Likelihood ratio test of two sample normal distributed with known means(which is 0) and unknown variance

Let $X_1,X_2,\cdots ,X_n$ be random sample form $N~(0,\theta_1)$ and Let $Y_1,Y_2,\cdots ,Y_m$ be random sample form $N~(0,\theta_2)$. Determine the $\lambda$, the likelihood ratio test in testing $H_0 : \theta_1=\theta_2$ and $H_1 : \theta_1 \neq \theta_2$. What F statistic is used in this test?

• i have already shown that $\lambda = (\frac{n+m}{\sum(x_i^2)+\sum(y_i^2)})^{\frac{n+m}{2}}$ $\times$ $\frac{\sum(x_i^2)^{n/2} \sum(y_i^2)^{m/2}}{n^{\frac{n}{2}}m^{\frac{m}{2}}}$ but i am in trouble to determine what F statistic is used in this test. Can i assume that $\frac{\sum(x_i^2)}{\theta_1}$ is chi square (n)? – cavvot May 12 '18 at 13:36
• and $\frac{\sum(x_i^2)}{\theta_1}$ divided by $\frac{\sum(y_i^2)}{\theta_1}$ is F (n.m) ? – cavvot May 12 '18 at 13:41