Reduced‐bias kernel estimators of a positive extreme value index. (11th July 2019)

• Record Type: Journal Article
• Authors: Caeiro, Frederico; Henriques‐Rodrigues, Lígia
• Other Names: Vigo-Aguiar Jesus guestEditor.; Kumam Poom guestEditor.
• Is Part Of: Mathematical methods in the applied sciences. Volume 42:Number 17(2019)
• Page Start: 5867
• Page End: 5880
• Publication Date: 2019-07-11
• Subjects: asymptotic behaviour -- bias estimation -- heavy tails -- optimal levels -- semi‐parametric estimation -- statistics of extremes; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.5761
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : 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.