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For multivariate non-Gaussian involving copulas, likelihood inference is dominated by the data in the middle, and fitted models might not be very good for joint tail inference, such as assessing the strength of tail dependence. When preliminary data and likelihood analysis suggest asymmetric tail dependence, a method is proposed to improve extreme value inferences based on the joint lower and upper tails. A prior that uses previous information on tail dependence can be used in combination with the likelihood. With the combination of the prior and the likelihood (which in practice has some degree of misspecification) to obtain a tilted log-likelihood, inferences with suitably transformed parameters can be based on Bayesian computing methods or with numerical optimization of the tilted log-likelihood to obtain the posterior mode and Hessian at this mode.

Multivariate models with parsimonious dependence have been used for a large number of variables, and have mainly been developed for multivariate Gaussian. Graphical dependence model representations include Bayesian networks, conditional independence graphs, and truncated vines. The class of Gaussian truncated vines is a subset of Gaussian Bayesian networks and Gaussian conditional independence graphs, but has an extension to non-Gaussian dependence with (i) combinations of continuous and discrete random variables with arbitrary univariate margins, and (ii) accommodation of latent variables. To illustrate the importance of graphical models with latent variables that do not rely on the Gaussian assumption, the combined factor-vine structure is presented and applied to a data set of stock returns. Des modèles multivariés à dépendance éparse ont été utilisés avec un nombre élevé de variables, mais ils ont surtout été développés dans un contexte multivarié gaussien. La représentation graphique de modèles de dépendance inclut les réseaux bayésiens, les graphes d’indépendance conditionnelle, et les vignes tronquées. La classe de vignes gaussiennes tronquées est un sous-ensemble des réseaux bayésiens gaussiens et des graphes d’indépendance conditionnelle dont une extension non-gaussienne peut être obtenue, permettant (i) des combinaisons de variables aléatoires continues et discrètes avec des marges univariées arbitraires, et (ii) la présence de variables latentes. L’auteur présente la structure combinée de vigne factorielle afin d’illustrer l’importance de disposer de modèles graphiques comportant des variables latentes mais n’étant pas basés sur l’hypothèse de normalité. Il applique la méthode à des données réelles de rendement boursier.

We introduce a family of goodness-of-fit statistics for testing composite null hypotheses in multidimensional contingency tables. These statistics are quadratic forms in marginal residuals up to order r . They are asymptotically chi-square under the null hypothesis when parameters are estimated using any asymptotically normal consistent estimator. For a widely used item response model, when r is small and multidimensional tables are sparse, the proposed statistics have accurate empirical Type I errors, unlike Pearson’s X 2 . For this model in nonsparse situations, the proposed statistics are also more powerful than X 2 . In addition, the proposed statistics are asymptotically chi-square when applied to subtables, and can be used for a piecewise goodness-of-fit assessment to determine the source of misfit in poorly fitting models.

For modeling count time series data, one class of models is generalized integer autoregressive of order p based on thinning operators. It is shown how numerical maximum likelihood estimation is possible by inverting the probability generating function of the conditional distribution of an observation given the past p observations. Two data examples are included and show that thinning operators based on compounding can substantially improve the model fit compared with the commonly used binomial thinning operator.

For multivariate data from an observational study, inferences of interest can include conditional probabilities or quantiles for one variable given other variables. For statistical modeling, one could fit a parametric multivariate model, such as a vine copula, to the data and then use the model-based conditional distributions for further inference. Some results are derived for properties of conditional distributions under different positive dependence assumptions for some copula-based models. The multivariate version of the stochastically increasing ordering of conditional distributions is introduced for this purpose. Results are explained in the context of multivariate Gaussian distributions, as properties for Gaussian distributions can help to understand the properties of copula extensions based on vines.