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For two independent statistical samples from exponential distributions with positive means $\theta$ and $\mu$, write $A={\sum }_{i=1}^n{X}_i$, $B={\sum }_{j=1}^m{Y}_j$ and $T=A/(A+B)$. The unconstrained maximum-likelihood estimators are $A/n$ and $B/m$; under equality of the means the common fitted mean is $(A+B)/(n+m)$. Dividing the constrained maximum likelihood function by its unrestricted maximum gives the displayed likelihood ratio. Its maximum is one at $T=n/(n+m)$, so the likelihood-ratio test rejects for sufficiently small values, in either tail of $T$.

Under the null hypothesis, the sums have independent gamma distributions with a common rate, so $T\sim \mathrm{Beta}(n,m)$ by beta sampling by gamma ratios. This permits exact calibration. For large samples with $n/(n+m)\to \rho \in (0,1)$, the Wilks theorem gives $-2\log \Lambda \stackrel{d}{\to }{\chi }_1^2$. Its upper-tail probability is the asymptotic p-value. The one statistical degree of freedom is the difference between the unrestricted and constrained parameter dimensions.

For Beta sampling by gamma ratios, use either preceding chi-squared distribution algorithm to generate independent $S\sim {\chi }_{2\alpha }^2$ and $T\sim {\chi }_{2\beta }^2$, and returnWrite $W=S+T$, so $S=WR$ and $T=W(1-R)$. The Jacobian determinant is $W$. The joint probability density function after this change of variables factors asIntegrating out $w$ proves the Beta distribution for $R$.

A new method is Beta sampling by uniform order statistics. Generate $m=\alpha +\beta -1$ independent uniforms, sort them, and return the $\alpha$th order statistic. There are $\alpha -1$ uniforms below the selected value and $\beta -1$ above it. Selecting their labels and placing the selected observation in $dr$ gives densitywhich is the Beta distribution density. The integrality of the shape parameters is what makes this order-statistic construction possible.