# Do I get the nice asymptotic properties of MLE when I restrict the parameter space?

I would like to know if the MLE is still consistent, asymptotically normal, and efficient when I put restrictions on the parameter space.

I think my confusion stems from the definition of the parameter space. I know that the theorems that give us these results depend on certain properties of the parameter space. But does the parameter space mean the "natural values" it can take or the parameter space I give it? For example, the parameter space for the true mean of a Bernouilli random variable is $[0,1]$ (this is what I refer to as the "natural values") but if, for some reason, the true mean can only take values over the interval $(.25, .75)$ what is my parameter space now? If I find the MLE over the interval $(.25,.75)$ do I get the nice asymptotic MLE properties?

The nice properties stop working if the true value is on the boundary of your parameter space --- that, and certain regularity conditions on the likelihood itself. I believe that all you need is for the true value of the parameter to be within an open set of the parameter space. In your example, if the true value of $p$ is 0.10, then it's impossible with respect to your restricted parameter space, so of course everything will fail. But if it's an interior point of (.25,.75), then the mle will still be the usual $\hat{p}$ and the nice asymptotic properties will hold. And if $p=0.25$, you won't get the nice asymptotics either.