= Single-parameter boundary likelihood-ratio test
{title2=$\tfrac12\delta_0+\tfrac12\chi^2_1$}
= One-sided boundary likelihood-ratio limit
{synonym}
For a scalar parameter restricted to $\psi\ge0$, 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. Consequently
$$
2(\ell_{\rm full}-\ell_{\rm null})\Rightarrow(\max(0,Z))^2
\sim\tfrac12\delta_0+\tfrac12\chi^2_1.
$$
For 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.
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