Implement Bootstrapping and confidence itnervals

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Bootstrap Methods

Bootstrap Confidence Intervals

Procedure:

1. Generate (B) bootstrap samples ({(\mathbf{X}^{(b)}, \mathbf{Y}^{(b)})}_{b=1}^B)
2. Compute ({\hat{I}^{(b)}}_{b=1}^B) for each bootstrap sample
3. Construct confidence interval:

$$ [\hat{I}_{(\alpha/2)}, \hat{I}_{(1-\alpha/2)}] $$

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Time Series Bootstrap

Standard bootstrap assumes i.i.d. data. For time series, specialized methods are needed.

Block Bootstrap:

$$ \text{Bootstrap Sample} = [B_1, B_2, \ldots, B_k] $$

where (B_i) are overlapping blocks of length (l).

Stationary Bootstrap:
Uses random block lengths with a geometric distribution to preserve temporal dependence.

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