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.

**Answer**

To develop StubbornAtom’s comment, if Xi is i.i.d. uniformly distributed on [0,θ]

and you have n samples then the maximum likelihood estimator of θ is ˆθn=max.

\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

**Attribution***Source : Link , Question Author : Zen , Answer Author : Henry*