Maximum likelihood estimation of extreme value index for irregular cases
主 题: Maximum likelihood estimation of extreme value index for irregular cases
报告人: 祁永成 教授 (University of Minnesota Duluth)
时 间: 2009-06-04 上午10:00 - 11:00
地 点: 理科一号楼 1490
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 -1/2.
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. In this paper, we study the
asymptotic distributions of MLE when the extreme value index is in
between -1 and -1/2. 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.
