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eduKateSG · Why Mathematics? · Basic Bootstrap Intervals: Reflected Errors and Bias Direction
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 8–14
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 15–21
Check assumptions
See where design and calibration matter.Route 4 · Chapters 22–28
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 29–34
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when bootstrap errors should be reflected around the observed estimate? The basic bootstrap interval, also called the reverse-percentile interval, uses quantiles of bootstrap deviations and maps them back in the opposite direction. A simple reflection reveals how bias direction changes an interval.
Did You Know? If the bootstrap distribution sits too high, the basic interval shifts lower. That is the opposite of the percentile interval, which follows the bootstrap distribution upward.
Full chapter index
Chapters 1–7 · Understand the question
Chapters 8–14 · Reproduce the example
Chapters 15–21 · Check assumptions
Chapters 22–28 · Explore applications
Chapters 29–34 · Study and verify
Bootstrap replicates approximate how θ_hat varies around the plug-in parameter θ_hat. The basic method treats the replicate errors θ_hat*−θ_hat as an empirical error distribution and subtracts their quantiles from the observed estimate.
SciPy names the basic interval the reverse-percentile method, while R boot.ci lists it as one of five nonparametric interval types. Read SciPy and R.
If q_low* and q_high* are α/2 and 1−α/2 quantiles of θ_hat*, the basic interval is [2θ_hat−q_high*, 2θ_hat−q_low*]. The high bootstrap quantile determines the lower endpoint.
Let θ_hat=10, q_low*=7.5 and q_high*=13.0. The basic interval is [20−13.0,20−7.5]=[7.0,12.5]. The corresponding percentile interval would be [7.5,13.0].
| Observed estimate | q low | q high |
|---|---|---|
| 10 | 7.5 | 13.0 |
The bootstrap error quantiles are q_low*−θ_hat and q_high*−θ_hat. If these bracket the estimation error θ_hat−θ, solve for θ by subtracting the upper error from θ_hat for the lower limit and the lower error for the upper limit.
| Lower | Upper |
|---|---|
| 7.0 | 12.5 |
The inequality e_low≤θ_hat−θ≤e_high becomes θ_hat−e_high≤θ≤θ_hat−e_low. Reversal follows from algebra. It is the same structural idea seen in bootstrap-t inversion without replicate standardisation.
If bootstrap estimates are shifted upward relative to θ_hat, percentile endpoints move upward. Basic endpoints reflect that shift downward. This can compensate for estimated bias in some settings, but the correction is first-order and can overreact.
The basic interval is generally not invariant under nonlinear transformations because reflection uses addition and subtraction on the chosen scale. Computing on θ and transforming endpoints may differ from computing directly on g(θ).
A rate ratio might be analysed on the original scale or log scale. The basic interval can differ after exponentiation because reflection on a multiplicative scale is not the same as reflection on an additive scale.
When the bootstrap distribution is symmetric around θ_hat, percentile and basic intervals can be mirror images that nearly coincide. Their disagreement grows with estimated bias and asymmetry.
Reflection can produce impossible values, such as a negative lower bound for a variance or probability. A transformation may help, but then the interval must be constructed consistently on that transformed scale.
Ties and coarse quantiles pass through the reflection formula. Endpoint discreteness or interpolation remains. A neat reflected interval does not guarantee correct coverage for a nonsmooth statistic.
Because basic endpoints depend on tail quantiles, they inherit quantile Monte Carlo error. Increasing B and rerunning with another seed helps reveal unstable digits.
A positively biased estimator of instrument response may produce a bootstrap distribution above the observed estimate. The basic interval reflects its errors, but calibration should be tested against a realistic measurement model.
Sample variance has a skewed distribution and a nonnegative parameter. A naive basic interval can cross zero. Chi-square, log-scale, BCa or model-based alternatives may be more suitable.
Percentile uses [q_low*,q_high*]. Basic uses [2θ_hat−q_high*,2θ_hat−q_low*]. Both use the same bootstrap distribution, but encode different error approximations.
BCa changes quantile levels using median bias and acceleration rather than a simple linear reflection. It is more adaptive and computationally involved.
Bootstrap-t reflects standardised errors rather than raw deviations. That can handle changing scale but requires s_hat* for every replicate.
Inspect bias, skewness, boundaries and transformation choice. Test coverage in simulation and compare pre-declared methods. Do not choose an interval because it alone crosses a desired decision threshold.
State θ_hat, q_low*, q_high*, B, seed, confidence level, scale, basic formula and final reflected endpoints.
Basic intervals may fail with unstable bias, nonlinear transformations, boundaries, nonsmooth statistics, dependence, weak identification or a poor resampling design.
