The $\text{Pareto}(1, \alpha)$ MLE is $\frac{n}{\sum_{i = 1}^{n} \log x_{i}}$, where $X_{1}, X_{2}, ..., X_{n}$ are i.i.d. with $f(x | \alpha) = \alpha x^{-\alpha - 1}$. Further, add the constraint that $\alpha > 2$.
This MLE may be less than 2. What's the proper way to handle that?
If the parameter space for $\alpha$ included 2, then we could simply say define $\hat{\alpha} = 2$ if $\frac{n}{\sum_{i = 1}^{n} \log x_{i}} \le 2$. But the parameter space is an open interval that doesn't include 2, so an MLE of 2 would be outside the parameter space.