One-sided likelihood-ratio test for two normal variances ($F=(S_X/(n_X-1))/(S_Y/(n_Y-1))$)

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 isIt 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.