Single-parameter boundary likelihood-ratio test ($\frac{1}{2}{\delta }_0+\frac{1}{2}{\chi }_1^2$)

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.