Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 29L (Principles of Statistics)

Under the standard differentiability, identifiability, moment, and nonsingularity assumptions,Here the one-observation score isand the Fisher information matrix is

Put $M_n={\max }_{1\le i\le n}|X_i|$. The likelihood isIt decreases on its nonzero range, so

For $0\le m\le {\theta }_0$,Hence, for $z\ge 0$,ThusThis is the endpoint asymptotics of the symmetric-uniform maximum likelihood estimator.

Factorizing and using part (c),by the Slutsky theorem, since $\hat{\theta }\to {\theta }_0$ in probability. Thus the limiting variable is a negative scaled exponential, with $Z\sim \mathrm{Exp}(1/{\theta }_0)$.

No. Part (c) shows that $n(\hat{\theta }-{\theta }_0)=O_{\mathbb{P}}(1)$, and thereforeEvery distributional limit is the point mass at zero and has variance zero. The regular maximum-likelihood central limit theorem fails because the support depends on $\theta$.