Consider these data:
\[ X_1,\,X_2,\,...,\,X_n\overset{\text{iid}}{\sim}\text{N}(\theta,\,1). \]
- What is the maximum likelihood estimator of \(\theta\in\mathbb{R}\)?
- What is the sampling distribution of the estimator?
- What is the MSE of the estimator?
- Based on the MSE, what are the statistical properties of this estimator?