Maximum likelihood estimation of extreme value index for irregular cases

A method in analyzing extremes is to fit a generalized Pareto distribution to the exceedances over a high threshold. By varying the threshold according to the sample size [Smith, R.L., 1987. Estimating tails of probability distributions. Ann. Statist. 15, 1174–1207] and [Drees, H., Ferreira, A., de Haan, L., 2004. On maximum likelihood estimation of the extreme value index. Ann. Appl. Probab. 14, 1179–1201] derived the asymptotic properties of the maximum likelihood estimates (MLE) when the extreme value index is larger than -12. Recently Zhou [2009. Existence and consistency of the maximum likelihood estimator for the extreme value index. J. Multivariate Anal. 100, 794–815] showed that the MLE is consistent when the extreme value index is larger than -1-1. In this paper, we study the asymptotic distributions of MLE when the extreme value index is in between -1-1 and -12 (including -12). Particularly, we consider the MLE for the endpoint of the generalized Pareto distribution and the extreme value index and show that the asymptotic limit for the endpoint estimate is non-normal, which connects with the results in Woodroofe [1974. Maximum likelihood estimation of translation parameter of truncated distribution II. Ann. Statist. 2, 474–488]. Moreover, we show that same results hold for estimating the endpoint of the underlying distribution, which generalize the results in Hall [1982. On estimating the endpoint of a distribution. Ann. Statist. 10, 556–568] to irregular case, and results in Woodroofe [1974. Maximum likelihood estimation of translation parameter of truncated distribution II. Ann. Statist. 2, 474–488] to the case of unknown extreme value index.