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Ordinary bootstrap failure for nonsmooth and first-order-degenerate mean functionals

Statement

Assume AC. For each integer $n \geq 1$ , let $X_{1}, \dots, X_{n}$ be IID $N (0, 1)$ and let ${\overset{ˉ}{X}}_{n} = n^{- 1} \sum_{i = 1}^{n} X_{i}$ . At $μ = 0$ , which is the boundary value of the nonnegative ranges of $g_{1} (μ) = ∣ μ ∣$ and $g_{2} (μ) = μ^{2}$ , consider these two functionals. For the ordinary bootstrap, draw $X_{1}^{*}, \dots, X_{n}^{*}$ conditionally IID from the empirical law and write $W_{n}^{*} = \sqrt{n} ({\overset{ˉ}{X}}_{n}^{*} - {\overset{ˉ}{X}}_{n})$ with conditional CDF $G_{n}^{*}$ . The conditional empirical-bootstrap CLT Conditional IID bootstrap CLT for an empirical law, specialized to $d = 1$ , $a = 1$ , and $σ^{2} = 1$ , gives $G_{n}^{*} (x) \to Φ (x)$ in probability for every fixed $x$ , where $Φ$ is the standard normal CDF. Here conditional weak consistency means that the conditional integral of every bounded continuous test function converges in probability to its sampling-law integral.

Then the ordinary bootstrap is inconsistent for both correctly centered roots $R_{n, 1} = \sqrt{n} ∣ {\overset{ˉ}{X}}_{n} ∣, R_{n, 1}^{*} = \sqrt{n} (∣ {\overset{ˉ}{X}}_{n}^{*} ∣ - ∣ {\overset{ˉ}{X}}_{n} ∣),$ and $R_{n, 2} = n {\overset{ˉ}{X}}_{n}^{2}, R_{n, 2}^{*} = n (({\overset{ˉ}{X}}_{n}^{*})^{2} - {\overset{ˉ}{X}}_{n}^{2}) .$ In both cases the true root CDF is continuous at zero with value zero, whereas the conditional bootstrap root CDF at zero stays away from zero with positive probability. The first functional is nonsmooth at the boundary; the second has zero first derivative there, so its nondegenerate sampling root is on the second-order scale.

Facts & Assumptions

Given: AC, an IID standard-normal sample, the ordinary empirical bootstrap, and the stated conditional CDF convergence assumption for its centered sample mean.

[A1]

The Axiom of Choice: Every family of nonempty sets has a choice function. This is the explicit axiom premise of the published normal-law and normal-sample-mean suppliers used here.

[F1]

m-out-of-n bootstrap and subsampling: The ordinary n-out-of-n bootstrap uses conditionally IID draws from the empirical law and centers the size- $n$ statistic at its observed value.

[F2]

Identical distribution and IID families: IID means identical laws and mutual independence of the whole family.

[F3]

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

[F4]

Standard normal and normal laws, Location-scale normal laws have their stated mean and variance, The mean of a finite independent normal sample is normal: Under AC, $N (0, 1)$ has density $ϕ (x) = (2 π)^{- 1 / 2} e^{- x^{2} / 2}$ , mean zero and variance one; the mean of $n$ IID $N (0, 1)$ variables has law $N (0, 1 / n)$ , so $Z_{n} : = \sqrt{n} {\overset{ˉ}{X}}_{n}$ has law $N (0, 1)$ for every $n$ .

[F5]

First- and second-order delta method for random vectors: If $r_{n} (T_{n} - θ) \Rightarrow Z$ and $g$ has zero first derivative and a second-order expansion $g (θ + h) - g (θ) = \frac{1}{2} H [h, h] + o (∥ h ∥^{2})$ , then $r_{n}^{2} (g (T_{n}) - g (θ)) \Rightarrow \frac{1}{2} H [Z, Z]$ ; this theorem's first-order differentiability hypothesis does not apply to $g_{1}$ at zero.

[F6]

Conditional IID bootstrap CLT for an empirical law: Under AC, IID $R^{d}$ observations with finite second moment, and a fixed projection of positive variance, the finite-index empirical bootstrap mean centered at ${\overset{ˉ}{X}}_{n}$ and scaled by $\sqrt{n}$ has conditional CDF converging uniformly in probability to the matching normal CDF. For standard-normal data, $d = 1$ , $a = 1$ , and variance $1$ satisfy these hypotheses, giving the pointwise convergence used here.

[F7]

Pointwise CDF convergence to a continuous law is uniform: Pointwise convergence in probability of CDFs to the continuous CDF $Φ$ implies uniform convergence in probability.

[F8]

Weak convergence of borel probability measures: Weak convergence of Borel laws requires convergence of integrals against every bounded continuous real test function.

[F9]

Portmanteau theorem: Weak convergence implies convergence of CDF values at every continuity point of the limiting CDF.

[F10]

Cumulative distribution function of a real random variable: The CDF of a real root $R$ under a law $P$ is $x \mapsto P (R \leq x)$ ; the bootstrap analogue uses its conditional law.

