Past exam of the mathematics course of the University of Cambridge / 2020 / ib / Paper 1 / 19H (Statistics)

Let $M=X_{(n)}={\max }_i{X}_i$. The likelihood function isOn its feasible range $\theta \ge M/2$, this is decreasing in $\theta$. Therefore the maximum likelihood estimator is

A statistic $T$ is sufficient for $\theta$ if the conditional distribution of the full sample given $T$ does not depend on $\theta$. It is minimal sufficient if it is a function of every sufficient statistic.

The likelihood factors asso the Fisher-Neyman factorization theorem shows that $M$, and hence its one-to-one transform $\hat{\theta }=M/2$, is sufficient. Moreover, for two samples $\mathbf{x}$ and $\mathbf{y}$, the ratio $L(\theta ;\mathbf{x})/L(\theta ;\mathbf{y})$ is independent of $\theta$ exactly when their maxima agree. The likelihood-ratio criterion for minimal sufficiency therefore shows that $M$ and $\hat{\theta }$ are minimal sufficient.

For $n\ge 2$, the sample mean $\tilde{\theta }$ is not sufficient: two samples can have the same mean but different maxima, and their likelihood ratio then depends on $\theta$ through the support indicators. It is consequently not minimal sufficient. For the degenerate special case $n=1$, the sample mean and maximum coincide and both conclusions reverse.

The scaled maximumhas cumulative distribution function $\mathbb{P}(U\le u)=u^n$ for $0\le u\le 1$. ThusThe mean squared error of the maximum likelihood estimator is therefore

Each $X_i$ from the uniform distribution on $[0,2\theta ]$ has mean $\theta$ and variance ${\theta }^2/3$. The sample mean is unbiased and has variance ${\theta }^2/(3n)$, hence

Because $0\le U\le 1$, for fixed $t$ and sufficiently large $n$,ThereforeEquivalently, $n(1-\hat{\theta }/\theta )$ converges in distribution to an exponential distribution of rate one.

Put $c=\log (1/\alpha )$. The limiting probabilityand the certain inequality $\hat{\theta }\le \theta$ suggest the approximate confidence intervalSince $\mathbb{E}[\hat{\theta }]=n\theta /(n+1)$, its expected length isas required.