Maximum Likelihood Estimation of the Lower Tail of a Probability Distribution
研究了基于n个观测值中最小k个观测值来估计分布函数下尾的三个参数，指出似然函数存在奇点但局部极大值可作为最大似然估计，并给出了渐近性质与区间估计方法。
SUMMARY We consider the estimation of the three parameters of the lower tail of a distribution function based on the k smallest out of n observations. The likelihood function has a singularity but it is argued that a local maximum, when it exists, should be taken as the m.l.e. Asymptotic results as k → ∞, n → ∞, k/n → 0 show that such a local maximum does exist, and provides consistent estimators whenever the shape parameter is greater than one; otherwise there is no local maximum and likelihood inference fails. We also discuss interval estimation and propose a test to distinguish between the Type I and Weibull limit laws.