Translator Disclaimer
November 2019 Gaussianization Machines for Non-Gaussian Function Estimation Models
T. Tony Cai
Statist. Sci. 34(4): 635-656 (November 2019). DOI: 10.1214/19-STS718

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).

Citation

Download Citation

T. Tony Cai. "Gaussianization Machines for Non-Gaussian Function Estimation Models." Statist. Sci. 34 (4) 635 - 656, November 2019. https://doi.org/10.1214/19-STS718

Information

Published: November 2019
First available in Project Euclid: 8 January 2020

zbMATH: 07240220
MathSciNet: MR4048595
Digital Object Identifier: 10.1214/19-STS718

Rights: Copyright © 2019 Institute of Mathematical Statistics

JOURNAL ARTICLE
22 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.34 • No. 4 • November 2019
Back to Top