In class, we talked about finding Maximum Likelihood Estimators but there is something I don't think we talked about. How is the MLE different for distributions with constrained bounds?
For example, for an iid random sample $X_1,\ldots,X_n$, if we compare the MLE for $\theta$ from a Bernoulli distribution versus the MLE for $\theta$ from another Bernoulli distribution but with 0 < $\theta$ < $\frac 12$, how is the MLE affected?
I know the MLE for a Bernoulli($\theta$) with 0 < $\theta$ < 1 is $\frac{\Sigma x_i}{n}$.
I thought it was the same for the 0 < $\theta$ < $\frac 12$ case but thinking back to a Uniform distribution, the MLE will vary depending on the bounds. However, I don't think order statistics is helpful here. Does anyone have any ideas?