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eduKateSG · Why Mathematics? · Parametric Bootstrap: Fitted Probability Models and Simulation
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 a fitted probability model describes how new data could be generated? The parametric bootstrap simulates complete datasets from that fitted model, refits each one and watches the estimate vary. Its power comes from a clear probability law; its risk comes from trusting the wrong law.
Did You Know? In a parametric bootstrap, the simulated datasets need not contain any original observations. They have the same structure, but their values are newly drawn from the fitted distribution.
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
Sometimes the observation mechanism has a natural distribution: counts may be Poisson, successes binomial and measurement errors approximately normal with a modelled variance. Parametric bootstrap asks what repeated estimation would look like if that fitted model generated new datasets.
The official R boot documentation supports sim="parametric" and requires a random-data generator returning a simulated dataset of the same form as the observed data. Read the R documentation.
Estimate parameters θ_hat from the observed data. For each replicate, draw D* from P_{θ_hat}, apply the same estimation method to D*, and store θ_hat*. The empirical distribution of estimates approximates sampling variation under the fitted family.
Suppose 6 successes occur in 10 independent trials, so p_hat=0.6. A parametric replicate draws X*~Binomial(10,0.6). One replicate might have X*=7 and p_hat*=0.7.
| n | Observed successes | p hat | Replicate successes | p hat star |
|---|---|---|---|---|
| 10 | 6 | 0.6 | 7 | 0.7 |
The probability of exactly seven successes under the fitted model is C(10,7)0.6^7 0.4^3. Since C(10,7)=120, this equals about 0.2150. That probability comes from the declared binomial assumptions.
| Combination | p power | q power | Probability |
|---|---|---|---|
| 120 | 0.6^7 | 0.4^3 | 0.215 |
Each simulated dataset must be analysed exactly like the original, including constraints, transformations and selection steps that are genuinely part of the estimator. Merely simulating outcomes without refitting does not reproduce estimator variability.
The true parameter is unknown, so simulation uses θ_hat. This plug-in step adds approximation error, especially in small samples or near boundaries. The bootstrap distribution is conditional on a model estimated from the same data.
A normal bootstrap assumes the normal family, not merely a mean and variance. A Poisson bootstrap links mean and variance. If tails, zero inflation or overdispersion differ materially, the replicate distribution can be too narrow or otherwise distorted.
Parametric simulation can include dependence if the model includes it: an AR process, random effects or spatial covariance can generate correlated observations. Simulating independent rows from marginal distributions would discard that structure.
Variance, dispersion, correlation and random-effect parameters are often estimated alongside the target. Decide whether each replicate refits them. Freezing nuisance values can understate the variability of the full estimation procedure.
Probabilities near zero or one, variance components near zero and rare-event counts create skewed or discrete bootstrap distributions. Standard normal summaries may fail, and some replicate fits may land on a boundary.
With B replicates, any reported percentile or p-value has simulation variability. Repeating with another seed and increasing B should stabilise meaningful digits. It cannot reveal misspecification outside the fitted family.
The standard deviation of θ_hat* estimates model-based standard error. The difference between the replicate mean and θ_hat estimates bootstrap bias under the fitted model. Bias correction is not automatically beneficial when the bias estimate is noisy.
Percentile, basic and bootstrap-t intervals use the simulated distribution differently. Discrete parameters may yield tied quantiles. Report the interval type, B, seed and any transformation used.
If arrival counts are plausibly Poisson over stable equal intervals, simulate counts from fitted rates, refit the rate model and assess uncertainty. Time variation or clustering would require a richer count process.
A fitted Weibull lifetime model can simulate complete failure times and censoring patterns. Censoring must be regenerated or conditioned on consistently; otherwise the bootstrap does not mimic the observed study.
Nonparametric bootstrap resamples observed empirical units and makes fewer distributional assumptions. Parametric bootstrap can be smoother and more efficient when the model is credible, especially for structured or sparse outcomes.
Residual bootstrap uses an empirical error distribution around fitted means. Parametric regression bootstrap draws errors from a stated family, perhaps with variance depending on x. The latter is more model dependent but can encode known structure.
Use residual diagnostics, goodness-of-fit checks, calibration and subject knowledge before simulation. Posterior predictive checks belong to a Bayesian framework; parametric bootstrap checks and uncertainty calculations use a fitted frequentist model.
State the fitted family, estimated parameters, generator, refitting rule, B, seed, target statistic, failed-fit handling and interval method.
Parametric bootstrap does not rescue a wrong distribution, missing dependence, selection bias, unmodelled censoring, weak identification or causal confounding. Results inherit every structural assumption in the generator.
