Confidence bands for multivariate and time dependent inverse regression models

Proksch, Katharina; Bissantz, Nicolai; Dette, Holger

doi:10.3150/13-bej563

Cited by 16 publications

(17 citation statements)

References 32 publications

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“…|F (1,1) (z)|dz < Ch r 2 +r 3 and therefore by incorporating the equidistant design in (20) one has |Z W n,1 (x, w, ψ) − Z W n,2 (x, w, ψ)| = O P log(n)n −1/2 h r 2 +r 3 −r 1 = O P log(n)n −1/2 h −1 ,…”

Section: C3 Proof Of Lemmamentioning

confidence: 99%

Asymptotic confidence sets for the jump curve in bivariate regression problems

Bengs

Eulert

Holzmann

2019

Journal of Multivariate Analysis

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We construct uniform and point-wise asymptotic confidence sets for the single edge in an otherwise smooth image function which are based on rotated differences of two one-sided kernel estimators. Using methods from Mestimation, we show consistency of the estimators of location, slope and height of the edge function and develop a uniform linearization of the contrast process. The uniform confidence bands then rely on a Gaussian approximation of the score process together with anti-concentration results for suprema of Gaussian processes, while point-wise bands are based on asymptotic normality. The finite-sample performance of the point-wise proposed methods is investigated in a simulation study. An illustration to real-world image processing is also given.

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Section: C3 Proof Of Lemmamentioning

confidence: 99%

Asymptotic confidence sets for the jump curve in bivariate regression problems

Bengs

Eulert

Holzmann

2019

Journal of Multivariate Analysis

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“…The estimations are very similar to the ones in the two-dimensional case and are therefore omitted. For the N -dimensional integration by parts formula or N -dimensional versions of the LIL see, e.g., Proksch et al (2012) or Paranjape and Park (1973), respectively.…”

Section: Kmentioning

confidence: 99%

“…The lengthy details of this, i.e., the definition of suitable approximating processes and detailed calculations can be found in Proksch et al (2012).…”

Section: Proof Of Corollarymentioning

confidence: 99%

“…The construction of multivariate confidence regions has received much less attention in the literature, see Rosenblatt (1976); Konakov and Piterbarg (1984); Rio (1994). However, in many cases, multivariate problems arise naturally such as in the analysis of astronomical or biological images taken with telescopes or microscopes, respectively, that also involves deconvolution (Proksch et al (2012)). Also in the problem of reconstructing a function from noisy observations of its Radon transform discussed in this paper we have to deal with a two-dimensional problem at least.…”

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Confidence regions for images observed under the Radon transform

Bissantz

Holzmann

Proksch

2014

Journal of Multivariate Analysis

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Recovering a function f from its integrals over hyperplanes (or line integrals in the two-dimensional case), that is, recovering f from the Radon transform Rf of f , is a basic problem with important applications in medical imaging such as computerized tomography (CT). In the presence of stochastic noise in the observed function Rf , we shall construct asymptotic uniform confidence regions for the function f of interest, which allows to draw conclusions regarding global features of f . Specifically, in a white noise model as well as a fixed-design regression model, we prove a Bickel-Rosenblatt-type theorem for the maximal deviation of a kernel-type estimator from its mean, and give uniform estimates for the bias for f in a Sobolev smoothness class. The finite sample properties of the proposed methods are investigated in a simulation study.

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“…Their bands are uniform with respect to a fixed but unspecified portion (smaller than one) of points in a possibly multidimensional set in contrast to the classical approach where uniformity is achieved on the complete set considered. Proksch et al (2015) proposed multivariate confidence bands for convolution type inverse regression models with fixed design.…”

Section: Introductionmentioning

confidence: 99%

Quantile Regression in Risk Calibration

Chao

Härdle

Wang

2014

Handbook of Financial Econometrics and Statistics

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Quantile regression studies the conditional quantile function Q Y |X (τ ) on X at level τ which satisfies F Y |X  Q Y |X (τ )  = τ , where F Y |X is the conditional CDF of Y given X, ∀τ ∈ (0, 1). Quantile regression allows for a closer inspection of the conditional distribution beyond the conditional moments. This technique is particularly useful in, for example, the Value-at-Risk (VaR) which the Basel accords (2011) require all banks to report, or the "quantile treatment effect" and "conditional stochastic dominance (CSD)" which are economic concepts in measuring the effectiveness of a government policy or a medical treatment.Given its value of applicability, to develop the technique of quantile regression is, however, more challenging than mean regression. It is necessary to be adept with general regression problems and M -estimators; additionally one needs to deal with non-smooth loss functions. In this dissertation, chapter 2 is devoted to empirical risk management during financial crises using quantile regression. Chapter 3 and 4 address the issue of high-dimensionality and the nonparametric technique of quantile regression.Chapter 2 applies nonparametric confidence bands for quantile functions to investigate the tail dependence of stock returns. It is shown that strong nonlinear correlation exists when stock prices drop, confirming the fact that in financial crises, firms are more dependent on each other than when the market is booming. This sheds light on the risk management of counterparty risk.In Chapter 3, motivated by applications in economics like quantile treatment effects, or conditional stochastic dominance, we focus on the construction of confidence corridors for nonparametric multivariate kernel quantile and expectile regression functions. Through an uniform kernel Bahadur representation for M -estimators, strong Gaussian approximation and asymptotic extreme value theory we derive the asymptotic confidence corridor for the nonparametric kernel conditional quantile/expectile functions. We find that the bands for quantile/expectile functions are wide when τ is close to 0 and 1 due to the variance of the estimator. The coverage ratios given by the asymptotic confidence corridors are meager. To deal with this issue, we propose a novel smoothing bootstrap which gives satisfactory coverage ratios while keeping the size of the confidence corridors in a reasonable range. Our method contributes to the differentiation between the "risk reduction CSD" and "potential enhancement CSD", which is not possible by using techniques based on previous research in CSD like Delgado and Escanciano (2013). This differentiation is crucial as the two types of CSD may induce different utility to the government and citizens. After applying our method to the data set from National Supported Work Demonstration, a temporary internship program offered to disadvantaged workers, it is found that this program tends to be "potential enhancement CSD" and it may not help foster the employment of less capable people as much...

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Confidence bands for multivariate and time dependent inverse regression models

Cited by 16 publications

References 32 publications

Asymptotic confidence sets for the jump curve in bivariate regression problems

Asymptotic confidence sets for the jump curve in bivariate regression problems

Confidence regions for images observed under the Radon transform

Quantile Regression in Risk Calibration

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