The Lagrange multiplier test, likelihood ratio test, and Wald test are all related and have similar power characteristics. By setting critical values to have equal type I error probabilities, type II error probabilities will also be equal, avoiding conflicting results. Exact critical values are important, as using asymptotic values can lead to discrepancies. While determining exact critical values for small samples can be challenging, under normality assumptions, the Wald test can be transformed into an F-test. This allows for exact finite sample testing without the risk of conflicting outcomes.