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eduKateSG · Why Mathematics? · Prepivoted Bootstrap: CDF Transformations and Higher-Order Calibration
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 29–35
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important when a statistic is not already a pivot with a stable, parameter-free distribution? Prepivoting transforms the statistic through an estimated cumulative distribution function. The transformed value is closer to uniform under the null, so tests and confidence sets can have smaller level error.
Did You Know? Rudolf Beran showed that prepivoting can be iterated. The first transformation is closely related to studentization when studentization is appropriate, while further transformations automate higher-order corrections.
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 29–35 · Study and verify
A perfect pivot has a distribution that does not depend on unknown nuisance parameters. Many useful roots are only approximately pivotal, so their nominal critical values can give inaccurate coverage or test size.
Beran defined prepivoting as transforming a confidence-set root by its estimated bootstrap cumulative distribution function and showed that iteration can reduce level error. Read the Biometrika abstract.
For an observed root R and estimated bootstrap CDF F-star, define U=F-star(R), with the tail convention chosen for the problem. If F-star were the true continuous distribution, the probability integral transform would make U uniform on [0,1].
Suppose 80 of 100 bootstrap roots are at or below the observed root. Then F-star(R)=80/100=0.80. The prepivoted value is 0.80 under the lower-tail CDF convention.
| Bootstrap roots | At or below observed | F star |
|---|---|---|
| 100 | 80 | 0.8 |
For a right-tail test using the same continuous convention, the estimated tail probability is 1−0.80=0.20. With discrete roots, equality and randomisation conventions must be stated.
| Lower-tail CDF | Right-tail value |
|---|---|
| 0.8 | 0.2 |
A uniform reference distribution has transparent critical values. If prepivoting makes the transformed statistic closer to uniform, comparing U with alpha or 1−alpha can improve level accuracy.
The empirical CDF is a step function. Finite B gives resolution about 1/B, so a 0.001 tail cannot be estimated credibly with only a few hundred replicates.
To prepivot again, generate an outer bootstrap root and estimate its own inner bootstrap CDF value. The distribution of those first-level transformed values supplies another transformation.
Studentization divides a centred statistic by an estimated standard error to reduce nuisance-scale dependence. Beran notes that first prepivoting is nearly equivalent when studentization is appropriate, but the operations are not literally identical in every problem.
A signed root suits a directional test; an absolute or quadratic root suits two-sided alternatives. The CDF and rejection tail must match the root definition.
For discrete data, the probability integral transform is not exactly uniform because the CDF has jumps. Mid-p, randomised or conservative conventions change the result and must be justified.
Prepivoting can reduce distributional dependence on nuisance parameters when the bootstrap estimates that dependence well. It cannot identify a nuisance parameter absent from the data.
With B=999, empirical tail probabilities change in increments near 0.001. Reporting 0.000347 from that run suggests false precision unless smoothing or importance methods are explicitly used.
For U-hat=0.80 from B=100 independent replicates, the simple standard error is sqrt(0.8×0.2/100)=0.04. The transformation itself has visible simulation noise.
A prepivoted test estimates how extreme the root is under its bootstrapped law, then may bootstrap that probability again. Simulation under known null models should verify actual size.
Confidence sets can be built by retaining parameter values whose prepivoted test does not reject. This may require repeated bootstrap calculations across candidate values.
A heteroscedastic regression root may retain nuisance dependence after simple scaling. A suitable wild-bootstrap CDF and prepivoting can target level accuracy while preserving the regression design.
A dependence-aware block bootstrap can estimate the CDF of a spatial root. Prepivoting does not excuse using independent resampling for correlated locations.
Percentile intervals read quantiles of the statistic directly. Prepivoting transforms a root by its estimated distribution and targets calibration of a test or confidence set.
Iterated prepivoting normally requires nested resampling and is therefore a conceptual route to double-bootstrap calibration. The specific root, CDF and iteration count define the actual method.
Check approximate uniformity with probability plots under simulated truth, vary B and inner C, test tail conventions, compare size and coverage, and record ties and failed fits.
Prepivoting can fail with an invalid bootstrap model, discontinuous or unstable roots, weak identification, insufficient tail resolution, dependence errors or excessive nested Monte Carlo noise.
Sort bootstrap roots, calculate empirical ranks, record equality convention and convert the observed root into U. For iteration, store outer U values separately from every inner CDF.
Unit-test 80/100=0.80 and right-tail 0.20. Use deterministic tie rules, independent random streams and a function that returns both the transformed value and its simulation resolution.
Prepivoting matters in econometrics, clinical trials, engineering tests and computational inference. Careers require asymptotic reasoning, resampling design and numerical validation.
Simulate a skewed null model, compare the raw root, studentized root and once-prepivoted root with Uniform(0,1), then measure rejection rates at 1%, 5% and 10%.
Prepivoting is not simply sorting p-values, an empirical CDF is not exact, iteration is not free, and uniform appearance in one simulation does not establish general validity.
CHAPTER 28 OF 35 · Explore applications
28. Connection to recent mathematics articles
The double-bootstrap guide explains the nested computation. The bootstrap-t guide explains studentized roots.
Students should calculate an empirical CDF by hand and state the tail convention. Parents can ask whether the transformed values were checked for uniform calibration under realistic simulations.
- Calculate 80/100.
- Convert it to a right-tail value.
- Compute the binomial Monte Carlo error.
- Explain one tie convention.
- Compare with studentization.
### What is transformed? A confidence-set or test root.
### What performs the transformation? An estimated bootstrap CDF.
### Can it be repeated? Yes.
### Is uniformity exact? Only under ideal conditions.
Report root, null or parameter value, resampling law, B, inner C, CDF and tie convention, iteration count, Monte Carlo error, uniformity diagnostic and measured size or coverage.
For a continuous true CDF F, F(R) is uniform when R follows F. Prepivoting replaces unknown F with a bootstrap estimate, so the gain depends on how accurately that estimate reproduces the root distribution.
Confirm the root and tail match, CDF resolution is adequate, ties are handled transparently, nested levels are preserved and improved calibration is measured rather than assumed.
Continue through the Mathematics Learning Hub. Reproduce one nested, transformed or distributed resampling calculation by hand before trusting a large automated run.
Extended practice laboratory
Practice case 1: Define root
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Choose CDF tail
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Generate bootstrap roots
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Sort roots
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Count below observed
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Calculate empirical CDF
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Calculate tail value
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Handle equality
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Check resolution
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Estimate Monte Carlo error
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Plot uniformity
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Iterate transformation
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Generate outer roots
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Generate inner CDFs
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Compare studentization
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Use signed root
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Use absolute root
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Add discreteness
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Add nuisance parameter
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Increase B
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Change inner C
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Check rejection size
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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: Check coverage
Setup: Start from the article’s prepivoting 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 probability or weighting identity, and state one diagnostic 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 prepivoting 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 probability or weighting identity, and state one diagnostic 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 prepivoting 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 probability or weighting identity, and state one diagnostic 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.