A New Look at $F$-Tests

TL;DR

This paper demonstrates that in several classical statistical testing scenarios, directional inference based on higher order likelihood approximations simplifies to the traditional F-test, unifying different testing approaches.

Contribution

It shows that in common statistical tests, directional inference aligns with the traditional F-test, simplifying calculations and providing new insights into their equivalence.

Findings

Directional inference simplifies to F-test in classical cases.

Examples include tests on variances, rates, and linear constraints.

Unifies likelihood-based and classical testing methods.

Abstract

Directional inference for vector parameters based on higher order approximations in likelihood inference has recently been developed in the literature. Here we explore examples of directional inference where the calculations can be simplified. It is found that in several classical situations the directional test is equivalent to the usual $F$-test. These situations include testing the ratio of group variances and rates from normal and exponential distributions, linear regression with a linear constraint and Hotelling's $T^2$-test.