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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Keywords Asymptotic normality, Extreme value index, Maximum likelihood
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Journal Journal of Multivariate Analysis
Zhou, C. (2010). The extent of the maximum likelihood estimator for the extreme value index. Journal of Multivariate Analysis, 101(4), 971–983. doi:10.1016/j.jmva.2009.09.013