An adaptation theory for nonparametric confidence intervals

Cai, Tianxi; Low, Mark G.

doi:10.1214/009053604000000049

Cited by 84 publications

(122 citation statements)

References 22 publications

(31 reference statements)

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“…, L) for t 0 > t), the construction of honest adaptive confidence bands is not possible; see [33]. Therefore, following [20], we will restrict the function class F ⇢ [ t2[t,t] ⌃(t, L) in a suitable way, as follows: Condition L2 (Bias bounds).…”

Section: Condition L1 (Density Estimator) the Density Estimatorfmentioning

confidence: 99%

Anti-concentration and honest, adaptive confidence bands

Chernozhukov¹,

Chetverikov²,

Kato³

2016

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Section: Condition L1 (Density Estimator) the Density Estimatorfmentioning

confidence: 99%

Anti-concentration and honest, adaptive confidence bands

Chernozhukov¹,

Chetverikov²,

Kato³

2016

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“…, the construction of honest adaptive confidence bands is not possible; see [29]. Therefore, following [19], we will restrict the function class F ⊂ ∪ t∈[t,t] Σ(t, L) in a suitable way, as follows:…”

Section: Honest and Adaptive Confidence Bands In Hölder Classesmentioning

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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“…Bjerve, Doksum and Yandell (1985), Hall (1992b), Hall and Owen (1993), Neumann (1995), Chen (1996), Neumann and Polzehl (1998), Picard and Tribouley (2000), Chen, Härdle and Li (2003) (in the context of hypothesis testing), Claeskens and Van Keilegom (2003), Härdle et al (2004) and McMurry and Politis (2008) employed methods that involve undersmoothing. There is also a theoretical literature which addresses the bias issue through consideration of the technical function class from which a regression mean or density came; see, for example, Low (1997) and Genovese and Wasserman (2008). This work sometimes involves confidence balls, rather than bands, and in that respect is connected to research such as that of Eubank and Wang (1994) and Genovese and Wasserman (2005).…”

Section: Motivationmentioning

confidence: 99%

A simple bootstrap method for constructing nonparametric confidence bands for functions

Horowitz¹,

Hall²

2013

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Standard approaches to constructing nonparametric confidence bands for functions are frustrated by the impact of bias, which generally is not estimated consistently when using the bootstrap and conventionally smoothed function estimators. To overcome this problem it is common practice to either undersmooth, so as to reduce the impact of bias, or oversmooth, and thereby introduce an explicit or implicit bias estimator. However, these approaches, and others based on nonstandard smoothing methods, complicate the process of inference, for example, by requiring the choice of new, unconventional smoothing parameters and, in the case of undersmoothing, producing relatively wide bands. In this paper we suggest a new approach, which exploits to our advantage one of the difficulties that, in the past, has prevented an attractive solution to the problem-the fact that the standard bootstrap bias estimator suffers from relatively high-frequency stochastic error. The high frequency, together with a technique based on quantiles, can be exploited to dampen down the stochastic error term, leading to relatively narrow, simple-toconstruct confidence bands.

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An adaptation theory for nonparametric confidence intervals

Cited by 84 publications

References 22 publications

Anti-concentration and honest, adaptive confidence bands

Anti-concentration and honest, adaptive confidence bands

Anti-concentration and honest, adaptive confidence bands

A simple bootstrap method for constructing nonparametric confidence bands for functions

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