Bounds for the asymptotic distribution of the likelihood ratio

TL;DR

This paper provides the first explicit bounds on how close the likelihood ratio statistic's distribution is to a chi-square distribution for i.i.d. data, depending on sample size and parameter dimension.

Contribution

It introduces a novel explicit bound for the chi-square approximation of the likelihood ratio statistic for i.i.d. data, applicable to various models.

Findings

Bound depends on sample size and parameter dimension.

Illustrated with exponential, normal, and logistic regression examples.

First explicit bound of its kind in the literature.

Abstract

In this paper we give an explicit bound on the distance to chisquare for the likelihood ratio statistic when the data are realisations of independent and identically distributed random elements. To our knowledge this is the first explicit bound which is available in the literature. The bound depends on the number of samples as well as on the dimension of the parameter space. We illustrate the bound with three examples: samples from an exponential distribution, samples from a normal distribution, and logistic regression.