Observe two values, 1 and 3. The empirical distribution assigns probability one half to each. Draw two independent values with replacement from that empirical law and calculate their mean. This is a nonparametric bootstrap replicate. The observed data are now held fixed, and randomness comes from resampling. The resulting conditional distribution approximates a sampling distribution only under additional assumptions; generating many replicates reduces simulation error, not necessarily the approximation error caused by an unsuitable resampling scheme.
Predict
Predict the possible bootstrap means and their weights before enumerating them. Then ask a different question: can any bootstrap observation exceed the largest original observation? It cannot. That restriction is harmless for some smooth statistics and damaging for others, especially sample extrema. A procedure that works for a mean does not automatically work for a maximum. The target statistic and the population’s tail or boundary structure determine which approximation theorem is relevant.
Experiment
Enumerate all four ordered bootstrap records. The conditional mean of the bootstrap mean is the original sample mean, two, and its conditional variance is one half.
Resampled record
Bootstrap mean
Conditional probability
(1,1)
1
1/4
(1,3)
2
1/4
(3,1)
2
1/4
(3,3)
3
1/4
Derive that variance both from the table and by dividing the empirical single-draw variance by two. Label it conditional bootstrap variance. It is not an exact statement about the unknown population variance, and two observations do not reliably reveal every feature of the generating distribution. Exact enumeration removes Monte Carlo error while leaving these statistical limitations intact.
Understand
For n observed values, the empirical law has mean X̄ and variance Σ(xᵢ−X̄)²/n. The mean of n independent bootstrap draws therefore has conditional variance Σ(xᵢ−X̄)²/n². In iid finite-variance settings, the centered bootstrap mean can consistently approximate the corresponding limiting sampling law under appropriate conditions. This lesson derives the finite empirical calculation and states the limiting result’s scope; a full bootstrap consistency proof requires additional empirical-process arguments beyond this unit.
Permutation inference has a different justification. A permutation test is exact when the joint null law is invariant under the allowed permutations, or when a specified randomized assignment mechanism supports the corresponding sharp-null calculation. Equality of two population means alone does not imply exchangeability: unequal variances or different distribution shapes can remain. Bootstrap resampling and permutation rearrangement should not be presented as interchangeable ways to manufacture a null distribution.
Apply
For iid continuous Uniform(0,θ) data, the observed maximum occurs at one sample position almost surely. The ordinary bootstrap maximum equals it whenever that position is selected at least once, with probability 1−(1−1/n)ⁿ, tending to 1−e⁻¹. Thus the scaled difference between bootstrap and observed maxima has a nonvanishing atom at zero. The true scaled maximum error has a continuous exponential limit. This visible mismatch proves a failure of the ordinary bootstrap approximation for that extreme statistic, even with infinitely many resampling replicates.
For dependent time-series observations, independent resampling destroys lag covariance. The variance of a mean includes covariance terms, so ignoring positive dependence can understate uncertainty. Block or model-based methods may be appropriate under additional stationarity, dependence-decay and tuning conditions; they are not automatic cures. Also separate inferential uncertainty from selection bias: repeatedly resampling an unrepresentative dataset does not make its target population representative.
Continue
Finish with a model audit: identify the population, sampling design, target, statistic, resampling law and claimed approximation. Compare an exact small-case calculation with the proposed numerical method and explain its limitations. Return to Mathematical statistics for mixed practice. This unit supplies derivations and bounded counterexamples; independent analysis, cumulative assessment, proof revision and source-specific course mapping remain necessary before claiming broader course readiness. Software agreement alone does not establish sound statistical judgment.
Apply · explain · revise
Practice and reasoning
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Resample two draws with replacement from observed values 0,2. What is the conditional variance of their mean?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
In that bootstrap experiment, what is P*(bootstrap mean=1)?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Derive the conditional variance of the iid nonparametric bootstrap mean from n observed values.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Explain the basic failure of ordinary bootstrap approximation for the maximum of iid Uniform(0,θ) data.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
State a condition making a permutation test exact and contrast it with a null hypothesis of equal means alone.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Why can iid resampling underestimate uncertainty in a positively correlated time series?
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
