# 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.

To develop StubbornAtom’s comment, if $$XiX_i$$ is i.i.d. uniformly distributed on $$[0,θ][0,\theta]$$

and you have $$nn$$ samples then the maximum likelihood estimator of $$θ\theta$$ is $$ˆθn=max\hat{\theta}_n=\max\limits_{1\le i \le n}X_i$$.

$$\hat{\theta}_n\hat{\theta}_n$$ has a $$\mathrm{Beta}(n,1)\mathrm{Beta}(n,1)$$ distribution scaled by $$\theta\theta$$.

As $$nn$$ increases, $$n\left(\theta-\hat \theta_n\right)n\left(\theta-\hat \theta_n\right)$$ converges in distribution to $$\mathrm{Exp}\left(\frac1\theta\right)\mathrm{Exp}\left(\frac1\theta\right)$$, not a normal distribution.

or in a handwaving sense, for large $$nn$$, the maximum likelihood estimator $$\hat{\theta}_n\hat{\theta}_n$$ approximately has a reversed and shifted exponential distribution with density $$\frac{n x^{n-1}}{\theta^n} \approx \frac n{\theta} \exp\left(\frac{nx}{\theta}-n\right)\frac{n x^{n-1}}{\theta^n} \approx \frac n{\theta} \exp\left(\frac{nx}{\theta}-n\right)$$ when $$0 < x \le \theta0 < x \le \theta$$ looking like 