Anti-concentration and honest, adaptive confidence bands

By Victor Chernozhukov, D. Chetverikov, Kengo Kato. 2013

184 citations recorded by the source corpus.

Modern construction of uniform confidence bands for non-parametric densities (and other functions) often relies on the classical Smirnov-Bickel-Rosenblatt (SBR) condition; see, for example, Gine and Nickl (2010). This condition requires the existence of a limit distribution of an extreme value type for the supremum of a studentized empirical process (equivalently, for the supremum of a Gaussian process with the same covariance function as that of the studentized empirical process). The principal contribution of this paper is to remove the need for this classical condition. We show that a considerably weaker sufficient condition is derived from an anti-concentration property of the supremum of the approximating Gaussian process, and we derive an inequality leading to such a property for separable Gaussian processes. We refer to the new condition as a generalized SBR condition. Our new result shows that the supremum does not concentrate too fast around any value. We then apply this result to derive a Gaussian multiplier bootstrap procedure for constructing honest confidence bands for nonparametric density estimators (this result can be applied in other nonparametetric problems as well). An essential advantage of our approach is that it applies generically even in those cases where the limit distribution of the supremum of the studentized empirical process does not exist (or is unknown). This is of particular importance in problems where resolution levels or other tuning parameters have been chosen in a data-driven fashion, which is needed for adaptive constructions of the confidence bands. Furthermore, our approach is asymptotically honest at a polynomial rate - namely, the error in coverage level converges to zero at a fast, polynomial speed (with respect to the sample size). In sharp contrast, the approach based on extreme value theory is asymptotically honest only at a logarithmic rate - the error converges to zero at a slow, logarithmic speed. Finally, of independent interest is our introduction of a new, practical version of Lepski's method, which computes the optimal, non-conservative resolution levels via a Gaussian multiplier bootstrap method.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/633b9860a10c744c860f0bc021c20eb36a4bebe3

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Citations in the corpus

Gaussian approximation of suprema of empirical processes (2012, background): “1), due essentially to the recent work of [4], complemented with a new “local” maximal inequality for the expectation of suprema of empirical processes that extends the work of [59] (Theorem 5.”
Simple and honest confidence intervals in nonparametric regression (2016, background): “Bull (2012) and Chernozhukov et al. (2014) have obtained rate optimal confidence bands under such conditions, which, like the CIs considered here, use a critical value based on an upper bound on the bias.”
Inference on Causal and Structural Parameters using Many Moment Inequalities (2013, background): “We call confidence regions Cn(α) for which (55) is verified asymptotically honest to Pn with a polynomial rate, where the term is inspired by Li (1989) and Chernozhukov, Chetverikov, and Kato (2014).”
Nonparametric Instrumental Variable Estimation Under Monotonicity (2015, unknown): “Our test is based on the ideas in Chetverikov (2012) who in turn builds on the methods for adaptive specification testing in Horowitz and Spokoiny (2001) and on the theoretical results on high dimensional distributional approximations in Chernozhukov, Chetverikov, and Kato (2013c) (CCK).”
Frequentist coverage and sup-norm convergence rate in Gaussian process regression (2017, methodology): “…together, we obtain that for any t ∈ R (notice that g is non-increasing) (by (63)) We have the anti-concentration bound for suprema of separable GP [14, Theorem 2.1], and the maximal inequality for GP [47, Corollary 2.2.8], where ρ X ( F ) = sup f,g ∈F ρ X ( f, g ) denotes the diameter of F…”
Testing Many Moment Inequalities (2013, methodology): “Hence by Theorem 2 in Chernozhukov, Chetverikov, and Kato (2013b), sup t∈R |P(W ≤ t | Xn1 )− P( max 1≤j≤p Yj ≤ t)| ≤ K∆̂1/3(1 ∨ log(p/∆̂))2/3, whereK is a universal constant.”
Improved central limit theorem and bootstrap approximations in high dimensions (2019, methodology): “For example, it appears in selecting the regularization parameters for the Lasso estimator and the Dantzig selector ([12]), in carrying out reality checks for data snooping and testing superior predictive ability ([45, 25]), in constructing model confidence sets ([26]), in testing conditional and/or”
Generic Machine Learning Inference on Heterogenous Treatment Effects in Randomized Experiments (2017, unknown): “Further assumptions such as ”self-similarity” are needed to bound the biases and expand the confidence bands by the size of bias (see [30, 22]) to produce partly adaptive confidence bands.”
Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression (2014, background): “Nonparametric confidence bands for an unknown function were considered by [1, 30] and more recently by [5, 6, 13].”
High-dimensional econometrics and regularized GMM (2018, background): “Useful references on this literature are provided in [48].”
Comparison and anti-concentration bounds for maxima of Gaussian random vectors (2013, background): “Our anti-concentration bounds do not require such positivity (or other) assumptions on covariances and hence are not implied by results [17].”
Conditional quantile processes based on series or many regressors (2011, methodology): “More recent contributions include [32], [48], [38], and [25], among many others.”
Nonparametric modal regression (2014, methodology): “We first recall a useful Theorem in Chernozhukov et al. (2013a): Theorem 13 (Theorem 3.1 in Chernozhukov et al. (2013a)).”
Empirical and multiplier bootstraps for suprema of empirical processes of increasing complexity, and related Gaussian couplings (2015, methodology): “These settings are not covered in our previous papers [5, 6, 7, 8, 9] and are new.”