Bayesian analysis of jointly heavy-tailed data
View/ Open
Palacios Ramírez2022.pdf (19.98Mb)
Date
29/11/2022Item status
Restricted AccessEmbargo end date
29/11/2023Author
Palacios Ramırez, Karla Vianey
Metadata
Abstract
This thesis develops novel Bayesian methodologies for statistical modelling of heavy-tailed data.
Heavy tails are often found in practice, and yet they are an Achilles heel of a variety of main-stream random probability measures such as the Dirichlet process. The first contribution of this
thesis is the study of random probability measures that can be used to model heavy-tailed data,
focusing on the characterization of the tails of the so-called Pitman–Yor process, which includes
the Dirichlet process as a particular case. We show that the right tail of a Pitman–Yor process,
known as the stable law process, is heavy-tailed, provided that the centering distribution is itself heavy-tailed. Empowered by this finding we developed two classes of heavy-tailed mixture
models and assessed their relative merits. Multivariate extensions of the proposed heavy-tailed
mixtures are also devised along with a predictor-dependent version so to learn about the effect
of covariates on a multivariate heavy-tailed response.
Another contribution of the thesis is a framework for learning about the frequency and
magnitude of extreme values in the joint tail as well as in other risk sets induced by a suitable
aggregation of variables on a common scale. The proposed framework reveals explicit links
between nonstationary multivariate extremes and heteroscedastic extremes, and hence can be
used to track the dynamics governing the degree of association between the extremes of a random
vector over time. Bayesian inference methods are conducted for the two targets of interest: the
structure scedasis function and the coefficient of tail dependence for structure processes. To
learn about the structure scedasis function from data, we resort to a Bayesian nonparametric
approach that defines a prior in the space of structure scedasis functions. A battery of numerical
experiments is used to study the finite sample performance of all methodological developments
made in this thesis. Data from the fields of neuroscience as well as from finance are used to
showcase the proposed methodologies in practice.