# 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.

• Does $X_i$ i.i.d Uniform on $(0,\theta)$ count as non-trivial? Oct 23 at 5:39
• It certainly does! And thank you for being such a stubborn and intelligent bunch of atoms. ;-)
– Zen
Oct 23 at 16:28

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

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.

• Thanks, Ben! I'll check the reference.
– Zen
Oct 23 at 16:02