MLE and non-normality What is a non-trivial example of an identifiable model whose MLE is consistent, but the MLE's asymptotic distribution is not normal? Parametric setting and IID sample would be desirable.
 A: Theorems showing asymptotic normality of the MLE generally hinge on the MLE being in in the interior of the parameter space and the information function existing.  Moran (1971) examines the MLE under non-standard conditions, including when the parameter is on the boundary.  That might be a good place to look if you would like a non-trivial example.
A: To develop StubbornAtom's comment,  if $X_i$ is i.i.d. uniformly distributed on $[0,\theta]$
and you have $n$ samples then the maximum likelihood estimator of $\theta$ is $\hat{\theta}_n=\max\limits_{1\le i \le n}X_i$.
$\hat{\theta}_n$ has a $\mathrm{Beta}(n,1)$ distribution scaled by $\theta$.
As $n$ increases, $n\left(\theta-\hat \theta_n\right)$ converges in distribution to $\mathrm{Exp}\left(\frac1\theta\right)$, not a normal distribution.
or in a handwaving sense, for large $n$, the maximum likelihood estimator $\hat{\theta}_n$ approximately has a reversed and shifted exponential distribution with density $\frac{n x^{n-1}}{\theta^n} \approx \frac n{\theta} \exp\left(\frac{nx}{\theta}-n\right)$ when $0 < x \le \theta$ looking like

