Section 3.4: Bootstrap Confidence Intervals using Percentiles

Percentile Method

A technique to construct a confidence interval by finding percentiles in an approximately symmetric bootstrap distribution such that the proportion of bootstrap statistics between them matches the desired confidence level.

Standard Error Method for a $95\%$ Confidence Interval

A method that estimates the standard error using the standard deviation of the bootstrap distribution to generate a $95\%$ confidence interval with the formula $\text{statistic} \pm 2 \times \text{SE}$.

$P\%$ Confidence Interval (Percentile Method)

A confidence interval constructed by keeping the middle $P\%$ of bootstrap statistics and trimming the remaining area equally from both tails.

$99\%$ Confidence Interval Percentiles

The interval spanning from the $0.5\text{th}$ percentile to the $99.5\text{th}$ percentile of the bootstrap distribution, which keeps the middle $99\%$ of bootstrap statistics and leaves $0.5\%$ in each tail.

Effect of Confidence Level on Interval Width

The relationship where a higher confidence level yields a wider confidence interval because it must encompass a larger middle percentage of the bootstrap distribution (for example, a $95\%$ interval is wider than a $90\%$ interval).

Effect of Sample Size on Interval Width

The relationship where a larger sample size results in a narrower confidence interval because larger sample sizes reduce the variability in the bootstrap distribution, providing a more precise estimate.

Smooth and Symmetric Condition

The required condition that a bootstrap distribution must be approximately smooth and symmetric for bootstrap confidence interval methods to be valid.

Spiky Bootstrap Distribution

A distribution displaying noticeable gaps and tall spikes (often occurring when calculating bootstrap statistics for the median), indicating that introductory bootstrap confidence interval methods cannot be used.

Recommended Number of Bootstrap Samples

The guideline stating that taking $1000$ bootstrap samples is sufficient for introductory coursework, whereas $10{,}000$ or $100{,}000$ samples should be drawn in real-world applications to maximize precision.

Selection Rule for Bootstrap CI Methods

The rule establishing that either the standard error method or the percentile method can