Maximum Likelihood Estimation of the Lower Tail of a Probability Distribution

Journal of the Royal Statistical Society: Series B (Statistical Methodology) · 1985 · Volume 47 · Issue 2

DOI: 10.1111/j.2517-6161.1985.tb01357.x

Abstract

We consider the estimation of the three parameters of the lower tail of a distribution function based on theksmallest out ofnobservations. 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 ask→ ∞,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.

Keywords

order statistics, extreme values, censored samples, maximum likelihood estimation

Enable JavaScript for citation tools and additional paper interactions.