. Multivariate Anal. 99 (2008) 2312–2327 copula construction and leads to the definition of the empirical beta copula. The latter turns out to be a particular case of the empirical Bernstein copula, the degrees of all Bernstein polynomials being equal to the sample size.
Necessary and sufficient conditions are given for a Bernstein polynomial to be a copula. These imply that the empirical beta copula is a genuine copula. Furthermore, the empirical process based on the empirical Bernstein copula is shown to be asymptotically the same as the ordinary empirical copula process under assumptions which are significantly weaker than those given in Janssen, Swanepoel and Veraverbeke J. Statist. Plann. Inference 142 (2012) 1189–1197.
A Monte Carlo simulation study shows that the empirical beta copula outperforms the empirical copula and the empirical checkerboard copula in terms of both bias and variance. Compared with the empirical Bernstein copula with the smoothing rate suggested by Janssen et al., its finite-sample performance is still significantly better in several cases, especially in terms of bias.
Extending sub-quasi-copulas Kokol Bukovšek, Damjana; Košir, Tomaž; Omladič, Matjaž ...
Journal of mathematical analysis and applications,
08/2021, Letnik:
500, Številka:
1
Journal Article
Recenzirano
The main result of this paper is a method that gives all possible quasi-copulas that extend a given sub-quasi-copula to the whole domain, i.e., unit square 0,1×0,1. It is known that extending ...quasi-copulas is a deeper challenge than extending copulas, so we need to develop new techniques to do that. Perhaps surprisingly, our method is at the same time elementary and universal – two seemingly contradicting properties that none of the known methods for extending copulas seem to possesses. We also give a construction of two quasi-copulas that unveil an interesting counterexample in imprecise probability theory.
In this paper we present a surprisingly general extension of the main result of a paper that appeared in this journal: Montes et al. (2015) 13. The main tools we develop in order to do so are: (1) a ...theory on quasi-distributions based on an idea presented in a paper by R. Nelsen with collaborators; (2) starting from what is called (bivariate) p-box in the above mentioned paper we propose some new techniques based on what we call restricted (bivariate) p-box; and (3) a substantial extension of a theory on coherent imprecise copulas developed by M. Omladič and N. Stopar in a previous paper in order to handle coherence of restricted (bivariate) p-boxes. A side result of ours of possibly even greater importance is the following: Every bivariate distribution whether obtained on a usual σ-additive probability space or on an additive space can be obtained as a copula of its margins meaning that its possible extraordinariness depends solely on its margins. This might indicate that copulas are a stronger probability concept than probability itself.
In this paper we solve in the negative the problem proposed in this journal (Montes et al. (2015) 13) whether an order interval defined by an imprecise copula contains a copula. Namely, if C is a ...nonempty set of copulas, then C_=inf{C}C∈C and C‾=sup{C}C∈C are quasi-copulas and the pair (C_,C‾) is an imprecise copula according to the definition introduced in the cited paper, following the ideas of p-boxes. We show that there is an imprecise copula (A,B) in this sense such that there is no copula C whatsoever satisfying A⩽C⩽B. So, it is questionable whether the proposed definition of the imprecise copula is in accordance with the intentions of the initiators. Our methods may be of independent interest: We upgrade the ideas of Dibala et al. (2016) 7 where possibly negative volumes of quasi-copulas as defects from being copulas were studied.
In this paper, we introduce patchwork constructions for multivariate quasi-copulas. These results appear to be new since the kind of approach has been limited to either copulas or only bivariate ...quasi-copulas so far. It seems that the multivariate case is much more involved, since we are able to prove that some of the known methods of bivariate constructions cannot be extended to higher dimensions. Our main result is to present the necessary and sufficient conditions both on the patch and the values of it for the desired multivariate quasi-copula to exist. We also give all possible solutions.
A complete and user-friendly directory of tails of Archimedean copulas is presented which can be used in the selection and construction of appropriate models with desired properties. The results are ...synthesized in the form of a decision tree: Given the values of some readily computable characteristics of the Archimedean generator, the upper and lower tails of the copula are classified into one of three classes each, one corresponding to asymptotic dependence and the other two to asymptotic independence. For a long list of single-parameter families, the relevant tail quantities are computed so that the corresponding classes in the decision tree can easily be determined. In addition, new models with tailor-made upper and lower tails can be constructed via a number of transformation methods. The frequently occurring category of asymptotic independence turns out to conceal a surprisingly rich variety of tail dependence structures.
•A systematic approach to the multivariate copula modeling of dependent degradation processes is provided.•A study of the effect of ignoring tail dependence on system failure probability assessment ...is conducted.•The applications on both system reliability evaluation and remaining useful life prediction are given.•A numerical example considering a three-dimensional degradation process is demonstrated.
Multivariate degradation processes have been observed in many engineering systems. Most existing multivariate degradation modeling techniques, such as multivariate general path models or multivariate Wiener process models, assume an underlying Gaussian dependence structure. Unfortunately, in reality, the dependencies among degradation processes are often nonlinear, asymmetric and greatly tail-skewed, and thus limit the usefulness of the conventional modeling techniques in practice. To overcome these limitations, in this paper, we develop a copula-based multivariate modeling framework. Three fundamental copula classes are applied to model the complex dependence structure among correlated degradation processes. Statistical inference and model selection techniques, including two graphical diagnostic tools, a test of independence and a goodness-of-fit test, are employed to identify the best model. The advantages of the proposed modeling framework are demonstrated through simulation studies. And we also discuss the effect of ignoring tail dependence on system failure probability assessment. Finally, the applications of the copula-based multivariate degradation models on both system reliability evaluation and remaining useful life prediction are provided. The proposed methodology is illustrated using a numerical example.
Weak convergence of the empirical copula process is shown to hold under the assumption that the first-order partial derivatives of the copula exist and are continuous on certain subsets of the unit ...hypercube. The assumption is non-restrictive in the sense that it is needed anyway to ensure that the candidate limiting process exists and has continuous trajectories. In addition, resampling methods based on the multiplier central limit theorem, which require consistent estimation of the first-order derivatives, continue to be valid. Under certain growth conditions on the second-order partial derivatives that allow for explosive behavior near the boundaries, the almost sure rate in Stute's representation of the empirical copula process can be recovered. The conditions are verified, for instance, in the case of the Gaussian copula with full-rank correlation matrix, many Archimedean copulas, and many extreme-value copulas.
Copula is a useful tool that captures the dependence structure among random variables. In practice, it is an important question which copula to choose depending on the given data and stochastic ...assumptions on the model in order to achieve an appropriate interpretation of the data at hand. This paper intends to help a practitioner to make a better decision about that. We concentrate on the study of the lack of exchangeability, a copulas’ attribute closely studied only recently. The main non-exchangeability measure μ∞ for a family of copulas is the supremum of the differences |C(x,y)−C(y,x)| over all (x,y) and all copulas C in the family. We give the sharp bound of μ∞ for the families of Marshall copulas, maxmin and reflected maxmin copulas (i.e. the main shock-model based copulas) as well as the families of positively and of negatively quadrant dependent copulas. A major contribution of this paper is also exact calculation of the maximal asymmetry function on each of the particular families of copulas. When restricted to special families of copulas considered, it helps us finding the sharp bound of μ∞ for each of the given families. And even more importantly, it helps us giving a stochastic interpretation of the extremal copulas and examples of shock models where the maximal asymmetry is attained.
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