James-Stein Shrinkage Estimator and Stein's Paradox

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Decision Theory / Shrinkage.

Your Task
Let $X \sim \mathcal{N}_p(\theta, I_p)$ be a single observation of a $p$-dimensional multivariate normal vector. Under the squared error loss $L(\hat{\theta}, \theta) = \|\hat{\theta} - \theta\|_2^2$, the maximum likelihood estimator is $\hat{\theta}_{\text{MLE}} = X$, with constant risk $\mathcal{R}(\hat{\theta}_{\text{MLE}}, \theta) = p$.
(1) State Stein's Paradox for $p \ge 3$.
(2) State the James-Stein estimator $\hat{\theta}_{\text{JS}} = \left(1 - \frac{p - 2}{\|X\|_2^2}\right) X$.
(3) Prove using Stein's Unbiased Risk Estimate (SURE) / integration by parts that $\mathcal{R}(\hat{\theta}_{\text{JS}}, \theta) < p$ for all $\theta \in \mathbb{R}^p$.

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.