Statistical inference of a threshold model in extreme value analysis

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 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.