# Asymptotic equivalence of non-parametric regression with spherical regressors and Gaussian white noise

## TL;DR

This paper proves that non-parametric regression on spherical data with different sampling designs becomes statistically equivalent to Gaussian white noise models as sample size grows, under certain smoothness conditions.

## Contribution

It establishes the asymptotic equivalence of regression experiments with spherical designs and Gaussian white noise, extending the Le Cam theory to spherical regressors.

## Key findings

- Regression experiments are asymptotically equivalent to Gaussian white noise models.
- Equivalence holds over spherical Sobolev and Besov balls under specified conditions.
- Sharpness of smoothness assumptions is demonstrated through non-equivalence results.

## Abstract

We study the asymptotic behaviour of both spherical $t$-designs and random uniform designs as the set of sampling points in non-parametric regression with spherical regressors of arbitrary dimension. We show that the corresponding regression experiments are asymptotically equivalent, in the sense of Le Cam, to the same sequence of Gaussian white noise experiments as the sample size tends to infinity. More precisely, global asymptotic equivalence is established over spherical Sobolev balls (for both the fixed and the random uniform design case) and over spherical Besov balls (for the fixed design case). Matching non-equivalence results demonstrate that the imposed smoothness assumptions are essentially sharp.