= One-sided likelihood-ratio test for two normal variances
{title2=$F=(S_X/(n_X-1))/(S_Y/(n_Y-1))$}
For independent samples from <normal distributions> with separate unknown means, test equal variances against $\sigma_X^2>\sigma_Y^2$. Profiling the means gives sample means. Under the null the variance maximum is $(S_X+S_Y)/(n_X+n_Y)$; the separate maxima are $S_X/n_X$ and $S_Y/n_Y$. If the latter obey the proposed order, their likelihood quotient is
$$
\Lambda=\frac{(S_X/n_X)^{n_X/2}(S_Y/n_Y)^{n_Y/2}}{[(S_X+S_Y)/(n_X+n_Y)]^{(n_X+n_Y)/2}}.
$$
It decreases strictly as $(S_X/n_X)/(S_Y/n_Y)>1$ increases. Otherwise the ordered alternative maximum lies on the equal-variance boundary and the quotient is one. Under the null, <Cochran's theorem> gives independent chi-squared residual sums with degrees $n_X-1,n_Y-1$, so rejection is in the upper tail of the displayed <F-distribution> statistic, calibrated at the desired significance level.
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