Consistency and asymptotic normality of MLE of $\theta$ in $N(\theta,1)$ when $\theta\ge 0$

Question

Question: Let $X_1,\ldots,X_n$ are iid $\sim N(\theta,1)$, $\theta\ge0$. Compute the MLE of $\theta$. Is the MLE consistent? Is it asymptotically normal?

I am able to find the MLE is $\hat{\theta}=\max\{\bar{X},0\}$. (see also The MLE of a $N(\theta, 1)$ distribution ) But I don't know whether it is consistent and asymptotically normal. To show a MLE is consistent, we may use Chebyshev's inequality and we need to show the variance $D\hat{\theta}\to 0$ as $n\to \infty$. I don't know if it is a correct way, since the distribution of $\hat{\theta}$ is a little complicated. And I have no idea about the asymptotic normality.

Answer 1

If $\theta > 0$, it's easy to see using asymptotics of $\Phi(-\sqrt{n}\theta)$ that $\sqrt{n}(\hat{\theta} - \overline{X}) \to 0$ in distribution, giving us consistency and asymptotic normality.

If $\theta = 0$, $|\overline{\theta} - \theta| < |\overline{X} - \theta|$, so we still have consistency, but since $\hat{\theta} - \theta \ge 0$ in this case we can't possibly have asymptotic normality.