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Models and diagnostics for parsimonious dependence with applications to multivariate extremes Lee, David


Statistical models with parsimonious dependence are useful for high-dimensional modelling as they offer interpretations relevant to the data being fitted and may be computationally more manageable. We propose parsimonious models for multivariate extremes; in particular, extreme value (EV) copulas with factor and truncated vine structures are developed, through (a) taking the EV limit of a factor copula, or (b) structuring the underlying correlation matrix of existing multivariate EV copulas. Through data examples, we demonstrate that these models allow interpretation of the respective structures and offer insight on the dependence relationship among variables. The strength of pairwise dependence for extreme value copulas can be described using the extremal coefficient. We consider a generalization of the F-madogram estimator for the bivariate extremal coefficient to the estimation of tail dependence of an arbitrary bivariate copula. This estimator is tail-weighted in the sense that the joint upper or lower portion of the copula is given a higher weight than the middle, thereby emphasizing tail dependence. The proposed estimator is useful when tail heaviness plays an important role in inference, so that choosing a copula with matching tail properties is essential. Before using a fitted parsimonious model for further analysis, diagnostic checks should be done to ensure that the model is adequate. Bivariate extremal coefficients have been used for diagnostic checking of multivariate extreme value models. We investigate the use of an adequacy-of-fit statistic based on the difference between low-order empirical and model-based features (dependence measures), including the extremal coefficient, for this purpose. The difference is computed for each of the bivariate margins and a quadratic form statistic is obtained, with large values relative to a high quantile of the reference distribution suggesting model inadequacy. We develop methods to determine the appropriate cutoff values for various parsimonious models, dimensions, dependence measures and methods of model fitting that reflect practical situations. Data examples show that these diagnostic checks are handy complements to existing model selection criteria such as the AIC and BIC, and provide the user with some idea about the quality of the fitted models.

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