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

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

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