Confidence Corridors for Multivariate Generalized Quantile Regression

We focus on the construction of confidence corridors for multivariate nonparametric generalized quantile regression functions. This construction is based on asymptotic results for the maximal deviation between a suitable nonparametric estimator and the true function of interest which follow after a series of approximation steps including a Bahadur representation, a new strong approximation theorem and exponential tail inequalities for Gaussian random fields. As a byproduct we also obtain confidence corridors for the regression function in the classical mean regression. In order to deal with the problem of slowly decreasing error in coverage probability of the asymptotic confidence corridors, which results in meager coverage for small sample sizes, a simple bootstrap procedure is designed based on the leading term of the Bahadur representation. The finite sample properties of both procedures are investigated by means of a simulation study and it is demonstrated that the bootstrap procedure considerably outperforms the asymptotic bands in terms of coverage accuracy. Finally, the bootstrap confidence corridors are used to study the efficacy of the National Supported Work Demonstration, which is a randomized employment enhancement program launched in the 1970s. This article has supplementary materials

Multiscale inference for a multivariate density with applications to X-ray astronomy

In this paper we propose methods for inference of the geometric features of a multivariate density. Our approach uses multiscale tests for the monotonicity of the density at arbitrary points in arbitrary directions. In particular, a significance test for a mode at a specific point is constructed. Moreover, we develop multiscale methods for identifying regions of monotonicity and a general procedure for detecting the modes of a multivariate density. It is is shown that the latter method localizes the modes with an effectively optimal rate. The theoretical results are illustrated by means of a simulation study and a data example. The new method is applied to and motivated by the determination and verification of the position of high-energy sources from X-ray observations by the Swift satellite which is important for a multiwavelength analysis of objects such as Active Galactic Nuclei.Comment: Keywords and Phrases: multiple tests, modes, multivariate density, X-ray astronomy AMS Subject Classification: 62G07, 62G10, 62G2

Simultaneous inference for Berkson errors-in-variables regression under fixed design

In various applications of regression analysis, in addition to errors in the dependent observations also errors in the predictor variables play a substantial role and need to be incorporated in the statistical modeling process. In this paper we consider a nonparametric measurement error model of Berkson type with fixed design regressors and centered random errors, which is in contrast to much existing work in which the predictors are taken as random observations with random noise. Based on an estimator that takes the error in the predictor into account and on a suitable Gaussian approximation, we derive %uniform confidence statements for the function of interest. In particular, we provide finite sample bounds on the coverage error of uniform confidence bands, where we circumvent the use of extreme-value theory and rather rely on recent results on anti-concentration of Gaussian processes. In a simulation study we investigate the performance of the uniform confidence sets for finite samples

Gromov-Wasserstein Distance based Object Matching: Asymptotic Inference

In this paper, we aim to provide a statistical theory for object matching based on the Gromov-Wasserstein distance. To this end, we model general objects as metric measure spaces. Based on this, we propose a simple and efficiently computable asymptotic statistical test for pose invariant object discrimination. This is based on an empirical version of a $\beta$-trimmed lower bound of the Gromov-Wasserstein distance. We derive for $\beta\in[0,1/2)$ distributional limits of this test statistic. To this end, we introduce a novel $U$-type process indexed in $\beta$ and show its weak convergence. Finally, the theory developed is investigated in Monte Carlo simulations and applied to structural protein comparisons.Comment: For a version with the complete supplement see [v2

Risk estimators for choosing regularization parameters in ill-posed problems - Properties and limitations

This paper discusses the properties of certain risk estimators that recently regained popularity for choosing regularization parameters in ill-posed problems, in particular for sparsity regularization. They apply Stein’s unbiased risk estimator (SURE) to estimate the risk in either the space of the unknown variables or in the data space. We will call the latter PSURE in order to distinguish the two different risk functions. It seems intuitive that SURE is more appropriate for ill-posed problems, since the properties in the data space do not tell much about the quality of the reconstruction. We provide theoretical studies of both approaches for linear Tikhonov regularization in a finite dimensional setting and estimate the quality of the risk estimators, which also leads to asymptotic convergence results as the dimension of the problem tends to infinity. Unlike previous works which studied single realizations of image processing problems with a very low degree of ill-posedness, we are interested in the statistical behaviour of the risk estimators for increasing ill-posedness. Interestingly, our theoretical results indicate that the quality of the SURE risk can deteriorate asymptotically for ill-posed problems, which is confirmed by an extensive numerical study. The latter shows that in many cases the SURE estimator leads to extremely small regularization parameters, which obviously cannot stabilize the reconstruction. Similar but less severe issues with respect to robustness also appear for the PSURE estimator, which in comparison to the rather conservative discrepancy principle leads to the conclusion that regularization parameter choice based on unbiased risk estimation is not a reliable procedure for ill-posed problems. A similar numerical study for sparsity regularization demonstrates that the same issue appears in non-linear variational regularization approaches