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As in 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 $$\Omega = \left\{ (\mu_1,\mu_2) : \mu_1 \geq 0, \mu_2 \geq 0\right\}$$

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What is the distribution of $\Lambda$?

We have known that $$\Lambda = x_1^2I_{(x_1>0, x_2<0)} + x_2^2I_{(x_1<0, x_2>0)} + (x_1^2+x_2^2)I_{(x_1>0, x_2>0)}$$

You also know the distribution of $x_1$ and $x_2$ when $H_0$ is true. It's a bivariate normal distribution centered at $0,0$ and with variance $I$. Use that to get the distribution of $\Lambda$.

Because of the indicator functions you will get to handle it as a mixture distribution. In 25% of the cases you get $I_{(x_1>0, x_2<0)} = 1$, in 25% of the cases you get $I_{(x_1<0, x_2>0)} = 1$, in 25% of the cases you get $I_{(x_1>0, x_2>0)} = 1$, and in 25% of the cases $\Lambda = 0$.

Below is an illustration that may help to visualize it. There's $10^4$ points simulated that illustrate the bivariate distribution of $x_1,x_2$. Two example points are drawn with isolines for which the value of $\Lambda$ is constant.

illustration

So for example the probability that $P(\Lambda \leq 1.92)$ consist of the sum of $$P(\Lambda \leq 1.92) = \underbrace{P(x_1<=0,x_2<=0)}_{= 0.25}+ \underbrace{P(x_1>0,x_2<=0 \text{ and } x_1^2 \leq 1.92)}_{= 0.25 \times P(\chi_1^2 <1.92)}+ \underbrace{P(x_1<=0,x_2>0 \text{ and } x_2^2 \leq 1.92 )}_{= 0.25 \times P(\chi_1^2 <1.92)}+ \underbrace{P(x_1>0,x_2>0 \text{ and } x_1^2 + x_2^2 \leq 1.92)}_{= 0.25 \times P(\chi_2^2 <1.92)} $$

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  • $\begingroup$ I think that $X_1^2 \sim \chi_1^2$ and $X_1^2+X_2^2\sim \chi_2^2$? Didn't you use that result? $\endgroup$
    – Hermi
    Commented Jan 16, 2022 at 20:35
  • $\begingroup$ I only gave a hint. Indeed you will get some mixture of 50% a $\chi_1^2$, 25% a $\chi_2^2$ distribution (and 25% a point mass). $\endgroup$ Commented Jan 16, 2022 at 20:37
  • $\begingroup$ Can you write the cdf function $F(x)=P(\Lambda\le x)$ in a formal way? I am not sure what do you mean $25\%$ cases. $\endgroup$
    – Hermi
    Commented Jan 16, 2022 at 20:37
  • $\begingroup$ Is this a piecewise function? I mean that $F(x)=0$ as $x_\le 0, x_2\le 0$, $F(x)=\chi_1^2$ as $x_1 \ge 0, x_2\le 0$ or $x_1 \le 0, x_2\ge 0$, and $F(x)=\chi_2^2$ as $x_1 \ge 0, x_2\ge 0$? Is this one right? $\endgroup$
    – Hermi
    Commented Jan 16, 2022 at 21:32
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    $\begingroup$ The latter one is close but there's two things: 1 you need to consider $x\leq0$ seperately, 2 you can combine two terms $$F(x)=\begin{cases}0.25+0.5P(\chi_1^2\le x)+0.25P(\chi_2^2\le x) &\quad \text{if $x>0$} \\ 0 &\quad \text{if $x\leq 0$} \end{cases}$$ $\endgroup$ Commented Jan 16, 2022 at 22:05

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