Past exam of the statistics course of the University of Oxford / 2021 / 1 / b

Put $A=-{\sum }_i\log x_i>0$, $B=-{\sum }_j\log y_j>0$, and $N=n+m$. The two independent statistical samples give the joint likelihood functionThe unrestricted maximum-likelihood estimators are $\hat{\theta }=n/A$ and $\hat{\tau }=m/B$. Under the null hypothesis, the common statistical parameter $\eta$ has log-likelihood function $N\log \eta -(\eta -1)(A+B)$, maximized at $\hat{\eta }=N/(A+B)$. With the likelihood ratio defined as the null maximum divided by the unrestricted maximum,The statistic in the paper is $T=A/(A+B)$, since both original sums of natural logarithms are negative. HenceIt has maximum one at $T=n/N$, corresponding to equal fitted rates. This is the likelihood-ratio test for equality of beta shape parameters.

A likelihood-ratio test rejects for small $\lambda$. The derivative of the natural logarithm of its variable factor isThus $t^n(1-t{)}^m$ increases up to $n/(n+m)$ and decreases thereafter, tending to zero at both endpoints. Every nontrivial lower level set of $\lambda$ therefore consists of two tails. For a prescribed test size $0<\alpha <1$, the exact critical region isWriting $F_{n,m}$ for the cumulative distribution function of the beta distribution with parameters $(n,m)$, the required two equations areThe first makes both cutoffs have the same likelihood ratio; the second calibrates the null hypothesis rejection probability. Equivalently, the second left-hand side isAs the common likelihood ratio threshold increases from zero to one, this probability increases continuously and strictly from zero to one; hence the equations determine unique cutoffs. The two tail probabilities need not each equal $\alpha /2$ when $n\ne m$. Exact calibration holds for every common rate because the given null distribution of $T$ is independent of that rate. Boundary equalities have zero probability.