The asymptotic normality theorem (see the answer to https://stats.stackexchange.com/questions/79655/simple-condition-for-asymptotic-normality-of-mle ) demands that the true parameter be an interior point of the parameter space.

Why do we need such a restriction? And, how can we check if the condition is met since we never know the real parameter?


Hand waving explanations:

  1. If $\theta$ lies on $\partial \Theta $, then the convergence of its ML estimator is only on the interior of $\Theta$, as such its limiting distribution is asymmetric, hence cannot be normal.

  2. We usually know or assume. Let us say that you are looking at some shifted distribution, then clearly the shifting parameter lies on the boundary of $\Theta$. Otherwise, for many application the $\Theta=\mathbb{R}$, where no such problem occurs. In other cases, boundary values of the parameter of interest will produce degenerate distributions resulting in no variation in your data set. Hence, in many applications it is clear whether $\theta$ is on $\partial \Theta$ or not.

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