Demo: Point Estimation
For samples from $\mathrm{Uniform}(0,\theta)$, compare a moment estimator with a likelihood estimator that is driven by the support constraint.
Mathematical setup
Let $X_1,\ldots,X_n\sim\mathrm{Uniform}(0,\theta)$ iid. Since $E[X_i]=\theta/2$, the method-of-moments estimator is
\[\hat\theta_{\mathrm{MOM}}=2\bar X.\]The likelihood is
\[L(\theta;x)=\theta^{-n}\mathbf 1\{\theta\geq x_{(n)}\}, \qquad x_{(n)}=\max_i x_i,\]so the maximum likelihood estimator is $\hat\theta_{\mathrm{MLE}}=X_{(n)}$. It is biased downward with $E[X_{(n)}]=n\theta/(n+1)$, while the MOM estimator is unbiased.
What to try
- Use small $n$ to see the MLE’s downward bias clearly. It often has lower spread but tends to sit below the true endpoint.
- Increase $n$ and compare RMSE. Both estimators improve, but they do so for different reasons: averaging versus the maximum moving toward the boundary.
- Change $\theta$ after fixing $n$. The relative behavior is scale-stable, while the absolute RMSE scales with the endpoint.
The MLE uses the support constraint: values of $\theta$ below the sample maximum have likelihood zero.