Second-order estimation procedures for complete and incomplete heavy-tailed data
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Date
2016
Authors
Maribe, Gaonyalelwe
Journal Title
Journal ISSN
Volume Title
Publisher
University of the Free State
Abstract
This thesis investigates the second-order re ned peaks over threshold model called
the Extended Pareto Distribution (EPD) introduced by Beirlant et al. (2009).
Focus is placed on estimation of the Extreme Value Index (EVI). Firstly we investigate
the e ectiveness of the EPD in modelling heavy-tailed distributions and
compare it to the Generalized Pareto Distribution (GPD) in terms of the bias,
mean squared error and variance of the EVI. This is done through a simulation
study and the Maximum Likelihood (ML) method of estimation is used to make
the comparison.
In practice, data can be tampered by some arbitrary process or study design.
We therefore investigate the performance of the EPD in estimating the EVI for
heavy-tailed data under the assumption that the data is completely observable
and uncontaminated, random right censored and contaminated respectively.
We suggest an improved ML numerical procedure in the estimation of EPD parameters
under the assumption that data is completely observable and uncontaminated.
We further propose a Bayesian EPD estimator of the EVI and show through
a simulation study that this estimator leads to much improved results as the ML
EPD estimator. A small case study is conducted to assess the performance of the
Bayesian EPD estimator and the ML EPD estimator using a real dataset from a
Belgian reinsurance rm.
We investigate the performance of some well known parametric and semi-parametric
estimators of the EVI adapted for censoring by a simulation study and further illustrate
their performance by applying them to a real survival dataset. A censored
Bayesian EPD estimator for right censored data is then proposed through an altered
expression of the posterior density. The censored Bayesian EPD estimator
is compared with the censored ML EPD estimator through a simulation study.
Behaviour of the minimum density power divergence estimator (MDPDE) is assessed
at uncontaminated and contaminated distributions respectively through an
exhaustive simulation study including other EPD estimators mentioned in this
thesis. The comparison is made in terms of the bias and mean squared error. EVI
estimates from the di erent estimators are then used to estimate quantiles, the
results are reported concurrently with the EVI estimates. We illustrate the performance
of all mentioned estimators on a real dataset from geopedology, in which
a few abnormal soil measurements highly in
uence the estimates of the EVI and
high quantiles.
Description
Keywords
Dissertation (M.Sc. (Mathematical Statistics and Actuarial Science))--University of the Free State, 2016, Extreme value theory