Nonparametric regression in exponential families

TL;DR

This paper introduces a unified method for nonparametric regression in exponential families using variance stabilizing transformations, enabling the application of Gaussian regression techniques like wavelet thresholding with proven adaptivity and near-optimal performance.

Contribution

It proposes a mean-matching variance stabilizing transformation to convert complex exponential family regression into Gaussian regression, facilitating the use of existing adaptive methods.

Findings

Estimators achieve near-optimal asymptotic rates.

Method demonstrates high adaptivity and spatial adaptivity.

Numerical results confirm strong practical performance.

Abstract

Most results in nonparametric regression theory are developed only for the case of additive noise. In such a setting many smoothing techniques including wavelet thresholding methods have been developed and shown to be highly adaptive. In this paper we consider nonparametric regression in exponential families with the main focus on the natural exponential families with a quadratic variance function, which include, for example, Poisson regression, binomial regression and gamma regression. We propose a unified approach of using a mean-matching variance stabilizing transformation to turn the relatively complicated problem of nonparametric regression in exponential families into a standard homoscedastic Gaussian regression problem. Then in principle any good nonparametric Gaussian regression procedure can be applied to the transformed data. To illustrate our general methodology, in this paper we use wavelet block thresholding to construct the final estimators of the regression function. The procedures are easily implementable. Both theoretical and numerical properties of the estimators are investigated. The estimators are shown to enjoy a high degree of adaptivity and spatial adaptivity with near-optimal asymptotic performance over a wide range of Besov spaces. The estimators also perform well numerically.