Three classical approaches to goodness-of-fit testing are Rao’s test, Wald’s test, and the likelihood-ratio test. The asymptotic equivalence of these three tests under the null hypothesis is a famous connection in statistical detection theory. We revisit these three likelihood-related tests from a non-asymptotic viewpoint under self-concordance assumptions. We recover the equivalence of the three tests and characterize the critical sample size beyond which the equivalence holds asymptotically. We also investigate their behavior under local alternatives. Along the way, we establish an estimation bound that matches the misspecified Cramér-Rao lower bound. We illustrate the interest of our results using generalized linear models and score matching with exponential families.