Cramer-Rao Lower Bound and Minimum Variance Unbiased Estimation

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Information Theory / Point Estimation.

Your Task
Let $X_1, \dots, X_n \sim \text{i.i.d. } \mathcal{N}(\mu, \sigma^2)$ where $\mu$ is known and $\theta = \sigma^2$ is unknown.
(1) Compute the Fisher Information $I_n(\theta)$ for sample size $n$.
(2) Compute the Cramér-Rao Lower Bound (CRLB) for any unbiased estimator of $\theta = \sigma^2$.
(3) Show that the sample variance $S_0^2 = \frac{1}{n} \sum_{i=1}^n (X_i - \mu)^2$ achieves the CRLB exactly.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.