## simulating from the joint cdf

Posted in Books, Kids, pictures, R, Statistics, University life with tags , , , , , , , , on July 13, 2022 by xi'an

An X validated question (what else?!) brought back (to me) the question of handling a bivariate cdf for simulation purposes. In the specific case of a copula when thus marginals were (well-)known…. And led me to an erroneous chain of thought, fortunately rescued by Robin Ryder! When the marginal distributions are set, the simulation setup is indeed equivalent to a joint Uniform simulation from a copula

$\mathbb P[U_1\leq u_1,U_2\leq u_2,\dots,U_d\leq u_d]=C(u_1,u_2,\dots,u_d)$

In specific cases, as for instance the obvious example of Gaussian copulas, there exist customised simulation algorithms. Looking for more generic solutions, I turn to the Bible, where Chapter XI[an], has two entire sections XI.3.2. and XI.3.3 on the topic (even though Luc Devroye does not use the term copula there despite them being introduced in 1959 by A, Sklar, in response to a query of M. Fréchet). In addition to a study of copulas, both sections contain many specific solutions (as for instance in the [unnumbered] Table on page 585) but I found no generic simulation method. My [non-selected] answer to the question was thus to propose standard solutions such as finding one conditional since the marginals are Uniform. Which depends on the tractability of the derivatives of C(·,·).

However, being dissatisfied with this bland answer, I thought further about the problem and came up with a fallacious scheme, namely to first simulate the value p of C(U,V) by drawing a Uniform, and second simulate (U,V) conditional on C(U,V)=p. Going as far as running an R code on a simple copula, as shown above. Fallacious reasoning since (as I knew already!!!), C(U,V) is not uniformly distributed! But has instead a case-dependent distribution… As a (connected) aside, I wonder if the generator attached with Archimedean copulas has any magical feature that help with the generation of the associated copula.

## Christian Robert is giving a talk in Jussieu tomorrow

Posted in Statistics, University life with tags , , , , , , , on September 26, 2019 by xi'an

My namesake Christian (Yann) Robert (CREST) is giving a seminar tomorrow in Jussieu (Université Pierre & Marie Curie, couloir 16-26, salle 209), between 2 and 3, on composite likelihood estimation method for hierarchical Archimedean copulas defined with multivariate compound distributions. Here is the abstract:

We consider the family of hierarchical Archimedean copulas obtained from multivariate exponential mixture distributions through compounding, as introduced by Cossette et al. (2017). We investigate ways of determining the structure of these copulas and estimating their parameters. An agglomerative clustering technique based on the matrix of Spearman’s rhos, combined with a bootstrap procedure, is used to identify the tree structure. Parameters are estimated through a top-down composite likelihood. The validity of the approach is illustrated through two simulation studies in which the procedure is explained step by step. The composite likelihood method is also compared to the full likelihood method in a simple case where the latter is computable.