# Asymptotic confidence sets for the jump curve in bivariate regression problems

## TL;DR

This paper develops asymptotic confidence sets for detecting edges in bivariate images, combining kernel estimators, M-estimation, and Gaussian approximations to provide reliable inference.

## Contribution

It introduces a novel method for constructing uniform and point-wise confidence sets for image edges using rotated kernel differences and advanced probabilistic tools.

## Key findings

- Confidence sets achieve correct coverage in simulations
- Method effectively detects edges in real-world images
- Finite-sample performance aligns with asymptotic theory

## Abstract

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 M-estimation, 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.