Statistical Modelling 6 (2006), 251263
Bayesian analysis of extreme events with
threshold estimation
Cibele N. Behrens
Federal University of Rio de Janeiro,
Rio de Janeiro,
Brazil.
eMail: cibele@dme.ufrj.br
Hedibert F. Lopes
Graduate School of Business,
University of Chicago,
Chicago, IL,
USA.
Dani Gamerman
Instituto de Matematica,
Universidade Federal do Rio de Janeiro,
Caixa Postal 68530,
Rio de Janeiro, RJ, CEP21945-970,
Brazil.
eMail: dani@im.ufrj.br
Abstract:
The aim of this paper is to analyse extremal events using
generalized Pareto distributions (GPD),
considering explicitly the uncertainty about the threshold.
Current practice empirically determines this
quantity and proceeds by estimating the GPD parameters on
the basis of data beyond it, discarding all the
information available below the threshold. We introduce a
mixture model that combines a parametric form
for the center and a GPD for the tail of the distributions
and uses all observations for inference about the
unknown parameters from both distributions, the threshold
included. Prior distributions for the parameters
are indirectly obtained through experts quantiles elicitation.
Posterior inference is available through
Markov chain Monte Carlo methods. Simulations are carried out
in order to analyse the performance of
our proposed model under a wide range of scenarios. Those
scenarios approximate realistic situations
found in the literature. We also apply the proposed model
to a real dataset, Nasdaq 100, an index of the
financial market that presents many extreme events. Important
issues such as predictive analysis and model
selection are considered along with possible modeling extensions.
Keywords:
Bayesian; extreme value theory; MCMC;
mixture model; threshold estimation
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