Interfaces between statistical learning and risk management.
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Date
2020
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Abstract
The recent hype on Artificial Intelligence, Data Science, and Machine Learning has been leading to a revolution in the industries of Banking and Finance. Motivated by this revolution, this
thesis develops novel statistical methodologies tailored for learning about financial risk in the
Big Data era. Specifically, the methodologies proposed in this thesis build over ideas, concepts,
and methods that relate to cluster analysis, copulas, and extreme value theory.
I start this thesis working on the framework of extreme value theory and propose novel statistical methodologies that identify time series which resemble the most in terms of magnitude
and dynamics of their extreme losses. A cluster analysis algorithm is proposed for the setup of
heteroscedastic extremes as a way to learn about similarity of extremal features of time series.
The proposed method pioneers the development of cluster analysis in a product space between
an Euclidean space and a space of functions.
In the second contribution of this thesis, I introduce a novel class of distributions—to which
we refer to as diagonal distributions. Similarly to the spectral density of a bivariate extreme value
distribution, the latter class consists of a mean-constrained univariate distribution function on
[0, 1], which summarizes key features on the dependence structure of a random vector. Yet,
despite their similarities, spectral and diagonal densities are constructed from very different
principles. In particular, diagonal densities extend the concept of marginal distribution—by
suitably projecting pseudo-observations on a segment line; diagonal densities also have a direct
link with copulas, and their variance has connections with Spearman’s rho.
Finally, I close the thesis by proposing a density ratio model for modeling extreme values of
non-indentically distributed observations. The proposed model can be regarded as a proportional tails model for multisample settings. A semiparametric specification is devised to link all
elements in a family of scedasis densities through a tilt from a baseline scedasis. Inference is
conducted by empirical likelihood inference methods.
Description
Tesis (Doctor in Statistics)--Pontificia Universidad Católica de Chile, 2020