Asymptotic approximation of nonparametric regression experiments with unknown variances

TL;DR

This paper develops an asymptotic approximation for nonparametric regression experiments accounting for unknown variances, separating information about the mean and variance, and extending to slowly varying variances.

Contribution

It introduces an asymptotic equivalence framework for nonparametric regression with unknown variances, a scenario common in practice but previously unaddressed.

Findings

Asymptotic equivalence separates variance and mean information.

Extension to regression with slowly varying variances.

Framework applicable to practical nonparametric regression problems.

Abstract

Asymptotic equivalence results for nonparametric regression experiments have always assumed that the variances of the observations are known. In practice, however the variance of each observation is generally considered to be an unknown nuisance parameter. We establish an asymptotic approximation to the nonparametric regression experiment when the value of the variance is an additional parameter to be estimated or tested. This asymptotically equivalent experiment has two components: the first contains all the information about the variance and the second has all the information about the mean. The result can be extended to regression problems where the variance varies slowly from observation to observation.