Suppose that $X_{1}, \ldots, X_{n}$ is a sample of size $n$ with common $N\left(\mu_{X}, 1\right)$ distribution, and $Y_{1}, \ldots, Y_{n}$ is an independent sample of size $n$ from a $N\left(\mu_{Y}, 1\right)$ distribution.

(i) Find (with careful justification) the form of the size- $\alpha$ likelihood-ratio test of the null hypothesis $H_{0}: \mu_{Y}=0$ against alternative $H_{1}:\left(\mu_{X}, \mu_{Y}\right)$ unrestricted.

(ii) Find the form of the size- $\alpha$ likelihood-ratio test of the hypothesis

$$H_{0}: \mu_{X} \geqslant A, \mu_{Y}=0$$

against $H_{1}:\left(\mu_{X}, \mu_{Y}\right)$ unrestricted, where $A$ is a given constant.

Compare the critical regions you obtain in (i) and (ii) and comment briefly.