Proof

Given: The objects and hypotheses in the statement.

1.1A1F2F3F4F5F8F9algebra

By [F2], the data form an IID sequence, and [F3] identifies ${\overset{ˉ}{X}}_{n}$ as its arithmetic sample mean. By [F4], $Z_{n} = \sqrt{n} {\overset{ˉ}{X}}_{n}$ is standard normal for every $n$ . For $g_{2} (μ) = μ^{2}$ , the derivative at zero vanishes and the second derivative is $2$ , so [F5] gives the second-order scale $R_{n, 2} = Z_{n}^{2} \Rightarrow Z^{2}$ ; directly, its law is exactly that of $Z^{2}$ for every $n$ . For $g_{1} (μ) = ∣ μ ∣$ , the first-order delta premise fails at zero, and directly $R_{n, 1} = ∣ Z_{n} ∣$ has the law of $∣ Z ∣$ for every $n$ . For $t > 0$ , the density bound in [F4] gives $P (∣ Z ∣ \leq t) \leq 2 t / \sqrt{2 π}$ and $P (Z^{2} \leq t) \leq 2 \sqrt{t} / \sqrt{2 π}$ ; for $t < 0$ both probabilities are zero. Therefore both true-root CDFs are continuous at zero and equal zero there. Their constant-law sequences converge weakly by [F8], and [F9] gives the same CDF limit at this continuity point.

1.2F1F6F7F10givenalgebra

By [F1], the bootstrap draws are conditionally IID from the empirical law and $W_{n}^{*}$ is the centered, $\sqrt{n}$ -scaled empirical-bootstrap mean. By [F10], $G_{n}^{*}$ is the CDF of this conditional root law. The specialized supplier fact [F6] and the local CDF lemma [F7] give $Δ_{n} : = \sup_{x} ∣ G_{n}^{*} (x) - Φ (x) ∣ \to 0$ in probability. For a realized value $z = Z_{n}$ , the event $R_{n, 1}^{*} \leq 0$ is $∣ z + W_{n}^{*} ∣ \leq ∣ z ∣$ , and the event $R_{n, 2}^{*} \leq 0$ is $(z + W_{n}^{*})^{2} \leq z^{2}$ ; both are the same interval event, with $W_{n}^{*} \in [- 2 z, 0]$ if $z \geq 0$ and $W_{n}^{*} \in [0, - 2 z]$ if $z < 0$ . A uniform CDF error $Δ_{n}$ also bounds the error at left limits, since $Φ$ is continuous. Thus each conditional probability differs by at most $2 Δ_{n}$ from its standard-normal value $Φ (2 ∣ Z_{n} ∣) - \frac{1}{2}$ .

2.1F4F7F8givenstep 1.2algebra∎

Put $b : = Φ (2) - \frac{1}{2} > 0$ and $c = b / 2$ . On $A_{n} : = {∣ Z_{n} ∣ \geq 1}$ , whose probability is the fixed positive number $P (∣ Z ∣ \geq 1)$ by [F4], the normal interval probability in step 1.2 is at least $b$ . On the additional event $Δ_{n} \leq b / 4$ , both conditional root CDFs at zero are therefore at least $c$ . The true roots have the laws of $∣ Z ∣$ and $Z^{2}$ for every $n$ by [F4]. Their atom at zero is zero, and the standard-normal density is bounded by $(2 π)^{- 1 / 2}$ , so for every $d > 0$ , $E [h_{d} (∣ Z ∣)] \leq P (∣ Z ∣ < d) \leq 2 d / \sqrt{2 π}$ and $E [h_{d} (Z^{2})] \leq P (∣ Z ∣ < \sqrt{d}) \leq 2 \sqrt{d} / \sqrt{2 π}$ , where $h_{d} (t) = 1$ for $t \leq 0$ , $h_{d} (t) = 1 - t / d$ for $0 < t < d$ , and $h_{d} (t) = 0$ for $t \geq d$ . Choose $d$ so both expectations are below $c / 2$ . For either bootstrap root, $E^{*} [h_{d} (R_{n, j}^{*})] \geq P^{*} (R_{n, j}^{*} \leq 0) \geq c$ on that event, while its true test integral is below $c / 2$ . The event has probability at least $P (∣ Z ∣ \geq 1) - P (Δ_{n} > b / 4)$ , whose limit is positive. Thus the conditional integral of a bounded continuous test function fails to converge in probability to the true integral, contradicting conditional weak consistency.

Depends on

Used by

Absolute-value boundary and squared-mean degeneracy Example

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Sources

P. J. Bickel, F. Götze, and W. R. van Zwet, Resampling Fewer Than n Observations: Gains, Losses, and Remedies for Losses, §2.3, Examples 5–6 (standard reference, not scraped)
Ryan Martin, Stat 451 Lecture Notes 0812: Bootstrap, pp. 15–17 (standard reference, not scraped)