Asymptotic Equivalence for Nonparametric Generalized Linear Models

TL;DR

This paper proves that nonparametric generalized linear models with non-Gaussian noise are asymptotically equivalent to Gaussian models, enabling simpler analysis and inference in complex statistical settings.

Contribution

It establishes the first rigorous asymptotic equivalence between nonparametric GLMs and Gaussian models for a broad class of functions, extending Gaussian approximation results.

Findings

Models are asymptotically equivalent in Le Cam's sense.

Equivalence holds for functions in Hölder spaces with exponent > 1/2.

Results facilitate simpler Gaussian-based analysis for complex models.

Abstract

We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance $\Delta $; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value $f(t_i)$ of a regression function $f$ at a grid point $t_i$ (nonparametric GLM). When $f$ is in a H\"{o}lder ball with exponent $\beta >\frac 12 ,$ we establish global asymptotic equivalence to observations of a signal $\Gamma (f(t))$ in Gaussian white noise, where $\Gamma $ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process.