Reduced‐bias kernel estimators of a positive extreme value index

• Published in: Mathematical methods in the applied sciences Vol. 42; no. 17; pp. 5867 - 5880
• Main Authors: Caeiro, Frederico, Henriques‐Rodrigues, Lígia
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 30.11.2019
• Subjects: asymptotic behaviour; Asymptotic properties; Bias; bias estimation; Estimators; Extreme values; heavy tails; Kernels; Normality; optimal levels; Parameter estimation; Parameter modification; semi‐parametric estimation; statistics of extremes; Value analysis
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.5761

In this paper, we deal with the semi‐parametric estimation of the extreme value index, an important parameter in extreme value analysis. It is well known that many classic estimators, such as the Hill estimator, reveal a strong bias. This problem motivated the study of two classes of kernel estimators. Those classes generalize the classical Hill estimator and have a tuning parameter that enables us to modify the asymptotic mean squared error and eventually to improve their efficiency. Since the improvement in efficiency is not very expressive, we also study new reduced bias estimators based on the two classes of kernel statistics. Under suitable conditions, we prove their asymptotic normality. Moreover, an asymptotic comparison, at optimal levels, shows that the new classes of reduced bias estimators are more efficient than other reduced bias estimator from the literature. An illustration of the finite sample behaviour of the kernel reduced‐bias estimators is also provided through the analysis of a data set in the field of insurance.