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First-order IID bootstrap validity for smooth functions of a mean

Statement

Assume AC. Let $X_{i}$ be IID $R^{d}$ -valued vectors with $E ∥ X_{1} ∥^{2} < \infty$ , mean $μ$ , and covariance $Σ$ . Let $g : R^{d} \to R$ be Borel and Fréchet differentiable at $μ$ , with derivative $a^{T}$ , and assume $0 < σ^{2} : = a^{T} Σ a < \infty .$ Set $θ = g (μ)$ , ${\hat{θ}}_{n} = g ({\overset{ˉ}{X}}_{n})$ , and ${\hat{θ}}_{n}^{*} = g ({\overset{ˉ}{X}}_{n}^{*})$ for the finite-index empirical bootstrap in IID empirical bootstrap sample and mean statistic. Then $\sqrt{n} ({\hat{θ}}_{n} - θ) \Rightarrow N (0, σ^{2}),$ and, for every fixed $x \in R$ , $P^{*} ⁣ (\sqrt{n} ({\hat{θ}}_{n}^{*} - {\hat{θ}}_{n}) \leq x ∣ X_{1}, \dots, X_{n}) ⟶ Φ_{σ} (x) in probability,$ where $Φ_{σ}$ is the CDF of $N (0, σ^{2})$ . Thus the ideal conditional bootstrap law and the sampling law have the same first-order limit with the stated centering and $\sqrt{n}$ scaling. This theorem does not imply coverage of any interval algorithm.

Facts & Assumptions

Given: AC; the IID data, finite second moment, Borel differentiable map, positive finite projected variance, and bootstrap law in the statement.

[A1]

AC is assumed because the multivariate IID CLT, multivariate normal construction, and conditional Lindeberg–Feller argument below use it. The Axiom of Choice

[F1]

The empirical bootstrap is the finite conditional IID law from $P_{n}$ , and ${\hat{θ}}_{n} = g ({\overset{ˉ}{X}}_{n})$ , ${\hat{θ}}_{n}^{*} = g ({\overset{ˉ}{X}}_{n}^{*})$ . IID empirical bootstrap sample and mean statistic

[F2]

Under AC and finite second moment, $\sqrt{n} ({\overset{ˉ}{X}}_{n} - μ) \Rightarrow N_{d} (0, Σ)$ , including singular $Σ$ . Multivariate iid central limit theorem

[F3]

If $r_{n} (T_{n} - θ) \Rightarrow Z$ and $g$ is differentiable at $θ$ with derivative $A$ , then $r_{n} (g (T_{n}) - g (θ)) \Rightarrow A Z$ . First- and second-order delta method for random vectors

[F4]

The conditional projected bootstrap root has CDF converging uniformly in probability to $N (0, a^{T} Σ a)$ . Conditional IID bootstrap CLT for an empirical law

[F5]

The sample covariance converges in probability to $Σ$ . Finite-second-moment IID sample covariance converges

[F6]

Finite-dimensional weak convergence implies boundedness in probability. Finite-dimensional stochastic-order calculus and tightness

[F7]

For a multivariate normal vector, $a^{T} Z \sim N (a^{T} m, a^{T} Σ a)$ . Multivariate normal law, including singular covariance

[F8]

Real convergence in distribution is convergence of CDFs at continuity points of the limiting CDF. Convergence in distribution for real random variables

[F9]

Conditional Markov bounds the bootstrap vector fluctuation by its conditional second moment. Markov's inequality for random variables

[F10]

The normal CDF $Φ_{σ}$ is continuous. Standard normal and normal laws The standard normal CDF is continuous and strictly increasing

[F11]

In finite-dimensional real spaces, differentiability means a linear first-order term with remainder $o (∥ h ∥)$ . The real Frechet derivative on real and complex matrix spaces with the Frobenius norm

Proof

1.1A1F2given

By [F2], $Z_{n} : = \sqrt{n} ({\overset{ˉ}{X}}_{n} - μ) \Rightarrow Z$ for $Z \sim N_{d} (0, Σ)$ .

1.2F1F2F4F5F6F9givenalgebrachoose

Put $H_{n}^{*} = \sqrt{n} ({\overset{ˉ}{X}}_{n}^{*} - {\overset{ˉ}{X}}_{n})$ and $V_{n}^{*} = a^{T} H_{n}^{*}$ . By [F4], the conditional CDF of $V_{n}^{*}$ converges uniformly in probability to $Φ_{σ}$ . Also $E^{*} (∥ H_{n}^{*} ∥^{2} ∣ X_{1}, \dots, X_{n}) = tr ({\hat{Σ}}_{n}) .$ By [F5], this trace converges in probability to $tr (Σ)$ . For any $M > 0$ , $P^{*} (∥ H_{n}^{*} ∥ > M ∣ X_{1}, \dots, X_{n}) \leq \frac{tr ({\hat{Σ}}_{n})}{M^{2}}$ by [F9]. Given $η > 0$ , choose $K > tr (Σ)$ ; with probability tending to one the trace is at most $K$ , and then choosing $M^{2} > K / η$ makes this conditional tail at most $η$ . Thus $H_{n}^{*}$ is bounded in conditional probability, in probability. By [F2] and [F6], $Z_{n}$ is bounded in probability, so $Z_{n} + H_{n}^{*} = \sqrt{n} ({\overset{ˉ}{X}}_{n}^{*} - μ)$ is also bounded in conditional probability, in probability.

