Asymptotic Behaviour of the Modified Likelihood Root

TL;DR

This paper investigates the asymptotic properties of the modified likelihood root, demonstrating its approximation accuracy and polynomial relation to the likelihood root within specific statistical families.

Contribution

It provides a higher-order asymptotic analysis of the modified likelihood root, revealing its polynomial structure and linearity in key statistical models.

Findings

$r^\star$ acts as a location and scale adjustment to $r$ up to $O_p(n^{-3/2})$

$r^\star$ can be expressed as a polynomial in $r$

Linearity of the modified likelihood root in the likelihood root for certain families

Abstract

We examine the normal approximation of the modified likelihood root, an inferential tool from higher-order asymptotic theory, for the linear exponential and location-scale family. We show that the $r^\star$ statistic can be thought of as a location and scale adjustment to the likelihood root $r$ up to $O_p(n^{-3/2})$, and more generally $r^\star$ can be expressed as a polynomial in $r$. We also show the linearity of the modified likelihood root in the likelihood root for these two families.