The extent of the maximum likelihood estimator for the extreme value index

In extreme value analysis, staring from Smith (1987) [1], the maximum likelihood procedure is applied in estimating the shape parameter of tails—the extreme value index γγ. For its theoretical properties, Zhou (2009) [12] proved that the maximum likelihood estimator eventually exists and is consistent for γ>−1γ>−1 under the first order condition. The combination of Zhou (2009) [12] and Drees et al (2004) [11] provides the asymptotic normality under the second order condition for γ>−1/2γ>−1/2. This paper proves the asymptotic normality for −1<γ≤−1/2−1<γ≤−1/2 and the non-consistency for γ<−1γ<−1. These results close the discussion on the theoretical properties of the maximum likelihood estimator.