In many data sets, a mixture distribution formulation applies when it is
known that each observation comes from one of the underlying categories. Even
if there are no apparent categories, an implicit categorical structure may justify
a mixture distribution. This thesis concerns the modeling of extreme values in
such a setting within the peaks-over-threshold (POT) approach. Specifically,
the traditional POT modeling using the generalized Pareto distribution is augmented
in the sense that, in addition to threshold exceedances, data below the
threshold are also modeled by means of the mixture exponential distribution.
In the first part of this thesis, the conventional frequentist approach is
applied for data modeling. In view of the mixture nature of the problem,
the EM algorithm is employed for parameter estimation, where closed-form
expressions for the iterates are obtained. A simulation study is conducted to
confirm the suitability of such method, and the observation of an increase in
standard error due to the variability of the threshold is addressed. The model
is applied to two real data sets, and it is demonstrated how computation time
can be reduced through a multi-level modeling procedure. With the fitted
density, it is possible to derive many useful quantities such as return periods
and levels, value-at-risk, expected tail loss and bounds for ruin probabilities.
A likelihood ratio test is then used to justify model choice against the simpler
model where the thin-tailed distribution is homogeneous exponential.
The second part of the thesis deals with a fully Bayesian approach to the
same model. It starts with the application of the Bayesian idea to a special
case of the model where a closed-form posterior density is computed for the
threshold parameter, which serves as an introduction. This is extended to
the threshold mixture model by the use of the Metropolis-Hastings algorithm
to simulate samples from a posterior distribution known up to a normalizing
constant. The concept of depth functions is proposed in multidimensional
inference, where a natural ordering does not exist. Such methods are then
applied to real data sets. Finally, the issue of model choice is considered
through the use of posterior Bayes factor, a criterion that stems from the
posterior density. / published_or_final_version / Statistics and Actuarial Science / Master / Master of Philosophy
| Identifer | oai:union.ndltd.org:HKU/oai:hub.hku.hk:10722/167221 |
| Date | January 2012 |
| Creators | Lee, David., 李大為. |
| Contributors | Li, WK |
| Publisher | The University of Hong Kong (Pokfulam, Hong Kong) |
| Source Sets | Hong Kong University Theses |
| Language | English |
| Detected Language | English |
| Type | PG_Thesis |
| Source | http://hub.hku.hk/bib/B4819945X |
| Rights | The author retains all proprietary rights, (such as patent rights) and the right to use in future works., Creative Commons: Attribution 3.0 Hong Kong License |
| Relation | HKU Theses Online (HKUTO) |
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