Local invariance principles and their application to density estimation

Rio, Emmanuel

doi:10.1007/bf01311347

Cited by 87 publications

(120 citation statements)

References 15 publications

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“…Then U is uniformly distributed in [0; 1] U. Thus, Theorem 1.1 of Rio (1994) can be applied to a normalized empirical process associated with ' n;u;x (U; X).…”

Section: A1 Proofs For Section 31mentioning

confidence: 99%

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Testing for stochastic monotonicity

Lee¹,

Linton²,

Whang³

2008

110

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We propose a test of the hypothesis of stochastic monotonicity. This hypothesis is of interest in many applications. Our test is based on the supremum of a rescaled U-statistic. We show that its asymptotic distribution is Gumbel. The proof is difficult because the approximating Gaussian stochastic process contains both a stationary and a nonstationary part and so we have to extend existing results that only apply to either one or the other case.Key words: Distribution function; Extreme Value Theory; Gaussian Process; Monotonicity.JEL Nos.: C14, C15.© The authors. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source.

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“…Then U is uniformly distributed in [0; 1] U. Thus, Theorem 1.1 of Rio (1994) can be applied to a normalized empirical process associated with ' n;u;x (U; X).…”

Section: A1 Proofs For Section 31mentioning

confidence: 99%

“…h n ' n;u;x (v; t), indexed by (u; x) 2 U X , is uniformly of bounded variation (UBV). By the de…nition of Rio (1994), it su¢ ces to show that…”

Section: A1 Proofs For Section 31mentioning

confidence: 99%

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Testing for stochastic monotonicity

Lee¹,

Linton²,

Whang³

2008

110

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show abstract

“…In addition, if the density of X exists, our coupling does not assume that this density is bounded away from zero on the support. See, for example, [34] for the construction of the Hungarian coupling and the use of aforementioned conditions.…”

Section: Appendix a Coupling Inequalities For Suprema Of Empirical Amentioning

confidence: 99%

Anti-concentration and honest, adaptive confidence bands

Kato¹,

Chetverikov²,

Chernozhukov³

2013

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Abstract. Modern construction of uniform confidence bands for nonparametric densities (and other functions) often relies on the the classical Smirnov-Bickel-Rosenblatt (SBR) condition; see, for example, . 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 nonparametric 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.

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“…I note that as of this writing google scholar lists 54 citations to our paper, which is pretty good for a theoretical paper. Rio (1994) has computed values for the constants in (29.8). Later, Castelle, and Laurent-Bonvalot (1998) obtained a similar refinement to the KMT Kiefer process approximation to an.…”

Section: Masonmentioning

confidence: 99%