Some Asymptotic Theory for the Bootstrap

By Peter J. Bickel, David Freedman. 1981

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In Statistics and Probability

Citations in the corpus

Deep Exploration via Bootstrapped DQN (2016, background): “The bootstrap is widely hailed as a great advance of 20th century applied statistics and even comes with theoretical guarantees [2].”
Distributional Reinforcement Learning with Quantile Regression (2017, background): “The p -Wasserstein metric W p , for p ∈ [1 , ∞ ] , also known as the Mallows metric (Bickel and Freedman 1981) or the Earth Mover’s Distance (EMD) when p = 1 (Levina and Bickel 2001), is an integral probability metric between distributions.”
A Distributional Perspective on Reinforcement Learning (2017, methodology): “The main tool for our analysis is the Wasserstein metric d p between cumulative distribution functions (see e.g. Bickel & Freedman, 1981, where it is called the Mallows metric).”
The Cramer Distance as a Solution to Biased Wasserstein Gradients (2017, methodology): “A proof of (I) and (S) for the Wasserstein metric is given by Bickel and Freedman (1981), while the lack of (U) is shown in the proof of Theorem 1.”
Deep Exploration via Randomized Value Functions (2017, neutral): “The bootstrap serves as a form of data-based simulation and, in certain cases, recovers strong convergence guarantees (Bickel and Freedman, 1981; Fushiki, 2005).”
Asymptotic Properties of Lasso+mLS and Lasso+Ridge in Sparse High-dimensional Linear Regression (2013, methodology): “On the asymptotic normality front, the authors in [4, 5] also adopted the OLS after model selection and derived the asymptotic normality for inference on the effect of a treatment variable on a scalar outcome in the presence of very many controls.”
ON THE CHOICE OF m IN THE m OUT OF n BOOTSTRAP AND CONFIDENCE BOUNDS FOR EXTREMA (2008, methodology): “Case 2: An upper bound for µ, when σ is unknown, is ¯ X n − t n−1,α s n / √ n; the n-bootstrap works (Singh (1981), Bickel and Freedman (1981)), and the tquantile is replaced by the quantile of the bootstrap distribution of √ m( ¯ X * m − ¯ X n )/s * m .”
Multivariate goodness-of-fit tests based on Wasserstein distance (2020, background): “Among them, distances between densities (after kernel smoothing) have attracted much interest, starting with Bickel and Rosenblatt (1973) in the univariate case.”
Statistical Aspects of Wasserstein Distances (2018, methodology): “Bickel & Freedman (1981) use this result in their study of the asymptotics of the bootstrap.”
Bootstrapping autoregressive processes with possible unit roots (2000, background): “By Lemma 8.3 of Bickel and Freedman (1981), it is su cient to show that ~FT ) F !”
A general methodology for bootstrapping in non-parametric frontier models (2000, background): “In boundary problems such as this, the naive bootstrap yields inconsistent estimates (see Bickel & Freedman, 1981 or Beran & Ducharme, 1991 for discussions of this problem in the context of estimating the support of a univariate density; see Gijbels et al., 1999 for a two-dimensional case).”
Dynamic treatment regimes: technical challenges and applications (2014, background): “The first two claims follow from weak law of large numbers [Bickel and Freedman 1981; Csörgő and Rosalsky 2003].”
Estimation and inference in two-stage, semi-parametric models of production processes (2007, methodology): “Step [3] is simply a parametric bootstrap of a (nonlinear) regression model; properties of the bootstrap in the context of regression models have been examined by Bickel and Freedman (1981), Wu (1986), and others.”
Central limit theorem and convergence to stable laws in Mallows distance (2004, methodology): “Theorem 1.1 implies the standard central limit theorem in the sense of weak convergence (Bickel and Freedman 1981, Lemma 8.3).”