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For a scalar parameter restricted to $\psi \ge 0$, test $\psi =0$ against $\psi >0$ with identifiable interior nuisance parameters. A locally quadratic regular likelihood has an efficient null score $Z\sim N(0,1)$, and the constrained fit projects the unconstrained local maximizer onto the nonnegative half-line. ConsequentlyFor a positive statistic, its asymptotic upper-tail p-value is half the ordinary one-degree chi-squared tail. Its 5% critical value is the 90th chi-squared percentile, approximately 2.7055. Additional boundary nuisance parameters or nonidentified mixture parameters can change this law.

Uncertainty associated with imperfect knowledge of calibration, efficiency, backgrounds or other persistent model inputs. Repeating the same measurement does not necessarily average it away. Correlated nuisance parameters can propagate it through a likelihood; estimating such an uncertainty is distinct from asserting that the systematic error has a known sign and magnitude.