I apologize for the basic question. If $\{p_\theta(x): \theta\in K\subseteq\mathbb{R}\}$ is a smooth family of distributions, then the MLE $\hat{\theta}_n,$ under suitable regularity conditions satisfies $\sqrt{n}(\hat{\theta}_n-\theta)\to\mathcal{N}(0,I(\theta)^{-1})$ where $I(\theta)$ is the Fisher information. We also know from the Cramer-Rao bound that no estimator can outperform this estimator in the sense of having a smaller mean squared error (except on a set of zero Lebesgue measure, the superefficiency phenomenon).

I am interested in knowing similar results for other families of distributions which do not satisfy the above conditions. The canonical examples are: $\{\mathrm{Uniform}[\theta,\theta+1]: \theta\in\mathbb{R}\}$ and $\{\mathrm{Uniform}[0,\theta]: \theta>0\}.$ In these cases, the MLE estimates the parameter to within $O(1/n),$ since in the former case, $\hat{\theta}_n = \min\{X_1,X_2,\ldots, X_n\}$ and $n(\theta-\hat{\theta}_n)\to -Z,$ where $Z\sim\mathrm{exp}(1)$ and in the latter case, $\hat{\theta}_n = \max\{X_1,X_2,\ldots, X_n\}$ and $n(\theta-\hat{\theta}_n)\to W,$ where $W\sim\mathrm{exp}(\frac{1}{\theta}).$

Do we know

(1) If MLE is always an order-optimal estimator in the sense that it estimates $\theta$ to $O(\frac{1}{n^\alpha})$ where $\alpha$ is the best possible over all estimators?

(2) If MLE is not just order-optimal but optimal in a suitable sense similar to that of the Cramer-Rao lower bound, in that, any other estimator can outperform MLE only on a set of Lebesgue measure 0?

(3) Any general results on the distribution of convergence of $n^\alpha(\theta-\hat{\theta}_n)$ where $O\left(\frac{1}{n^\alpha}\right)$ is the rate of the convergence of the estimator?


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