Improved likelihood inference in generalized linear models

TL;DR

This paper introduces a Bartlett-type correction for the gradient test in generalized linear models, significantly improving small-sample inference accuracy compared to traditional tests.

Contribution

It derives a new correction factor for the gradient test, enhancing small-sample inference in generalized linear models, and compares its performance with existing tests.

Findings

Corrected gradient test reduces size distortion in small samples.

Simulation shows improved accuracy over traditional tests.

Empirical example demonstrates practical applicability.

Abstract

We address the issue of performing testing inference in generalized linear models when the sample size is small. This class of models provides a straightforward way of modeling normal and non-normal data and has been widely used in several practical situations. The likelihood ratio, Wald and score statistics, and the recently proposed gradient statistic provide the basis for testing inference on the parameters in these models. We focus on the small-sample case, where the reference chi-squared distribution gives a poor approximation to the true null distribution of these test statistics. We derive a general Bartlett-type correction factor in matrix notation for the gradient test which reduces the size distortion of the test, and numerically compare the proposed test with the usual likelihood ratio, Wald, score and gradient tests, and with the Bartlett-corrected likelihood ratio and score tests. Our simulation results suggest that the corrected test we propose can be an interesting alternative to the other tests since it leads to very accurate inference even for very small samples. We also present an empirical application for illustrative purposes.