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Take two independent statistical samples from beta distributions with parameters $(\theta ,1)$ and $(\tau ,1)$, of positive sizes $n,m$. Put $A=-{\sum }_i\log X_i$, $B=-{\sum }_j\log Y_j$, and $T=A/(A+B)$. To test the null hypothesis $\theta =\tau$, divide the maximized common-rate likelihood function by the unrestricted maximum to obtain the displayed likelihood ratio. Indeed the unrestricted fitted rates are $n/A,m/B$, and the common fitted rate is $(n+m)/(A+B)$; substitution cancels the exponential factors.

Under the null hypothesis, $A$ and $B$ have independent gamma distributions with a common rate. Beta-gamma independence gives $T\sim \mathrm{Beta}(n,m)$. The derivative of $\log (T^n(1-T{)}^m)$ changes sign at $n/(n+m)$, so the exact critical region of size $\alpha$ is $T<k_1$ or $T>k_2$, with $k_1<n/(n+m)<k_2$. Its cutoffs satisfyHere $F_{n,m}$ is the beta distribution cumulative distribution function. Equal likelihood ratios at the cutoffs do not generally mean equal tail probabilities. Increasing the ratio threshold continuously adds positive null distribution mass on both sides, proving uniqueness for $0<\alpha <1$.

Set $U=X/(X+Y)$ and $V=X+Y$. Their inverse change of variables is $X=UV$, $Y=(1-U)V$, with domain $0<U<1$, $V>0$. The absolute Jacobian determinant of this inverse transformation isUse the independence of random variables to multiply the two original Gamma distribution densities. The change of variables formula then gives the joint probability densityFactor this into two normalized densities:The support is a product domain, and the factors are precisely a Beta distribution density and a Gamma distribution density. ThereforeThis beta-gamma independence uses the common rate $\theta$; it is a rate parameter, not the scale parameter $1/\theta$.