Maximum Likelihood Estimation for Uniform Distribution Extrema

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Point Estimation / Non-Regular Models.

Your Task
Let $X_1, \dots, X_n \sim \text{i.i.d. Uniform}(0, \theta)$ with unknown $\theta > 0$.
(1) Find the Maximum Likelihood Estimator (MLE) $\hat{\theta}_{\text{MLE}}$.
(2) Compute the bias and Mean Squared Error (MSE) of $\hat{\theta}_{\text{MLE}}$.
(3) Find an unbiased estimator based on the MLE.

Constraints

• Samples are independent and identically distributed on their respective support.
• Boundary conditions are defined strictly within standard real intervals.