The maximum likelihood estimate that is not normal

Question

For most models the maximum likelihood estimator is asymptotically normal with variance set by the Fisher information. But consider i.i.d. observations $X_1, \dots, X_n$ from Uniform $(0, \theta)$ , where the maximum likelihood estimator is the sample maximum $\hat\theta = X_{(n)}$ .

Show that $\hat\theta$ is not asymptotically normal, find its convergence rate and limiting distribution, and explain why the usual theory fails.

Accepted Answer

Why the standard theory does not apply. The asymptotic-normality result for maximum likelihood estimators requires regularity conditions, chief among them that the support of the distribution does not depend on the parameter. Here the support is [0, 	heta], which moves with 	heta, so the Fisher-information machinery simply does not apply. Something different happens. Find the limiting distribution. Look at how far the maximum falls below the ceiling. For y 0, P\big(n(	heta - X{(n)}) y\big) = P\Big(X{(n)} < 	heta - 	frac{y}{n}\Big) = \left(\frac{	heta - y/n}{	heta}ight)^n = \left(1 - …

The maximum likelihood estimate that is not normal

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