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Statistical Inference for a New Class of Multivariate Pareto Distributions
Various solutions to the parameter estimation problem of a recently introduced multivariate Pareto distribution are developed and exemplified numerically. Namely, a density of the aforementioned multivariate Pareto distribution with respect to a dominating measure, rather than the corresponding Lebesgue measure, is specified and then employed to investigate the maximum likelihood estimation (MLE) approach. Also, in an attempt to fully enjoy the common shock origins of the multivariate model of interest, an adapted variant of the expectation-maximization (EM) algorithm is formulated and studied. The method of moments is discussed as a convenient way to obtain starting values for the numerical optimization procedures associated with the MLE and EM methods
Statistical inference for radially-stable generalized Pareto distributions and return level-sets in geometric extremes
We obtain a functional analogue of the quantile function for probability
measures admitting a continuous Lebesgue density on , and use it
to characterize the class of non-trivial limit distributions of radially
recentered and rescaled multivariate exceedances in geometric extremes. A new
class of multivariate distributions is identified, termed radially stable
generalized Pareto distributions, and is shown to admit certain stability
properties that permit extrapolation to extremal sets along any direction in
. Based on the limit Poisson point process likelihood of the
radially renormalized point process of exceedances, we develop parsimonious
statistical models that exploit theoretical links between structural
star-bodies and are amenable to Bayesian inference. The star-bodies determine
the mean measure of the limit Poisson process through a hierarchical structure.
Our framework sharpens statistical inference by suitably including additional
information from the angular directions of the geometric exceedances and
facilitates efficient computations in dimensions and . Additionally,
it naturally leads to the notion of the return level-set, which is a canonical
quantile set expressed in terms of its average recurrence interval, and a
geometric analogue of the uni-dimensional return level. We illustrate our
methods with a simulation study showing superior predictive performance of
probabilities of rare events, and with two case studies, one associated with
river flow extremes, and the other with oceanographic extremes.Comment: 65 pages, 34 figure
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