Jackknife Estimator of Parameter Bias and Variance

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Resampling Methods.

Your Task
Given a sample $x_1, \dots, x_n$ and an estimator $\hat{\theta} = T(x_1, \dots, x_n)$. Let $\hat{\theta}_{(i)} = T(x_1, \dots, x_{i-1}, x_{i+1}, \dots, x_n)$ be the leave-one-out estimate.
(1) State Quenouille's Jackknife estimator of bias $\widehat{\text{Bias}}_{\text{jack}} = (n - 1)(\bar{\theta}_{(\cdot)} - \hat{\theta})$.
(2) State Tukey's Jackknife estimator of variance $\widehat{\text{Var}}_{\text{jack}} = \frac{n-1}{n} \sum_{i=1}^n (\hat{\theta}_{(i)} - \bar{\theta}_{(\cdot)})^2$.
(3) Prove that for the sample mean $T(\mathbf{x}) = \bar{x}$, the Jackknife variance estimator exactly equals the unbiased sample variance $s^2/n$.

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.