Profile Likelihood Intervals for Quantiles in Extreme Value Distributions

TL;DR

This paper advocates for using profile likelihood intervals over asymptotic intervals for estimating large quantiles in Extreme Value distributions, especially for small to moderate sample sizes, due to better coverage and fewer estimation issues.

Contribution

It demonstrates the effectiveness of profile likelihood intervals for quantile estimation in small to moderate samples and highlights issues with maximum likelihood estimation in Weibull models.

Findings

Profile likelihood intervals have better coverage than mla intervals for sample sizes 25-100.

Maximum likelihood estimation can face singularities in Weibull models with small shape parameters.

Using the exact likelihood avoids estimation problems associated with the continuous approximation.

Abstract

Profile likelihood intervals of large quantiles in Extreme Value distributions provide a good way to estimate these parameters of interest since they take into account the asymmetry of the likelihood surface in the case of small and moderate sample sizes; however they are seldom used in practice. In contrast, maximum likelihood asymptotic (mla) intervals are commonly used without respect to sample size. It is shown here that profile likelihood intervals actually are a good alternative for the estimation of quantiles for sample sizes $25 \leq n\leq 100$ of block maxima, since they presented adequate coverage frequencies in contrast to the poor coverage frequencies of mla intervals for these sample sizes, which also tended to underestimate the quantile and therefore might be a dangerous statistical practice. In addition, maximum likelihood estimation can present problems when Weibull models are considered for moderate or small sample sizes due to singularities of the corresponding density function when the shape parameter is smaller than one. These estimation problems can be traced to the commonly used continuous approximation to the likelihood function and could be avoided by using the exact or correct likelihood function, at least for the settings considered here. A rainfall data example is presented to exemplify the suggested inferential procedure based on the analyses of profile likelihoods.