Optimal estimation and confidence sets for discontinuities in noisy, blurred regression functions
Final Report Abstract
Discontinuity points and other irregularities are important features of signals that are of interest in various areas such as economics, medicine or the physical sciences. For example, a discontinuity in the mean function of a time series indicates a change in the trend, or a change curve - called edge - in a two-dimensional image function may correspond to the location of a particular object. The main focus of the project was on the construction of confidence sets for such irregularities. In a first subproject, we considered the estimation of the location and of the height of the jump in the γ-th derivative (a kink of order γ) of a regression curve, which was assumed to be Hölder smooth of order s ≥ γ + 1 away from the kink. Optimal convergence rates as well as the joint asymptotic normal distribution of estimators based on the so-called zero-crossing-time technique were established. Surprisingly the estimates for the two parameters turned out to be asymptotically independent, and the discretization bias from the fixed-design regression turned out to be asymptotically negligible. We further constructed joint as well as marginal asymptotic confidence sets for these parameters which are honest and adaptive with respect to the smoothness parameter s over subsets of the Hölder classes. We applied the method to a time series of annual global surface temperatures. In analogy to change points it would be of interest in future work to construct segmentation algorithms based on kinks. In the time series example these would signify segments of different trend periods. In a second subproject, we constructed uniform and pointwise asymptotic confidence sets for the edge in an otherwise smooth image function which were based on rotated differences of two onesided kernel estimators. For this problem, no methods were previously available. We developed a uniform linearization of the contrast process. The uniform confidence bands then relied on a Gaussian approximation of the score process together with anti-concentration results for suprema of Gaussian processes, while pointwise bands were based on asymptotic normality. We also gave an illustration to real-world image processing with a relatively high noise level. It would be of interest in future work to combine the confidence sets with an edge-detection step. In the remaining parts of the project, we shifted the focus of study from deblurring - that is, deconvolution - to signal recovery in other statistical inverse problems which appeared to be more timely during the phase of the project. In a third subproject, we investigated rate-optimal estimation in random coefficient regression models, which have seen renewed interest in the recent econometric literature. We obtained novel optimal pointwise convergence rate for estimating the density over Hölder smoothness classes, and in particular showed how the tail behavior of the design density impacts this rate. The optimal choice of the tuning parameters in our estimator depends on the tail parameter of the design density and on the smoothness parameter of the Hölder class, and we also showed how to make the estimator adaptive with respect to these two parameters. In a final fourth subproject, we considered a nonparametric measurement error model of Berkson type with fixed design regressors, in contrast to much existing work with random predictors containing random noise. We derived uniform confidence statements for the function of interest under weaker assumptions on error density than are usually imposed in the literature, using methods which are closely related to those in the second subproject. Overall, the project contributed to enhancing the understanding of the construction of estimators and confidence sets for nonparametric models with irregularities.
Publications
- (2019) Asymptotic confidence sets for the jump curve in bivariate regression problems. Journal of Multivariate Analysis 173 291–312
Bengs, Viktor; Eulert, Matthias; Holzmann, Hajo
(See online at https://doi.org/10.1016/j.jmva.2019.02.017) - (2017). The Triangular Model with Random Coefficients. Journal of Econometrics 201 144-169
Hoderlein, S., Holzmann, H., and Meister, A
(See online at https://doi.org/10.1016/j.jeconom.2017.05.018)
