Nonparametric Bootstrap Confidence Intervals

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

Your Task
Given an i.i.d. sample $\mathbf{x} = (x_1, \dots, x_n)$ and an estimator $\hat{\theta} = T(\mathbf{x})$.
(1) Formulate the non-parametric bootstrap algorithm to approximate the sampling distribution of $\hat{\theta}$.
(2) Derive the Percentile Bootstrap confidence interval and the Studentized (bootstrap-$t$) confidence interval.
(3) Explain why the bootstrap-$t$ interval achieves second-order accuracy ($\mathcal{O}(1/n)$ coverage error) compared to $\mathcal{O}(1/\sqrt{n})$ for standard normal intervals.

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.