Higher-order asymptotic normality of approximations to the modified signed likelihood ratio statistic for regular models

TL;DR

This paper establishes higher-order asymptotic normality for approximations to the modified signed likelihood ratio statistic in regular models, removing the need for difficult-to-verify conditions on the sufficient statistic.

Contribution

It provides new conditions ensuring the asymptotic normality of the statistic without assuming a specific form of the sufficient statistic.

Findings

Asymptotic normality of the statistic is achieved to order n^{-1}

Conditions are derived that do not require the sufficient statistic to be expressed as (hatθ, a)

The results simplify the application of likelihood ratio tests in complex models

Abstract

Approximations to the modified signed likelihood ratio statistic are asymptotically standard normal with error of order $n^{-1}$, where $n$ is the sample size. Proofs of this fact generally require that the sufficient statistic of the model be written as $(\hat{\theta},a)$, where $\hat{\theta}$ is the maximum likelihood estimator of the parameter $\theta$ of the model and $a$ is an ancillary statistic. This condition is very difficult or impossible to verify for many models. However, calculation of the statistics themselves does not require this condition. The goal of this paper is to provide conditions under which these statistics are asymptotically normally distributed to order $n^{-1}$ without making any assumption about the sufficient statistic of the model.