Likelihood ratio interpretation I have $X_1, \ldots, X_n$ and $Y_1, \ldots, Y_n$ as as random samples from two normal distributions with means $0$ and variances $\theta_1$ and $\theta_2$ respectively.The null hypothesis is $\theta_1 = \theta_2$ and the alternative is $\theta_1$ not equal to $\theta_2$ I calculated the likelihood ratio (which is shown below) and now I am trying to figure out what this likelihood ratio is a function of. I believe it is a function of $F$ but I am unsure how to show that it is $F$-distributed with $v_1 = n$, and $v_2 = m$. Thanks for the help.
$$
\lambda={   
{  
\left\{
{\textstyle 1\over\textstyle  2\pi\bigl[\,(\,\sum x_i^2+\sum y_i^2\,)/(n+m)\, \bigr]}
\right\}^{n+m\over2}
}
\over
\biggl[{    {\textstyle1\over\textstyle 2\pi(\sum x_i^2 /n)}      }\biggl]^{n/2}
\biggl[{    {\textstyle1\over\textstyle 2\pi(\sum y_i^2 /m)}      }\biggl]^{m/2}
}
$$
 A: It is just algebra.
Ok, you have
$$\lambda = \frac{\left[\frac{\sum x_i^2}{n}\right]^{n/2} \left[\frac{\sum y_i^2}{m}\right]^{m/2}}
{\left[\frac{\sum x_i^2 + \sum y_i^2}{m+n}\right]^{\frac{m+n}{2}}}$$
We can easily factor this
$$\lambda = \left[\frac{\frac{\sum x_i^2}{n}}
{\frac{\sum x_i^2 + \sum y_i^2}{m+n}}\right]^{n/2}
\left[\frac{\frac{\sum y_i^2}{m}}
{\frac{\sum x_i^2 + \sum y_i^2}{m+n}}\right]^{m/2}$$
Now multiply  by the appropriate  power of $\frac{\sum y_i^2}{\sum y_i^2}$ to get
$$\lambda = \left[\frac{(m+n)\frac{\sum x_i^2}{\sum y_i^2}}{n\left(1+\frac{\sum x_i^2}{\sum y_i^2}\right)}\right]^{n/2}
\left[\frac{(m+n)}{m\left(1+\frac{\sum x_i^2}{\sum y_i^2}\right)}\right]^{m/2}$$
Now we know that $\frac{m}{n}\frac{\sum x_i^2}{\sum y_i^2}$ has a $F_{n,m}$ distribution (it is a ratio of two independent random variables having $\chi^2$ distributions) so we can write  $\lambda$ as
$$\lambda = \left[\frac{\frac{(n+m)n}{m}F_{n,m}}{
\frac{n^2}{m}\left(F_{n,m}+\frac{m}{n}\right)
}\right]^{n/2}\left[\frac{m+n}{n\left(F_{n,m}+\frac{m}{n}\right)}\right]^{m/2}$$
Now this has $\lambda$ as a function of $F$.  Check my algebra.
To be useful it should be a monotone function of $F$.  That it is is not immediately clear to me.
