Boundary likelihood-ratio test for two Gaussian means ($T=n[{\bar{Y}}^2+(A-\bar{X}{)}_{+}^2]$)

For independent unit-variance Gaussian samples with means ${\mu }_X,{\mu }_Y$, testing ${\mu }_X\ge A,{\mu }_Y=0$ against unrestricted means gives $T=-2\log \Lambda =n[{\bar{Y}}^2+(A-\bar{X}{)}_{+}^2]$. The unrestricted maximum-likelihood estimators are the two sample means, while the null estimators are $\max (A,\bar{X})$ and zero. Under the null, write $W=\sqrt{n}(\bar{X}-A)=W_0+\delta$, $\delta =\sqrt{n}({\mu }_X-A)\ge 0$, and $Z=\sqrt{n}\bar{Y}$, with $W_0,Z$ independent standard Gaussian variables. Then $T=Z^2+(-W_0-\delta {)}_{+}^2$ decreases pointwise with $\delta$, so the largest rejection probability occurs at the boundary ${\mu }_X=A$. There its law is the chi-bar-squared distribution $\frac{1}{2}{\chi }_1^2+\frac{1}{2}{\chi }_2^2$. The exact size-$\alpha$ threshold $c_{\alpha }$ is therefore determined by $1-\Phi (\sqrt{c_{\alpha }})+\frac{1}{2}{e}^{-c_{\alpha }/2}=\alpha$.