Second-Order Properties of Distribution Tails and Estimation of Tail Exponents in Random Difference Equations

• Activity Number: 120
• Type: Contributed
• Date/Time: Monday, August 3, 2009 : 8:30 AM to 10:20 AM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Changryong Baek*+ and Vladas Pipiras and Patrice Abry and Herwig Wendt
• Companies: The University of North Carolina at Chapel Hill and The University of North Carolina at Chapel Hill and ENS Lyon and Purdue University
• Keywords: random difference equations ; tail exponent and its estimation ; extreme value theory ; second order regular vatiation ; bias reduction ; GARCH models

Abstract

It is well known that random difference equations (RDEs) have a power-law distribution tail in the asymptotic sense. However, empirical evidence shows that classical estimators of tail exponent of RDEs, such as Hill estimator, are extremely biased for larger values of tail exponents. It is argued that the bias occurs because the power-tail region is too far in the tail from a practical perspective. This is supported by proving a weaker form of the second order regular variation of distribution tails of RDEs, which measures deviations from the asymptotic power tail. To reduce bias, several least squares estimators, generalizing rank-based and QQ-estimators, and conditional maximum likelihood estimators, based on the exact form of second order regular variation, are introduced and the their basic asymptotics are established. Motivating examples include GARCH models in finance.