On Location Invariant Heavy Tail Index Estimation

Extreme value theory constitutes a fundamental part in the study of rare events,and which has been used extensively modeling extremal phenomena in nature and society such as teletraffic,finance,insurance,enviromental and engineering science.More information is needed on the heavy tail distribution and its tail index estimation.Based on the asymptotic convergence behavior and distributional representation of the statistics M_n^(α)(k₀,k),two kinds of semi-parametric location invariant heavy tailed estimators are proposed:which are(?)_n⁽¹⁾(k₀,k,α) and(?)_n⁽²⁾(k₀,k,α). The asymptotic distributional representation and asymptotic normality of the two heavy tail index estimators are studied respectively in Chapter 2 and Chapter 3 under second order regular variation.Futhermore,the optimal choice of k₀ by means of minimizing Mean Square Error is also discussed.Simulation studies are provided in Chapter 4.Choosing suitable turning parameterα,we compare the properties of the proposed estimators(?)_n⁽¹⁾(k₀,k,α) and(?)_n⁽²⁾(k₀,k,α) with the one proposed by Fraga Alves in terms of mean value, mean square error,coverage probability and confidence interval length through some heavy tail distributions.Simulation indicates that both proposed heavy tail index estimators have smaller bias and comparable mean square error,and the preferable estimator(?)_n⁽¹⁾(k₀,k,α) shows shorter confidence interval and higher coverage probability.