Suppose $X_1, X_2, \dots, X_n$ are i.i.d $N(\theta, 1),\theta_0 \lt\theta$ , Find the MLE of $\theta$ and show that it is better than the sample mean $\bar X$ in the sense of having smaller mean squared error.

Trial: Here MLE of $\theta$ is $$\hat\theta = \begin{cases} \theta_0, &\bar X \lt \theta_0\\ \bar X, &\theta_0 \le \bar X \\ \end{cases}$$

I can't show that $E[(\hat\theta -\theta)^2] \lt Var(\bar X)=\dfrac{1}{n}$.

  • $\begingroup$ @TrynnaDoStat I think that the other question really is a duplicate of this one if we allow $\theta_1$ in the other question to increase without bound. I am voting to close this one. $\endgroup$ Commented Apr 2, 2015 at 14:07
  • $\begingroup$ @DilipSarwate I agree $\endgroup$ Commented Apr 2, 2015 at 14:18