Question

please help with Maximum likelihood estimation

Answer 1

Let X1,...,Xn∼N(θ,1). Suppose θ≥0.

Show that MLE of θ is\[\hat{\theta}_n\] = max{0,X¯}

Why is \[\hat{\theta}_n\] consistent estimator of

Answer 2

\[\hat{\theta_n}=\max{(0, \overline{X})}\]
To show that \(\hat{\theta_n}\) is a maximum likelihood estimator, look at the likelihood function:
\[\LARGE L(\theta)=\prod_{i=1}^n \frac{1}{\sqrt{2\pi}}e^{\frac{1}{2}(X_i-\theta)^2}=(2\pi)^{-n/2}e^{\sum_{i=1}^n \frac{1}{2}(X_i-\theta)^2}\]
As with most problems of this type, instead of maximizing L(theta), we maximize ln(L(theta)) wrt theta:
\[\ln{L(\theta)}=-\frac{n}{2}\ln{2\pi}+\frac{1}{2}\sum_{i=1}^n (X_i-\theta)^2\]
Let me know what you get after you maximize this.