CONFIDENCE BANDS IN DENSITY ESTIMATION

By Evarist Gin'e, Richard Nickl. 2010

186 citations recorded by the source corpus.

Given a sample from some unknown continuous density f: ℝ → ℝ, we construct adaptive confidence bands that are honest for all densities in a "generic" subset of the union of t-Holder balls, 0 < t ≤ r, where r is a fixed but arbitrary integer. The exceptional ("nongeneric") set of densities for which our results do not hold is shown to be nowhere dense in the relevant Holder-norm topologies. In the course of the proofs we also obtain limit theorems for maxima of linear wavelet and kernel density estimators, which are of independent interest.

Subfields: Applied Mathematics, Computer Vision and Pattern Recognition, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c9f56d5cfb9a2d6a83577276ff5b98d3e4ecc28c

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In Applied Mathematics

In Computer Vision and Pattern Recognition

In Statistics and Probability

Citations in the corpus

GLOBAL UNIFORM RISK BOUNDS FOR WAVELET DECONVOLUTION ESTIMATORS (2011, result): “We do not detail this here, nor do we address the more difficult problem of adaptive confidence bands: using Proposition 3, such results can be obtained in the same way as in the case of density estimation considered in [20].”
On adaptive inference and confidence bands (2011, methodology): “The following is Proposition 4 in [12].”
Honest adaptive confidence bands and self-similar functions (2011, methodology): “2); those in Giné and Nickl (2010) are not, for the model (1.”
Simple and honest confidence intervals in nonparametric regression (2016, unknown): “Another approach to data-driven choices of M is to use “self-similarity” conditions, as suggested by Giné and Nickl (2010), which relate the maximum and minimum bias at different bandwidths.”
Adaptive test of conditional moment inequalities (2011, methodology): “An example is Lepski’s method combined with the sample splitting where a part of the sample is used to select the smoothing parameter according to the Lepski’s algorithm and the other part is used for testing; see, for example, [18].”
Anti-concentration and honest, adaptive confidence bands (2013, methodology): “The authors would like to thank Emre Barut, Enno Mammen and Richard Nickl for very helpful discussions.”
Confidence sets in sparse regression (2012, methodology): “To formalise these ideas, we take the separation approach to adaptive confidence sets introduced in Giné and Nickl (2010), Hoffmann and Nickl (2011), Bull and Nickl (2013) in the framework of nonparametric function estimation.”
Adaptive Bernstein–von Mises theorems in Gaussian white noise (2014, methodology): “See Giné and Nickl [21] and also Bull [6] for further discussion about this condition.”
Nearly optimal central limit theorem and bootstrap approximations in high dimensions (2020, background): “In particular, results in [37] and [27] imply that one can construct a centered Gaussian random vector G in Rd such that ‖W −G‖∞ .”
Uncertainty Quantification for the Horseshoe (with Discussion) (2016, methodology): “…preceding self-similarity condition plays the same role as the assumptions of “self-similarity” or “polished tail” used by Picard and Tribouley (2000); Giné and Nickl (2010); Bull (2012); Nickl and Szabo (2016); Szabó et al. (2015b); Sniekers and van der Vaart (2015c); Rousseau and Szabo (2016)…”
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].”
Conditional quantile processes based on series or many regressors (2011, background): “More recent contributions include [32], [48], [38], and [25], among many others.”
On the Effect of Bias Estimation on Coverage Accuracy in Nonparametric Inference (2015, background): “, Hall (1993), Xia (1998), Gine and Nickl (2010) and Chernozhukov, Chetverikov, and Kato (2013).”
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.”
Gaussian approximation of suprema of empirical processes (2012, background): “Conditions like (10) are typically referred to as under-smoothing (see [29], page 1130 for related discussion).”