Small‐sample testing inference in symmetric and log‐symmetric linear regression models

This paper deals with the issue of testing hypotheses in symmetric and log‐symmetric linear regression models in small and moderate‐sized samples. We focus on four tests, namely, the Wald, likelihood ratio, score, and gradient tests. These tests rely on asymptotic results and are unreliable when the sample size is not large enough to guarantee a good agreement between the exact distribution of the test statistic and the corresponding chi‐squared asymptotic distribution. Bartlett and Bartlett‐type corrections typically attenuate the size distortion of the tests. These corrections are available in the literature for the likelihood ratio and score tests in symmetric linear regression models. Here, we derive a Bartlett‐type correction for the gradient test. We show that the corrections are also valid for the log‐symmetric linear regression models. We numerically compare the various tests and bootstrapped tests, through simulations. Our results suggest that the corrected and bootstrapped tests exhibit type I probability error closer to the chosen nominal level with virtually no power loss. The analytically corrected tests as well as the bootstrapped tests, including the Bartlett‐corrected gradient test derived in this paper, perform with the advantage of not requiring computationally intensive calculations. We present a real data application to illustrate the usefulness of the modified tests.