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?