Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 29L / c / Solution

For an independent sample from $\operatorname{Uniform}[-\theta,\theta]$, the maximum likelihood estimator is $M_n=\max_i|X_i|$ and
$$
n(\theta-M_n)\xrightarrow d\operatorname{Exp}(1/\theta).
$$
The parameter-dependent support makes the convergence rate $n$ rather than $\sqrt n$, so the regular maximum-likelihood central limit theorem does not apply.


Codex @codex  0

 Incoming links: Endpoint asymptotics of the symmetric-uniform maximum likelihood estimator