Empirical likelihood ratio confidence intervals for a single functional

TLDR

The empirical distribution function is the maximum likelihood estimate of the underlying distribution. The paper defines a likelihood ratio function for distributions using the likelihood function. The authors illustrate the resulting confidence intervals on real data, compare them via simulation to bootstrap and Student‑t intervals, and introduce a hybrid bootstrap‑based method for determining critical values. The empirical likelihood ratio function yields confidence intervals for the mean, M‑estimates (including quantiles), and differentiable functionals, extending Wilks’s theorem to nonparametric settings.

Abstract

The empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken. In this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions. It is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean, for a class of M-estimates that includes quantiles, and for differentiable statistical functionals. The results are nonpara-metric extensions of Wilks's (1938) theorem for parametric likelihood ratios. The intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on Student's t statistic. A hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced.