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Bootstrap failure, m-out-of-n methods, and subsampling — Examples

1 · Prerequisites

2 · Summary

These worked cases compare the ordinary and smaller-resample roots at a uniform support endpoint, exhibit a Bernoulli sequence whose bootstrap root laws are weakly close while uniform quantile coverage fails, and compute the random limiting bootstrap law for absolute- and squared-mean functionals at zero.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

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Uniform endpoint: ordinary bootstrap failure and the m-out-of-n remedy

Example

Take IID $U (0, 1)$ observations and estimate the unknown upper endpoint $1$ by the sample maximum $M_{n}$ . Compare the ordinary bootstrap root $n (M_{n}^{*} - M_{n})$ with the size- $m$ roots $m (M_{m, n}^{*} - M_{n})$ and $m (M_{I} - M_{n})$ when $m \to \infty$ and $m / n \to 0$ .

Facts & Assumptions

Given: The IID uniform sample and the three roots in the statement.

[F1]

Ordinary bootstrap failure for a uniform support endpoint, Proof step 1.2: the unique sample maximum is selected by at least one ordinary bootstrap draw with conditional probability $1 - (1 - 1 / n)^{n} \to 1 - e^{- 1}$ , giving the bootstrap root a conditional atom at zero.

[F2]

m-out-of-n bootstrap and subsampling recover the uniform endpoint root, Proof step 1.1: for every sample size $k$ , the actual root $k (M_{k} - 1)$ has CDF $0$ for $x < - k$ , $(1 + x / k)^{k}$ for $- k \leq x < 0$ , and $1$ for $x \geq 0$ ; these CDFs converge to the continuous limit $F (x) = e^{x}$ for $x < 0$ and $1$ for $x \geq 0$ .

[F3]

m-out-of-n bootstrap and subsampling: With-replacement and without-replacement resampling are distinct laws, and the resampled root uses the size- $m$ normalization.

[F4]

m-out-of-n bootstrap and subsampling recover the uniform endpoint root, Statement: for either resampling method, the conditional CDF converges uniformly in probability to $F$ when $m \to \infty$ and $m / n \to 0$ .

Verification

1.1F1F2F3given

By [F3], the ordinary root is $n (M_{n} - 1)$ and its bootstrap root is centered at $M_{n}$ with the same $n$ -scale. By [F1], the conditional bootstrap root has an atom at zero with limiting mass $1 - e^{- 1}$ ; by [F2], the sampling-root law converges to the continuous CDF $F$ .

1.2F3given

For the smaller-resample methods the corresponding endpoint root is $m (M_{m} - 1)$ , and the conditional roots are centered at the full-sample maximum but scaled by $m$ .

2.1F2F4step 1.2∎

By [F4], for either replacement design the conditional CDF converges uniformly in probability to the CDF of the actual size- $m$ sampling-root limit, so the smaller-resample approximation is first-order valid in this model. This conclusion is specific to the displayed uniform endpoint model and does not give uniform coverage over other laws.

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Weakly close root laws can still give failed uniform coverage

Example

Assume AC and let $n \geq 1$ . Let $P = {Bernoulli (p) : 0 \leq p \leq 1}$ , and use the sample mean with root $R_{n, p} = \sqrt{n} ({\overset{ˉ}{X}}_{n} - p)$ . Under the ordinary bootstrap, the conditional root law is the conditional law of $R_{n, p}^{*} = \sqrt{n} ({\overset{ˉ}{X}}_{n}^{*} - {\overset{ˉ}{X}}_{n})$ given the observed sample; write $L^{*}$ for this law and $L$ for the sampling law. For every $ϵ > 0$ , $\sup_{p \in [0, 1]} P_{p} ⁣ (π (L^{*} (R_{n, p}^{*}), L (R_{n, p})) > ϵ) ⟶ 0,$ so the root laws are uniformly close in a metric for weak convergence. Nevertheless, for any fixed $λ > 0$ and $α \in (0, e^{- λ})$ , the one-sided bootstrap lower confidence set based on the $(1 - α)$ -quantile has coverage below $1 - α$ along $p_{n} = 1 - λ / n$ . Thus weak-metric closeness alone does not ensure uniform quantile coverage.

Facts & Assumptions

Given: AC, the Bernoulli model class, the sample-mean estimator, the ordinary bootstrap root, and the generalized quantile convention from CDF ordering and uniform error control quantile coverage.

[A1]

The Axiom of Choice: AC is the assertion that every family of nonempty sets has a choice function.

[F1]

m-out-of-n bootstrap and subsampling: The ordinary bootstrap draws a size- $n$ sample conditionally IID from $P_{n}$ and centers the resampled statistic at ${\hat{θ}}_{n}$ .

[F2]

Identical distribution and IID families: The original observations are mutually independent and have one common law.

[F3]

Partial sums, row sums and sample means: ${\overset{ˉ}{X}}_{n} = n^{- 1} \sum_{i = 1}^{n} X_{i}$ .

[F4]

Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials: A Bernoulli $(p)$ variable is $1$ with probability $p$ and $0$ otherwise; a binomial variable is a finite sum of independent Bernoulli variables.

[F5]

A Bernoulli $(p)$ variable has mean $p$ and variance $p (1 - p)$ ; a binomial $(n, p)$ variable has mean $n p$ and variance $n p (1 - p)$ : A binomial $(n, p)$ variable has mean $n p$ and variance $n p (1 - p)$ .

[F6]

Chebyshev's inequality for random variables: For a square-integrable real random variable $Y$ and $t > 0$ , $P (∣ Y - E Y ∣ \geq t) \leq Var (Y) / t^{2}$ .

[F7]

Bernoulli root laws are uniformly close in Levy–Prokhorov distance: For every $n$ and $p, q \in [0, 1]$ , $π (μ_{n, p}, μ_{n, q}) \leq ∣ p - q ∣^{1 / 3}$ for the centered, $\sqrt{n}$ -scaled Bernoulli sample-mean roots.

[F8]

Confidence sets and coverage: A confidence procedure has measurable membership sections, and its coverage at a fixed parameter is the probability that the procedure contains that parameter.

[F9]

Levy prokhorov metric: $π$ is the infimum of radii satisfying the two closed-set enlargement inequalities in the statement of the definition.

[F10]

Weak convergence of borel probability measures: Weak convergence of Borel laws means convergence of integrals against all bounded continuous real functions.

[F11]

Levy prokhorov metric metrizes weak convergence: Under AC, convergence in Levy–Prokhorov distance on a separable metric space is equivalent to weak convergence of Borel probability laws.

[F12]

Uniform subsampling coverage under root-law control: Uniform quantile coverage follows from the stated one-sided root-CDF comparisons and uniform feasible-root CDF control; a weak metric alone is not among those premises.

[F13]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t: For $x > 0$ , $\log^{'} (x) = 1 / x$ and $\log x = \int_{1}^{x} d t / t$ .

[F14]

CDF ordering and uniform error control quantile coverage: The generalized quantile is $q_{G} (u) = \inf {x : G (x) \geq u}$ for $u \in (0, 1)$ , with $q_{G} (0) = - \infty$ and $q_{G} (1) = + \infty$ .

[F15]

The Euclidean metric space $R$ is separable because $Q$ is countable and dense in $R$ . Separability: the existence of an at most countable dense subset $Q$ is countably infinite Both $Q$ and $R ∖ Q$ are dense in $R$ , and every nonempty open subset of $R$ is uncountable

Verification

1.1F1F2F3F4F7

Under $P_{p}$ , ${\overset{ˉ}{X}}_{n}$ has law $Binomial (n, p) / n$ . Conditional on the sample, the bootstrap observations are IID Bernoulli $({\overset{ˉ}{X}}_{n})$ , so their centered root law is $μ_{n, {\overset{ˉ}{X}}_{n}}$ , while the true root law is $μ_{n, p}$ . By [F7], $π (L^{*} (R_{n, p}^{*}), L (R_{n, p})) \leq ∣ {\overset{ˉ}{X}}_{n} - p ∣^{1 / 3}$ .

1.2F1F3F8

Fix $λ > 0$ , take $p_{n} = 1 - λ / n$ for $n > λ$ , and let $q_{n}$ be the conditional $(1 - α)$ -quantile of $R_{n, p_{n}}^{*}$ . Define the lower interval $I_{n} = [{\overset{ˉ}{X}}_{n} - q_{n} / \sqrt{n}, 1] \cap [0, 1]$ . It is measurable because the data take only finitely many values, and $p_{n} \in I_{n}$ is exactly the coverage event for root threshold $q_{n}$ .

2.1A1F5F6F9F10F11F15step 1.1algebra

From [F5], ${Var}_{p} ({\overset{ˉ}{X}}_{n}) = p (1 - p) / n \leq 1 / (4 n)$ . For each $ϵ > 0$ , Chebyshev's inequality [F6] and step 1.1 give $\sup_{p \in [0, 1]} P_{p} {π (L^{*} (R_{n, p}^{*}), L (R_{n, p})) > ϵ} \leq \sup_{p} P_{p} {∣ {\overset{ˉ}{X}}_{n} - p ∣ > ϵ^{3}} \leq 1 / (4 n ϵ^{6}) \to 0$ . By [F15], $R$ is a separable metric space, so [F11] shows this is uniform closeness in a metric that induces weak convergence.

2.2F1F13F14step 1.2algebra

On the event $X_{1} = \dots = X_{n} = 1$ , the empirical law is $δ_{1}$ , every bootstrap root is zero, and [F14] gives $q_{n} = 0$ because $1 - α \in (0, 1)$ ; then $I_{n} = {1}$ and $p_{n} \notin I_{n}$ . This event has probability $p_{n}^{n} = (1 - λ / n)^{n}$ . By the integral formula in [F13], $n \log (1 - λ / n) = - n \int_{1 - λ / n}^{1} d t / t \to - λ$ , since the integrand converges uniformly to $1$ on that shrinking interval; exponentiating gives $p_{n}^{n} \to e^{- λ}$ . Hence ${lim sup}_{n} P_{p_{n}} (p_{n} \in I_{n}) \leq 1 - e^{- λ} < 1 - α$ , so ${lim inf}_{n} \inf_{p \in [0, 1]} P_{p} (p \in I_{n}) < 1 - α$ .

3.1A1F7F11F12step 1.1step 2.1step 1.2step 2.2∎

Steps 1.1 and 2.1 establish uniform weak-metric closeness, while steps 1.2 and 2.2 show the coverage failure. Thus weak-metric closeness alone does not imply uniform quantile coverage or the stronger root-CDF and feasible-root conditions in [F12]. The coupling and coverage calculation use no choice; AC enters only through the metric-to-weak-topology equivalence [F11].

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Absolute-value boundary and squared-mean degeneracy

Example

For IID standard-normal data at mean zero, compare the two estimators $∣ {\overset{ˉ}{X}}_{n} ∣$ and ${\overset{ˉ}{X}}_{n}^{2}$ . The first has a kink at the boundary value zero. The second has zero first derivative and a nondegenerate second-order limit. Both ordinary bootstrap roots retain a random conditional distribution.

Facts & Assumptions

Given: AC, the IID normal model at mean zero, and the two statistic sequences.

[A1]

The Axiom of Choice: AC is the axiom that every family of nonempty sets has a choice function.

[F1]

Ordinary bootstrap failure for nonsmooth and first-order-degenerate mean functionals: Under AC and IID standard-normal observations, both correctly centered bootstrap root CDFs at zero stay separated in probability from the corresponding true root CDF value zero.

[F2]

m-out-of-n bootstrap and subsampling: The ordinary bootstrap is the size- $n$ with-replacement method, centered at the observed estimator and scaled by the actual root normalization.

Verification

1.1A1F1given

Under AC and the IID standard-normal model in [F1], the boundary mean is zero. The target for $∣ μ ∣$ is zero and $∣ \cdot ∣$ is not differentiable there; the target for $μ^{2}$ is also zero and its first derivative vanishes. The root normalizations are respectively $\sqrt{n}$ and $n$ .

2.1A1F1F2given∎

The ordinary bootstrap roots therefore are $\sqrt{n} (∣ {\overset{ˉ}{X}}_{n}^{*} ∣ - ∣ {\overset{ˉ}{X}}_{n} ∣)$ and $n (({\overset{ˉ}{X}}_{n}^{*})^{2} - {\overset{ˉ}{X}}_{n}^{2})$ , as follows from the centering and size- $n$ scale in [F2]. Theorem [F1] proves that each conditional CDF fails at the continuity point zero, despite the nondegenerate true limits $∣ Z ∣$ and $Z^{2}$ .

Sources

J. P. Romano and A. M. Shaikh, On the Uniform Asymptotic Validity of Subsampling and the Bootstrap, §2.2, Remark 2.7 and Example 2.1