Confidence distributions from likelihoods by median bias correction

TL;DR

This paper introduces a new method for constructing accurate confidence distributions using median bias correction and tail-symmetric confidence curves, offering higher-order accuracy in scalar parameter inference.

Contribution

It proposes a novel approach based on inversion of the log-likelihood ratio at the median MLE, achieving third-order accuracy for confidence distributions.

Findings

Provides third-order accurate confidence intervals via median bias correction.

Shows the method yields exact confidence distributions in regular models.

Offers an alternative second-order approximation in exponential families.

Abstract

By the modified directed likelihood, higher order accurate confidence limits for a scalar parameter are obtained from the likelihood. They are conveniently described in terms of a confidence distribution, that is a sample dependent distribution function on the parameter space. In this paper we explore a different route to accurate confidence limits via tail-symmetric confidence curves, that is curves that describe equal tailed intervals at any level. Instead of modifying the directed likelihood, we consider inversion of the log-likelihood ratio when evaluated at the median of the maximum likelihood estimator. This is shown to provide equal tailed intervals, and thus an exact confidence distribution, to the third-order of approximation in regular one-dimensional models. Median bias correction also provides an alternative approximation to the modified directed likelihood which holds up to the second order in exponential families.