The Classical Trinity of Tests

Derives Wald's test, the likelihood ratio test, and the score test from a single picture: the log-likelihood curve's local quadratic (inverted-parabola) approximation around the MLE, with an annotated figure showing Wald as a horizontal distance, score as a tangent slope, and LRT as a vertical drop, all on one curve. A worked binomial example (n=20, x=3, testing p=0.5) shows real finite-sample disagreement, then two verified failure modes for Wald specifically: it is not invariant to reparameterization (testing on the log-odds scale changes the statistic from 19.2 to 7.7 on identical data, while score and LRT don't move), and it diverges to infinity at a data boundary that score and LRT read as weak evidence. A 300,000-rep simulation shows Wald over-rejecting a true null 43.5% of the time at n=8 versus a nominal 5%, and a local-alternative convergence table confirms the promised asymptotic equivalence actually emerging as n grows. Closes with practical guidance on which test to reach for and when, plus a brief multivariate generalization.