Confidence balls in Gaussian regression

Baraud, Yannick

doi:10.1214/009053604000000085

Cited by 35 publications

(100 citation statements)

References 6 publications

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“…Their condition is shown to be necessary and sufficient. Similar important conclusions concerning adaptivity in terms of confidence statements are obtained under Hilbert space geometry with corresponding L 2 -loss, see Juditsky and Lambert-Lacroix (2003), Baraud (2004), Genovese and Wasserman (2005), Cai and Low (2006), Robins and van der Vaart (2006), Bull and Nickl (2013), and Nickl and Szabó (2016). Concerning L p -loss, we also draw attention to Carpentier (2013).…”

supporting

confidence: 78%

Locally adaptive confidence bands

Patschkowski¹,

Rohde²

2019

Ann. Statist.

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We develop honest and locally adaptive confidence bands for probability densities. They provide substantially improved confidence statements in case of inhomogeneous smoothness, and are easily implemented and visualized. The article contributes conceptual work on locally adaptive inference as a straightforward modification of the global setting imposes severe obstacles for statistical purposes. Among others, we introduce a statistical notion of local Hölder regularity and prove a correspondingly strong version of local adaptivity. We substantially relax the straightforward localization of the self-similarity condition in order not to rule out prototypical densities. The set of densities permanently excluded from the consideration is shown to be pathological in a mathematically rigorous sense. On a technical level, the crucial component for the verification of honesty is the identification of an asymptotically least favorable stationary case by means of Slepian's comparison inequality. P ⊗n p sup t∈ [a,b] |C n (t, α)| ≥ c · r n (β) < ε.

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confidence: 78%

Locally adaptive confidence bands

Patschkowski¹,

Rohde²

2019

Ann. Statist.

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“…considering substantially smaller sets Θ n . For instance, Baraud (2004) developed nonasymptotic confidence regions which perform well on finitely many New input to the related problem in sup-norm loss has come very recently by Giné and Nickl (2010) who demonstrate in the context of density estimation that honest confidence bands can be achieved over Hölder balls if a set of only first Baire category is removed, see also Hoffmann and Nickl (2011).…”

Section: Introductionmentioning

confidence: 99%

Statistical inference for the optimal approximating model

Rohde

Dümbgen

2012

Probab. Theory Relat. Fields

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In the setting of high-dimensional linear models with Gaussian noise, we investigate the possibility of confidence statements connected to model selection. Although there exist numerous procedures for adaptive (point) estimation, the construction of adaptive confidence regions is severely limited (cf. Li, 1989). The present paper sheds new light on this gap. We develop exact and adaptive confidence regions for the best approximating model in terms of risk. One of our constructions is based on a multiscale procedure and a particular coupling argument. Utilizing exponential inequalities for noncentral χ 2 -distributions, we show that the risk and quadratic loss of all models within our confidence region are uniformly bounded by the minimal risk times a factor close to one.AMS 2000 subject classifications: 62G15, 62G20.

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“…419-420, 423, for some results in that direction). Note that some progress has recently been made concerning confidence sets for the entire parameter vector in high-dimensional models (see, e.g., Beran and Dümbgen 1998;Beran 2000;Baraud 2004).…”

Section: Introductionmentioning

confidence: 99%

On the Large-Sample Minimal Coverage Probability of Confidence Intervals After Model Selection

Kabaila

Leeb

2006

Journal of the American Statistical Association

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We give a large-sample analysis of the minimal coverage probability of the usual confidence intervals for regression parameters when the underlying model is chosen by a "conservative" (or "overconsistent") model selection procedure. We derive an upper bound for the large-sample limit minimal coverage probability of such intervals that applies to a large class of model selection procedures including the Akaike information criterion as well as various pretesting procedures. This upper bound can be used as a safeguard to identify situations where the actual coverage probability can be far below the nominal level. We illustrate that the (asymptotic) upper bound can be statistically meaningful even in rather small samples.

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Confidence balls in Gaussian regression

Cited by 35 publications

References 6 publications

Locally adaptive confidence bands

Locally adaptive confidence bands

Statistical inference for the optimal approximating model

On the Large-Sample Minimal Coverage Probability of Confidence Intervals After Model Selection

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