On confidence bands for multivariate nonparametric regression

Proksch, Katharina

doi:10.1007/s10463-014-0494-5

Cited by 7 publications

(5 citation statements)

References 29 publications

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“…Recalling the definition of Λ K in (3.2) and n (A) in (4.2) we obtain by similar arguments as in the proof of Theorem 2 in Proksch (2016) that…”

Section: A1 Preliminariesmentioning

confidence: 60%

“…, have been investigated by several authors, mostly in the case of independent observations (see Johnston, 1982;Xia, 1998;Proksch, 2016, and the references therein), but also for stationary data (see Wu and Zhao, 2007;Zhao and Wu, 2008, among others). The most prominent statistical application of results of this type concerns the construction of simultaneous asymptotic confidence bands for the mean function.…”

Section: The General Testing Problem and Mathematical Preliminariesmentioning

confidence: 99%

“…The representation of A as a union of finitely many intervals is needed to ensure the blowing up property of h −1 n A in order to apply Theorem 14.1 of Piterbarg (2012) in the proof of Theorem 2 of Proksch (2016). In fact, A can be replaced by any sequence (A n ) n∈N of subsets of I n with (h −1 n A n ) n∈N satisfying the blowing up property (cf.…”

Section: A1 Preliminariesmentioning

confidence: 99%

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Are deviations in a gradually varying mean relevant? A testing approach based on sup-norm estimators

Bücher¹,

Dette²,

Heinrichs³

2020

Preprint

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Classical change point analysis aims at (1) detecting abrupt changes in the mean of a possibly non-stationary time series and at (2) identifying regions where the mean exhibits a piecewise constant behavior. In many applications however, it is more reasonable to assume that the mean changes gradually in a smooth way. Those gradual changes may either be non-relevant (i.e., small), or relevant for a specific problem at hand, and the present paper presents statistical methodology to detect the latter. More precisely, we consider the common nonparametric regression model Xi = µ(i/n) + εi with possibly non-stationary errors and propose a test for the null hypothesis that the maximum absolute deviation of the regression function µ from a functional g(µ) (such as the value µ(0) or the integral 1 0 µ(t)dt) is smaller than a given threshold on a given interval [x0, x1] ⊆ [0, 1]. A test for this type of hypotheses is developed using an appropriate estimator, say d∞,n, for the maximum deviation d∞ = sup t∈[x 0 ,x 1 ] |µ(t) − g(µ)|. We derive the limiting distribution of an appropriately standardized version of d∞,n, where the standardization depends on the Lebesgue measure of the set of extremal points of the function µ(•) − g(µ). A refined procedure based on an estimate of this set is developed and its consistency is proved. The results are illustrated by means of a simulation study and a data example.

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“…Recalling the definition of Λ K in (3.2) and n (A) in (4.2) we obtain by similar arguments as in the proof of Theorem 2 in Proksch (2016) that…”

Section: A1 Preliminariesmentioning

confidence: 60%

Section: The General Testing Problem and Mathematical Preliminariesmentioning

confidence: 99%

Section: A1 Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

Are deviations in a gradually varying mean relevant? A testing approach based on sup-norm estimators

Bücher¹,

Dette²,

Heinrichs³

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Constructing confidence bands for nonparametric regression functions is overall a very challenging problem (see. e.g., [30]) and we hope to report results on this under our shape constraint framework in a future work.…”

Section: Discussionmentioning

confidence: 98%

A projection approach for multiple monotone regression

Lin,

Thomas,

Piegorsch

et al. 2019

Preprint

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Shape-constrained inference has wide applicability in bioassay, medicine, economics, risk assessment, and many other fields. Although there has been a large amount of work on monotone-constrained univariate curve estimation, multivariate shape-constrained problems are much more challenging, and fewer advances have been made in this direction. With a focus on monotone regression with multiple predictors, this current work proposes a projection approach to estimate a multiple monotone regression function. An initial unconstrained estimator -such as a local polynomial estimator or spline estimator -is first obtained, which is then projected onto the shape-constrained space. A shape-constrained estimate is obtained by sequentially projecting an (adjusted) initial estimator along each univariate direction. Compared to the initial unconstrained estimator, the projection estimate results in a reduction of estimation error in terms of both L p (p ≥ 1) distance and supremum distance. We also derive the asymptotic distribution of the projection estimate. Simple computational algorithms are available for implementing the projection in both the unidimensional and higher dimensional cases. Our work provides a simple recipe for practitioners to use in real applications, and is illustrated with a joint-action example from environmental toxicology.

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“…Härdle and Song (2010) constructed uniform confidence bands for a quantile regression curve with a onedimensional predictor, whereas Cai et al (2014) constructed adaptive confidence bands based on nonparametric regression functions. Massé and Meiniel (2014) developed adaptive confidence bands for the case of nonparametric fixed design regression models, and Proksch (2016) developed uniform confidence bands in a nonparametric regression setting with deterministic and multivariate predictor. Another related result is that of Gu and Yang (2015).…”

Section: Introductionmentioning

confidence: 99%

A simple approach to construct confidence bands for a regression function with incomplete data

Al-Sharadqah

Mojirsheibani

2019

AStA Adv Stat Anal

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A long-standing problem in the construction of asymptotically correct confidence bands for a regression function m(where Y is the response variable influenced by the covariate X, involves the situation where Y values may be missing at random, and where the selection probability, the density function f (x) of X, and the conditional variance of Y given X are all completely unknown. This can be particularly more complicated in nonparametric situations. In this paper we propose a new kerneltype regression estimator and study the limiting distribution of the properly normalized versions of the maximal deviation of the proposed estimator from the true regression curve. The resulting limiting distribution will be used to construct uniform confidence bands for the underlying regression curve with asymptotically correct coverages. The focus of the current paper is on the case where X ∈ R. We also perform numerical studies to assess the finite-sample performance of the proposed method. In this paper, both mechanics and the theoretical validity of our methods are discussed.

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On confidence bands for multivariate nonparametric regression

Cited by 7 publications

References 29 publications

Are deviations in a gradually varying mean relevant? A testing approach based on sup-norm estimators

Are deviations in a gradually varying mean relevant? A testing approach based on sup-norm estimators

A projection approach for multiple monotone regression

A simple approach to construct confidence bands for a regression function with incomplete data

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