Second-order estimation procedures for complete and incomplete heavy-tailed data

[摘要] This thesis investigates the second-order re ned peaks over threshold model calledthe Extended Pareto Distribution (EPD) introduced by Beirlant et al. (2009).Focus is placed on estimation of the Extreme Value Index (EVI). Firstly we investigatethe e ectiveness of the EPD in modelling heavy-tailed distributions andcompare 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 simulationstudy and the Maximum Likelihood (ML) method of estimation is used to makethe 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 forheavy-tailed data under the assumption that the data is completely observableand uncontaminated, random right censored and contaminated respectively.We suggest an improved ML numerical procedure in the estimation of EPD parametersunder the assumption that data is completely observable and uncontaminated.We further propose a Bayesian EPD estimator of the EVI and show througha simulation study that this estimator leads to much improved results as the MLEPD estimator. A small case study is conducted to assess the performance of theBayesian EPD estimator and the ML EPD estimator using a real dataset from aBelgian reinsurancerm.We investigate the performance of some well known parametric and semi-parametricestimators of the EVI adapted for censoring by a simulation study and further illustratetheir performance by applying them to a real survival dataset. A censoredBayesian EPD estimator for right censored data is then proposed through an alteredexpression of the posterior density. The censored Bayesian EPD estimatoris compared with the censored ML EPD estimator through a simulation study.Behaviour of the minimum density power divergence estimator (MDPDE) is assessedat uncontaminated and contaminated distributions respectively through anexhaustive simulation study including other EPD estimators mentioned in thisthesis. The comparison is made in terms of the bias and mean squared error. EVIestimates from the di erent estimators are then used to estimate quantiles, theresults are reported concurrently with the EVI estimates. We illustrate the performanceof all mentioned estimators on a real dataset from geopedology, in whicha few abnormal soil measurements highly inuence the estimates of the EVI andhigh quantiles.

[发布机构] University of the Free State