Asymptotic Equivalence for Nonparametric Regression with Non-Regular Errors

TL;DR

This paper extends the concept of asymptotic equivalence in nonparametric regression to models with non-regular errors that have jump discontinuities, linking them to Poisson point processes.

Contribution

It establishes asymptotic equivalence between regression models with non-regular errors and Poisson point processes, a novel result for models with jump discontinuities.

Findings

Proves asymptotic equivalence for non-regular error models

Characterizes the Poisson process intensity involving jump sizes and design density

Highlights differences from Gaussian error models

Abstract

Asymptotic equivalence in Le Cam's sense for nonparametric regression experiments is extended to the case of non-regular error densities, which have jump discontinuities at their endpoints. We prove asymptotic equivalence of such regression models and the observation of two independent Poisson point processes which contain the target curve as the support boundary of its intensity function. The intensity of the point processes is of order of the sample size $n$ and involves the jump sizes as well as the design density. The statistical model significantly differs from regression problems with Gaussian or regular errors, which are known to be asymptotically equivalent to Gaussian white noise models.