Past exam of the statistics course of the University of Oxford / 2017 / 2 / c

Put $A={\sum }_i{X}_i$, $B={\sum }_j{Y}_j$ and $N=n+m$. Independence gives the likelihood functionThe unrestricted maximum-likelihood estimators are $\hat{\theta }=A/n$ and $\hat{\mu }=B/m$. Under the null hypothesis, the common mean $\eta =\theta =\mu$ maximizes ${\eta }^{-N}{e}^{-(A+B)/\eta }$ at $\hat{\eta }=(A+B)/N$. These are maxima because each one-parameter score function has the sign of its observed sum minus its sample size times its proposed mean.

Define the generalized likelihood ratio to be the constrained maximum divided by the unrestricted maximum. The exponential factors cancel, givingSince $T=A/(A+B)$ and $1-T=B/(A+B)$,It has maximum one at $T=n/N$, when the two fitted means agree, and is small toward either endpoint. A likelihood-ratio test rejects for small $\Lambda$, or equivalently large $W=-2\log \Lambda$. This is the likelihood-ratio test for equality of exponential means.

Under the null hypothesis, the full model has two free positive statistical parameters and the constrained model has one. The Wilks theorem therefore givesas both sample sizes tend to infinity, with $n/(n+m)$ tending to a number strictly between zero and one. For the observed $t$, calculateThe large-sample p-value is the upper-tail probabilitywhere $\Phi$ is the cumulative distribution function of the standard normal distribution. The last equality uses the fact that the square of a standard normal random variable has a chi-squared distribution with one statistical degree of freedom. Values of $T$ on either side of $n/(n+m)$ are already included by the transformation to $W$.