Store θ_hat and bootstrap estimates. Calculate lower and upper quantiles, reflect them with 2θ_hat−q, and display percentile and basic endpoints side by side.
Implement the two endpoint formulas directly and unit-test θ_hat=10, q_low=7.5, q_high=13 to obtain [7,12.5]. Save the scale and quantile convention.
Reverse-percentile reasoning appears across applied statistics, calibration, economics and reliability. Strong practice requires knowing when additive reflection matches the parameter scale.
Simulate biased estimators on original and log scales. Compare percentile and basic coverage, impossible endpoints and transformation sensitivity across sample sizes.
Basic is not merely another name for percentile, reflected bias is not automatically corrected bias, and a mathematically valid endpoint can still be scientifically impossible.
CHAPTER 27 OF 34 · Explore applications
27. Connection to recent mathematics articles
The percentile interval guide uses raw quantiles. The bootstrap-t guide standardises replicate errors.
Students should derive the reflection from an inequality. Parents can ask why a high bootstrap quantile becomes the low endpoint and whether the scale is meaningful.
- Compute [7,12.5].
- Derive reversed endpoints.
- Compare percentile endpoints.
- Diagnose a negative variance limit.
- Explain transformation sensitivity.
### Is basic the same as percentile? No.
### Why reverse the quantiles? Error inversion.
### Is it transformation invariant? Generally no.
### Can it cross a boundary? Yes.
Report resampling scheme, θ_hat, B, seed, quantile convention, scale, formulas, boundary checks, simulation evidence and failed replicates.
On a number line, the map x↦2θ_hat−x reflects each bootstrap quantile around θ_hat. Distances are preserved on that chosen additive scale, which explains both the method and its transformation limitation.
Confirm correct quantiles, reversed endpoints, consistent scale, boundary plausibility, enough tail replicates, coverage checks and transparent comparison with alternatives.
Continue through the Mathematics Learning Hub. Reproduce one interval or resampling array by hand, then compare its assumptions with the study before trusting automated output.
Extended practice laboratory
Practice case 1: Name observed estimate
Setup: Start from the article’s basic-interval mechanism for case 1. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 2: Generate replicates
Setup: Start from the article’s basic-interval mechanism for case 2. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 3: Sort estimates
Setup: Start from the article’s basic-interval mechanism for case 3. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 4: Read q low
Setup: Start from the article’s basic-interval mechanism for case 4. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 5: Read q high
Setup: Start from the article’s basic-interval mechanism for case 5. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 6: Reflect upper quantile
Setup: Start from the article’s basic-interval mechanism for case 6. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 7: Reflect lower quantile
Setup: Start from the article’s basic-interval mechanism for case 7. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 8: Verify interval
Setup: Start from the article’s basic-interval mechanism for case 8. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 9: Compare percentile
Setup: Start from the article’s basic-interval mechanism for case 9. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 10: Shift bootstrap upward
Setup: Start from the article’s basic-interval mechanism for case 10. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 11: Shift bootstrap downward
Setup: Start from the article’s basic-interval mechanism for case 11. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 12: Add symmetry
Setup: Start from the article’s basic-interval mechanism for case 12. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 13: Add skewness
Setup: Start from the article’s basic-interval mechanism for case 13. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 14: Change parameter scale
Setup: Start from the article’s basic-interval mechanism for case 14. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 15: Apply log scale
Setup: Start from the article’s basic-interval mechanism for case 15. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 16: Check boundary
Setup: Start from the article’s basic-interval mechanism for case 16. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 17: Increase B
Setup: Start from the article’s basic-interval mechanism for case 17. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 18: Change seed
Setup: Start from the article’s basic-interval mechanism for case 18. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 19: Use a variance
Setup: Start from the article’s basic-interval mechanism for case 19. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 20: Use a ratio
Setup: Start from the article’s basic-interval mechanism for case 20. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 21: Compare BCa
Setup: Start from the article’s basic-interval mechanism for case 21. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 22: Compare bootstrap-t
Setup: Start from the article’s basic-interval mechanism for case 22. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 23: Audit formula
Setup: Start from the article’s basic-interval mechanism for case 23. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 24: Name limits
Setup: Start from the article’s basic-interval mechanism for case 24. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.
Practice case 25: Write conclusion
Setup: Start from the article’s basic-interval mechanism for case 25. Task: Change the named quantity, recompute the dependent result and compare with the original.
Check: Preserve the resampling unit, verify the quantile or counting identity, and state one calibration check before interpretation. Show the arithmetic, name the response and predictor units, and separate the numerical result from the claim it supports. Then vary one defensible assumption, calculate again and explain whether the conclusion changed because of the data, the fitted model or the validation purpose. Finish with one compact table or graph that could reveal a failure hidden by the summary.