Given a sample $X_1,\dots, X_n$, where each random variable has a Bernoulli distribution $$P(X_i=1)=\theta, \qquad \text{ with }0<\theta<\frac12$$ what is the Method of Moments Estimator (MME) of $\theta$?
I know the MME without the constraints is $\overline{X}_n$. But is it still the same one with such constraints?
The method of moments works in the following way:
We have some kind of parameter $\theta$ that we want to estimate and we can represent this parameter as a function of (finitely many) moments of our distribution, i.e. $\theta = f(E[X], \ldots, E[X^k])$.
Now we try to estimate $\theta$ by estimating the moments and then plugging the estimated moments into $f$. Then we hope that under the right conditions we get a good estimate of our parameter.
Now the function $f$ is not uniquely determined. In your example you might want to choose $f(x) = x$, which will give you your standard estimator. But you might as well choose $f(x) = \min\{x, \tfrac{1}{2}\}$, this is pretty much up to you. Typically you should just choose a function that is continuous on some reasonably large set, so that your estimate is "asymptotically good".