2.1F3F7givenstep 1.1

Apply [F3] with $r_{n} = \sqrt{n}$ and derivative $a^{T}$ . Then $\sqrt{n} {g ({\overset{ˉ}{X}}_{n}) - g (μ)} \Rightarrow a^{T} Z$ , and [F7] identifies this law as $N (0, σ^{2})$ . This proves the sampling limit.

2.2F1F6F11step 1.2givenalgebra

By [F11], $g (μ + h) = g (μ) + a^{T} h + r (h)$ with $∣ r (h) ∣ / ∥ h ∥ \to 0$ . Fix $ε, η > 0$ . By [F6], choose $R_{0}$ so that ${lim sup}_{n} P (∥ Z_{n} ∥ > R_{0}) < η$ . The conditional tightness established in step 1.2 gives $R_{1}$ such that, for all sufficiently large $n$ , the data probability that $P^{*} (∥ H_{n}^{*} ∥ > R_{1} ∣ X_{1}, \dots, X_{n}) > η$ is below $η$ . On the complementary data event and $∥ Z_{n} ∥ \leq R_{0}$ , the conditional probability that $∥ \sqrt{n} ({\overset{ˉ}{X}}_{n}^{*} - μ) ∥ > R_{0} + R_{1}$ is at most $η$ . Put $R = R_{0} + R_{1}$ and choose $δ > 0$ so $∣ r (h) ∣ \leq ε ∥ h ∥ / (4 R)$ for $0 < ∥ h ∥ < δ$ . For $n$ large enough that $R / \sqrt{n} < δ$ , the true remainder $\sqrt{n} ∣ r ({\overset{ˉ}{X}}_{n} - μ) ∣$ is at most $ε / 4$ on $∥ Z_{n} ∥ \leq R_{0}$ (enlarge $R$ to at least $R_{0}$ ), and the bootstrap remainder $\sqrt{n} ∣ r ({\overset{ˉ}{X}}_{n}^{*} - μ) ∣$ is at most $ε / 4$ whenever its root norm is at most $R$ . The exceptional data probability has limiting upper bound at most $2 η$ , and the exceptional conditional probability is at most $η$ on the remaining data event. Letting $η ↓ 0$ proves that the difference of these two remainders is $o_{P^{*}} (1)$ in probability. Consequently $\sqrt{n} ({\hat{θ}}_{n}^{*} - {\hat{θ}}_{n}) = V_{n}^{*} + Δ_{n}^{*},$ where $P^{*} (∣ Δ_{n}^{*} ∣ > η ∣ X_{1}, \dots, X_{n}) \to 0$ in probability for every $η > 0$ .

3.1F4F8F10step 2.1step 2.2∎

Fix $x \in R$ and $h > 0$ . With $G_{n}^{*}$ the conditional CDF of the left side in step 2.2 and $F_{n}^{*}$ that of $V_{n}^{*}$ , the event inclusion gives $F_{n}^{*} (x - h) - q_{n} (h) \leq G_{n}^{*} (x) \leq F_{n}^{*} (x + h) + q_{n} (h),$ where $q_{n} (h) = P^{*} (∣ Δ_{n}^{*} ∣ > h ∣ X_{1}, \dots, X_{n})$ . By [F4] and step 2.2, for each fixed $h$ these bounds converge in probability to $Φ_{σ} (x - h)$ and $Φ_{σ} (x + h)$ . Given any tolerance, [F10] lets us choose $h$ so both constants differ from $Φ_{σ} (x)$ by less than that tolerance; convergence of the bounds then gives $G_{n}^{*} (x) \to Φ_{σ} (x)$ in probability. By [F8], this is conditional weak convergence in probability to the same first-order law as the sampling limit in step 2.1.

Depends on

Used by

The IID bootstrap for a smooth mean functional Example

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Sources

University of Washington STAT 581, Chapter 8: Bootstrap and Jackknife Estimation of Sampling Distributions, §5, Theorem 5.1 and full proof (standard reference, not scraped)
Bradley Efron and Robert Tibshirani, The Bootstrap Method for Assessing Statistical Accuracy, Behaviormetrika 12 (1985), §§1–2 and §7 (standard reference, not scraped)
P. J. Bickel, F. Götze, and W. R. van Zwet, Resampling Fewer Than n Observations: Gains, Losses, and Remedies for Losses, Statistica Sinica 7 (1997), §2.3 (standard reference, not scraped)