Put fitted parameters in named cells. Use distribution functions to generate a dataset of the original shape, refit with the same formulas and collect the estimate and convergence flag for each replicate.
Write separate fit(data) and simulate(fit,rng) functions. Check shape and units, seed the generator, unit-test the binomial probability 0.2150 and log convergence and boundary events.
Model-based simulation supports biostatistics, reliability, risk, operations, ecology and machine learning. Careers require probability modelling, software validation and domain judgement.
Generate binomial data at p=0.6 for n=10, 50 and 200. Compare parametric and nonparametric intervals, then replace the data-generating process with beta-binomial overdispersion.
Parametric does not mean normal, model-based does not mean true, and more simulation does not fix misspecification. Refitting is essential, and a seed gives reproducibility rather than validity.
CHAPTER 27 OF 34 · Explore applications
27. Connection to recent mathematics articles
The residual-bootstrap article uses empirical residuals. The standard-error guide explains sampling precision.
Students should write the probability model in words before generating data. Parents can ask which assumption lets the software draw a value and how that assumption was checked.
- Calculate P(X*=7).
- Explain plug-in simulation.
- Name a nuisance parameter.
- Diagnose overdispersion.
- Compare parametric and empirical resampling.
### Are original rows resampled? Usually no.
### Must the model be normal? No.
### Is refitting required? Yes.
### Does a large B prove fit? No.
Report data model, dependence and censoring rules, fitted parameter values, simulator version, refit procedure, B, seed, failures, target, interval and model checks. Separate Monte Carlo error from model uncertainty.
The R boot documentation requires the generator to return data in the same form as the original. This practical rule prevents subtle changes in dimensions, labels or observation units from changing the estimator.
Confirm that the family, dependence, variance, censoring and boundaries are plausible; the same estimator is refit; failures are counted; and B, seed, software version and interval construction are recorded.
Continue through the Mathematics Learning Hub. The quickest way to understand a bootstrap is to reproduce one resample by hand, state the implied data-generating story, and then compare that story with the real study.
Extended practice laboratory
Practice case 1: Name probability family
Setup: Start from the article’s parametric-bootstrap mechanism for case 1. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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: Estimate parameters
Setup: Start from the article’s parametric-bootstrap mechanism for case 2. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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: Write generator
Setup: Start from the article’s parametric-bootstrap mechanism for case 3. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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: Preserve data shape
Setup: Start from the article’s parametric-bootstrap mechanism for case 4. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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: Draw binomial count
Setup: Start from the article’s parametric-bootstrap mechanism for case 5. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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: Compute exact probability
Setup: Start from the article’s parametric-bootstrap mechanism for case 6. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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: Refit estimator
Setup: Start from the article’s parametric-bootstrap mechanism for case 7. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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: Repeat B times
Setup: Start from the article’s parametric-bootstrap mechanism for case 8. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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: Estimate standard error
Setup: Start from the article’s parametric-bootstrap mechanism for case 9. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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 bias
Setup: Start from the article’s parametric-bootstrap mechanism for case 10. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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: Build an interval
Setup: Start from the article’s parametric-bootstrap mechanism for case 11. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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 nuisance parameter
Setup: Start from the article’s parametric-bootstrap mechanism for case 12. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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: Refit nuisance parameter
Setup: Start from the article’s parametric-bootstrap mechanism for case 13. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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: Approach a boundary
Setup: Start from the article’s parametric-bootstrap mechanism for case 14. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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: Add overdispersion
Setup: Start from the article’s parametric-bootstrap mechanism for case 15. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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: Add dependence
Setup: Start from the article’s parametric-bootstrap mechanism for case 16. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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: Add censoring
Setup: Start from the article’s parametric-bootstrap mechanism for case 17. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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: Log convergence
Setup: Start from the article’s parametric-bootstrap mechanism for case 18. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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: Compare empirical bootstrap
Setup: Start from the article’s parametric-bootstrap mechanism for case 19. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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: Compare residual bootstrap
Setup: Start from the article’s parametric-bootstrap mechanism for case 20. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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 family
Setup: Start from the article’s parametric-bootstrap mechanism for case 21. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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: Set a seed
Setup: Start from the article’s parametric-bootstrap mechanism for case 22. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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 model fit
Setup: Start from the article’s parametric-bootstrap mechanism for case 23. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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 parametric-bootstrap mechanism for case 24. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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 parametric-bootstrap mechanism for case 25. Task: Change the named quantity, recompute the dependent result and compare it with the original.
Check: Preserve the correct sampling unit, verify the probability or algebraic identity, and state one diagnostic before interpreting the result. 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.