A Note on Conditional Versus Joint Unconditional Weak Convergence in Bootstrap Consistency Results

Abstract

The consistency of a bootstrap or resampling scheme is classically validated by weak convergence of conditional laws. However, when working with stochastic processes in the space of bounded functions and their weak convergence in the Hoffmann-Jørgensen sense, an obstacle occurs: due to possible non-measurability, neither laws nor conditional laws are well-defined. Starting from an equivalent formulation of weak convergence based on the bounded Lipschitz metric, a classical circumvent is to formulate bootstrap consistency in terms of the latter distance between what might be called a conditional law of the (nonmeasurable) bootstrap process and the law of the limiting process. The main contribution of this note is to provide an equivalent formulation of bootstrap consistency in the space of bounded functions which is more intuitive and easy to work with. Essentially, the equivalent formulation consists of (unconditional) weak convergence of the original process jointly with an arbitrary large number of bootstrap replicates. As a by-product, we provide two equivalent formulations of bootstrap consistency for R d -valued statistics: the first in terms of (unconditional) weak convergence of the statistic jointly with its bootstrap replicates, the second in terms of convergence in probability of the empirical distribution function of the bootstrap replicates. Finally, the asymptotic validity of bootstrap-based confidence intervals and tests is briefly revisited, with particular emphasis on the, in practice unavoidable, Monte Carlo approximation of conditional quantiles.

Combining Cumulative Sum Change‐Point Detection Tests for Assessing the Stationarity of Univariate Time Series

“…Notice that, by Lemma 2.2 of Bücher and Kojadinovic () and the continuity of the d.f. of (see Proposition above), the last statement of Proposition is equivalent to the following, more classical, formulation of bootstrap consistency: Furthermore, Lemma 4.2 in Bücher and Kojadinovic () ensures that the test based on with approximate p ‐value holds its level asymptotically under the null hypothesis of stationarity as n and M tend to the infinity. By Corollary 4.3 in the same reference, this implies that when n → ∞ for any sequence M n → ∞ .…”

“…In addition, recall thatF T j (x) = ℙ(T j ≥ x), x ∈ ℝ, j ∈ {1, … , r}. Combining either (2.7) or (2.8) with Lemma 2.2 in Bücher and Kojadinovic (2018) and Problem 23.1 in van der Vaart (1998), we obtain:…”

“…Notice that, by Lemma 2.2 of Bücher and Kojadinovic (2018) and the continuity of the d.f. of S C (h) (see Proposition 3.3 above), the last statement of Proposition 3.4 is equivalent to the following, more classical, formulation of bootstrap consistency:…”

“…Finally, let us show that holds. Since are identically distributed and independent conditionally on the data, using Lemma 2.2 in Bücher and Kojadinovic (), we have that implies Whence is proven if we show that Again using , the term on the left of the previous display is smaller than …”

“…A small simulation showing how the automatically chosen bandwidth )}, i ∈ {1, … , N}. The previous display has the following two consequences: first, by Problem 23.1 in van der Vaart (1998),sup x∈ℝ |ℙ(W n ≤ x) − ℙ(W ≤ x)| , … , W [N]n are identically distributed and conditionally independent on the data, using Lemma 2.2 inBücher and Kojadinovic (2018), we have BÜCHER, J.-D. FERMANIAN AND I. KOJADINOVIC…”