Adaptive Confidence Bands for Nonparametric Regression Functions

Cai, Tommaso; Low, Mark G.; Ma, Zongming

doi:10.1080/01621459.2013.879260

Cited by 33 publications

(38 citation statements)

References 26 publications

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“…For instance, one can impose shape restrictions that break the centrosymmetry, as in Cai, Low, and Xia (2013) or Armstrong (2015), or self-similarity assumptions that break the convexity, as in Giné and Nickl (2010) or Chernozhukov, Chetverikov, and Kato (2014). Alternatively, one can weaken the coverage requirement in the definition of a CI, by, say, only requiring average coverage as in Cai, Low, and Ma (2014) or Hall and Horowitz (2013).…”

Section: Introductionmentioning

confidence: 99%

Optimal Inference in a Class of Regression Models

2018

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We consider the problem of constructing confidence intervals (CIs) for a linear functional of a regression function, such as its value at a point, the regression discontinuity parameter, or a regression coefficient in a linear or partly linear regression. Our main assumption is that the regression function is known to lie in a convex function class, which covers most smoothness and/or shape assumptions used in econometrics. We derive finite‐sample optimal CIs and sharp efficiency bounds under normal errors with known variance. We show that these results translate to uniform (over the function class) asymptotic results when the error distribution is not known. When the function class is centrosymmetric, these efficiency bounds imply that minimax CIs are close to efficient at smooth regression functions. This implies, in particular, that it is impossible to form CIs that are substantively tighter using data‐dependent tuning parameters, and maintain coverage over the whole function class. We specialize our results to inference on the regression discontinuity parameter, and illustrate them in simulations and an empirical application.

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Section: Introductionmentioning

confidence: 99%

Optimal Inference in a Class of Regression Models

2018

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“…This would parallel the Holder differentiability conditions often made in the adaptive nonparametric estimation literature (Giné and Nickl (2010), Cai, Low, and Xia (2013), Hoffmann andNickl (2011), Cai, Low, andMa (2014), Armstrong (2014), Chernozhukov, Chetverikov, and Kato (2014)). Although allowing for general power laws for the rates of decay for the Fourier transform is straightforward, it may be difficult to formulate corresponding transparent sufficient conditions in real space.…”

Section: Extensionsmentioning

confidence: 67%

“…This follows from the fact that our bias bound estimator is adaptive as well, since it consistently estimates the decay exponent  reflecting the true smoothness of the unknown function to be estimated. This feature is shared by existing wavelet-based adaptive methods which have recently been proposed (for instance, Giné and Nickl (2010), Hoffmann andNickl (2011), Cai, Low, andMa (2014)), but the underlying assumptions regarding the generating process differ, as explained at the end of Section 3.…”

Section: Extensionsmentioning

confidence: 98%

“…While the classic topic of nonparametric inference is well-established and has been the subject of many reviews (for instance, Härdle and Linton (1994), Pagan and Ullah (1999), Ichimura and Todd (2007), Li and Racine (2007), Horowitz (2009)), the topic is still receiving considerable ongoing attention (Lewbel and Linton (2002), Giné and Nickl (2010), Low (1997), Cai, Low, and Xia (2013), Hoffmann andNickl (2011), Cai, Low, andMa (2014), Armstrong (2014), Chernozhukov, Chetverikov, and Kato (2014), Calonico, Cattaneo, and Titiunik (2014), Armstrong and Kolesár (2014), Calonico, Cattaneo, and Titiunik (2015), Pinkse (2001), Su and Ullah (2008), among many others), due to numerous open problems that have not found theoretical or practical solutions. One of these remaining problems is the following undesirable dilemma.…”

Section: Introductionmentioning

confidence: 99%

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A bias bound approach to nonparametric inference

Schennach

2015

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in AbstractThe traditional approach to obtain valid confidence intervals for nonparametric quantities is to select a smoothing parameter such that the bias of the estimator is negligible relative to its standard deviation. While this approach is apparently simple, it has two drawbacks: First, the question of optimal bandwidth selection is no longer well-defined, as it is not clear what ratio of bias to standard deviation should be considered negligible. Second, since the bandwidth choice necessarily deviates from the optimal (mean squares-minimizing) bandwidth, such a confidence interval is very inefficient. To address these issues, we construct valid confidence intervals that account for the presence of a nonnegligible bias and thus make it possible to perform inference with optimal mean squared error minimizing bandwidths. The key difficulty in achieving this involves finding a strict, yet feasible, bound on the bias of a nonparametric estimator. It is well-known that it is not possible to consistently estimate the pointwise bias of an optimal nonparametric estimator (for otherwise, one could subtract it and obtain a faster convergence rate violating Stone's bounds on optimal convergence rate). Nevertheless, we find that, under minimal primitive assumptions, it is possible to consistently estimate an upper bound on the magnitude of the bias, which is sufficient to deliver a valid confidence interval whose length decreases at the optimal rate and which does not contradict Stone's results.

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“…We report here how the proposed confidence band based on the empirical Bayes posterior performs in terms of this new formulation and compare the performance with the adaptive confidence band procedure considered in Cai, Low and Ma (2014). We consider five test functions.…”

Section: University Of Pennsylvaniamentioning

confidence: 99%

Discussion of “Frequentist coverage of adaptive nonparametric Bayesian credible sets”

Low¹,

Ma²

2015

Ann. Statist.

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We congratulate the authors for this very interesting article focused on the frequentist coverage of Bayesian credible sets in the context of an infinitedimensional signal in white noise models. In such settings the construction of honest confidence sets is especially complicated, at least when the goal is to construct confidence sets that have a size that adapts to the unknown parameters in the model, while maintaining coverage probability.The focus of the present paper is on constructing l 2 balls as confidence sets. There are some advantages that come with the focus on balls for confidence sets. For bands results in Low (1997) rule out the possibility of adaptation over even a pair of Lipschitz or Sobolev spaces at least for confidence bands that have a guaranteed coverage level. On the other hand, fully rate adaptive confidence balls which do maintain coverage probability can be constructed over Sobolev smoothness levels that range over an interval [α, 2α]. However, this range of models where such adaptation is possible is still quite limited and here the authors develop a theory that applies over a broader class of models. The approach taken, following Giné andNickl (2010) andBull (2012), is to focus on parameters that are in some sense typical and removing a set of parameter values that cause difficulties at least when constructing adaptive sets. The goal is then to construct fully adaptive confidence sets over the remaining collection of parameter values. In the present paper the parameter values that are kept belong to a class of parameters that they call polished tail sequences and the authors develop results that show that a particular empirical Bayes credible ball is both honest when restricted to such sequences and adaptive in size.There are of course many settings where it is more natural to focus on the construction of confidence bands rather than confidence balls and, typically,

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Adaptive Confidence Bands for Nonparametric Regression Functions

Cited by 33 publications

References 26 publications

Optimal Inference in a Class of Regression Models

Optimal Inference in a Class of Regression Models

A bias bound approach to nonparametric inference

Discussion of “Frequentist coverage of adaptive nonparametric Bayesian credible sets”

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