# Likelihood ratio test for $H_0:(\mu_1,\mu_2)=(0,0)$ vs $H_1:(\mu_1,\mu_2) \neq (0,0)$

There are $$X_1, X_2$$ where $$X_i \sim N(\mu_i,1), i=1,2$$. They are independent. The question is

Find the likelihood ratio test with $$H_0:(\mu_1,\mu_2)=(0,0), H_1:(\mu_1,\mu_2) \neq (0,0)$$. The significance level is $$\alpha (0< \alpha <1)$$ and parameter space $$\Omega$$ is $$\Omega = \left\{ (\mu_1,\mu_2) : \mu_1 \geq 0, \mu_2 \geq 0\right\}$$

My solution is $$X_1^2 \geq \chi_p^2(1)$$ or $$X_2^2 \geq \chi_q^2(1)$$ or $$X_1^2+X_2^2 \geq \chi_r^2(2)$$ where $$p+q+r=\alpha$$. Is it right?

Detail of my solution :

Let $$\mu = (\mu_1, \mu_2)^T$$. Then $$\hat{\mu}^{\Omega_0}=(0,0)$$ and $$\hat{\mu}^{\Omega} = \left(\max\{x_1, 0\}, \max\{x_2,0\}\right)$$ because parameter space is not $$\mathbb{R}^2$$.

Then I calculated $$\Lambda = 2[l(\hat{\mu}^{\Omega}) - l(\hat{\mu}^{\Omega_0})]$$ to find the rejection region from $$\Lambda \geq \lambda (\lambda > 0)$$.

After some algebra, I got $$\Lambda = x_1^2I_{(x_1>0, x_2<0)} + x_2^2I_{(x_1<0, x_2>0)} + (x_1^2+x_2^2)I_{(x_1>0, x_2>0)}$$.

Under the null hypothesis, $$X_i^2 \sim \chi^2(1)(i=1,2)$$ so $$X_1^2+X_2^2 \sim \chi^2(2)$$.

Finally I got the above rejection region.

• Thanks. I added the tag and detail. Oct 25, 2020 at 4:29
• Can you say something about how do you get the MLE of $(\mu_1, \mu_2)$? Jan 15, 2022 at 21:23

Your solution seems to be correct. The strange shape of your parameter space (it's not a open subset of $$\mathbb R^2$$) creates this ambiguity in the final result: each combination of $$(p,q,r)$$ gives a different LRT. Some are more powered for $$\mu_1=0$$, some are more powered for $$\mu_2=0$$ and some are more powered for $$\mu_1\neq0,\mu_2\neq0$$, but all of them are valid LRTs with significance level $$\alpha$$.
• @flossy that's correct. also, there is no uniformly most powerful test in this setting, since the LRT with $p=1$ is MP if $\mu_2=0$ and the LRT with $q=1$ is MP if $\mu_1=0$. Oct 27, 2020 at 4:00