On testing the extreme value index via the pot-method

• 成果类型: Article
• 署名作者: Falk, M
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• 发表日期: 1995
• 页码: 2013-2035
• 关键词: quantile estimation; models

摘要

Consider an lid sample Y-1, ..., Y-n of random variables with common distribution function F, whose upper tail belongs to a neighborhood of the upper tail of a generalized Pareto distribution H-beta, beta is an element of R. We investigate the testing problem beta = beta(0) against a sequence beta = beta(n) of contiguous alternatives, based on the point processes N-n of the exceedances among Y-i over a sequence of thresholds t(n). It turns out that the (random) number of exceedances tau(n) over t(n) is the central sequence for the log-likelihood ratio dL(beta n)(N-n)/dL(beta 0)(N-n), yielding its local asymptotic normality (LAN). This result implies in particular that tau(n) carries asymptotically all the information about the underlying parameter beta, which is contained in N-n. We establish sharp bounds for the rate at which tau(n) becomes asymptotically sufficient, which show, however, that this is quite a poor rate. These results remain true if we add an unknown scale parameter.