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postgraduate thesis: Statistical inference of a threshold model in extreme value analysis
Title  Statistical inference of a threshold model in extreme value analysis 

Authors  
Advisors  Advisor(s):Li, WK 
Issue Date  2012 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Lee, D. [李大為]. (2012). Statistical inference of a threshold model in extreme value analysis. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4819945 
Abstract  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 peaksoverthreshold (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 closedform
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 multilevel modeling procedure. With the fitted
density, it is possible to derive many useful quantities such as return periods
and levels, valueatrisk, 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 thintailed 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 closedform 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 MetropolisHastings 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. 
Degree  Master of Philosophy 
Subject  Inference. Extreme value theory. Multivariate analysis. 
Dept/Program  Statistics and Actuarial Science 
Persistent Identifier  http://hdl.handle.net/10722/167221 
HKU Library Item ID  b4819945 
DC Field  Value  Language 

dc.contributor.advisor  Li, WK   
dc.contributor.author  Lee, David.   
dc.contributor.author  李大為.   
dc.date.issued  2012   
dc.identifier.citation  Lee, D. [李大為]. (2012). Statistical inference of a threshold model in extreme value analysis. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4819945   
dc.identifier.uri  http://hdl.handle.net/10722/167221   
dc.description.abstract  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 peaksoverthreshold (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 closedform 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 multilevel modeling procedure. With the fitted density, it is possible to derive many useful quantities such as return periods and levels, valueatrisk, 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 thintailed 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 closedform 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 MetropolisHastings 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.   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.rights  This work is licensed under a Creative Commons AttributionNonCommercialNoDerivatives 4.0 International License.   
dc.source.uri  http://hub.hku.hk/bib/B4819945X   
dc.subject.lcsh  Inference.   
dc.subject.lcsh  Extreme value theory.   
dc.subject.lcsh  Multivariate analysis.   
dc.title  Statistical inference of a threshold model in extreme value analysis   
dc.type  PG_Thesis   
dc.identifier.hkul  b4819945   
dc.description.thesisname  Master of Philosophy   
dc.description.thesislevel  Master   
dc.description.thesisdiscipline  Statistics and Actuarial Science   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.5353/th_b4819945   
dc.date.hkucongregation  2012   