Gaussianization Machines for Non-Gaussian Function Estimation Models

Abstract

A wide range of nonparametric function estimation models have been studied individually in the literature. Among them the homoscedastic nonparametric Gaussian regression is arguably the best known and understood. Inspired by the asymptotic equivalence theory, Brown, Cai and Zhou (Ann. Statist. 36 (2008) 2055–2084; Ann. Statist. 38 (2010) 2005–2046) and Brown et al. (Probab. Theory Related Fields 146 (2010) 401–433) developed a unified approach to turn a collection of non-Gaussian function estimation models into a standard Gaussian regression and any good Gaussian nonparametric regression method can then be used.

These Gaussianization Machines have two key components, binning and transformation. When combined with BlockJS, a wavelet thresholding procedure for Gaussian regression, the procedures are computationally efficient with strong theoretical guarantees. Technical analysis given in Brown, Cai and Zhou (Ann. Statist. 36 (2008) 2055–2084; Ann. Statist. 38 (2010) 2005–2046) and Brown et al. (Probab. Theory Related Fields 146 (2010) 401–433) shows that the estimators attain the optimal rate of convergence adaptively over a large set of Besov spaces and across a collection of non-Gaussian function estimation models, including robust nonparametric regression, density estimation, and nonparametric regression in exponential families. The estimators are also spatially adaptive.

The Gaussianization Machines significantly extend the flexibility and scope of the theories and methodologies originally developed for the conventional nonparametric Gaussian regression. This article aims to provide a concise account of the Gaussianization Machines developed in Brown, Cai and Zhou (Ann. Statist. 36 (2008) 2055–2084; Ann. Statist. 38 (2010) 2005–2046), Brown et al. (Probab. Theory Related Fields 146 (2010) 401–433).