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Nonparametric regression in natural exponential families

T. Toni Cai and Harrison H. Zhou

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Abstract

Theory and methodology for nonparametric regression have been particularly well developed in the case of additive homoscedastic Gaussian noise. Inspired by asymptotic equivalence theory, there have been ongoing efforts in recent years to construct explicit procedures that turn other function estimation problems into a standard nonparametric regression with Gaussian noise. Then in principle any good Gaussian nonparametric regression method can be used to solve those more complicated nonparametric models. In particular, Brown, Cai and Zhou [3] considered nonparametric regression in natural exponential families with a quadratic variance function.

In this paper we extend the scope of Brown, Cai and Zhou [3] to general natural exponential families by introducing a new explicit procedure that is based on the variance stabilizing transformation. The new approach significantly reduces the bias of the inverse transformation and as a consequence it enables the method to be applicable to a wider class of exponential families. Combining this procedure with a wavelet block thresholding estimator for Gaussian nonparametric regression, we show that the resulting estimator enjoys a high degree of adaptivity and spatial adaptivity with near-optimal asymptotic performance over a broad range of Besov spaces.

Chapter information

Source
James O. Berger, T. Tony Cai and Iain M. Johnstone, eds., Borrowing Strength: Theory Powering Applications – A Festschrift for Lawrence D. Brown (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2010), 199-215

Dates
First available in Project Euclid: 26 October 2010

Permanent link to this document
https://projecteuclid.org/euclid.imsc/1288099021

Digital Object Identifier
doi:10.1214/10-IMSCOLL614

Mathematical Reviews number (MathSciNet)
MR2798520

Zentralblatt MATH identifier
1202.62050

Subjects
Primary: 62J08
Secondary: 62G20: Asymptotic properties

Keywords
adaptivity asymptotic equivalence exponential family James-Stein estimator Gaussian nonparametric regression quantile coupling wavelets

Rights
Copyright © 2010, Institute of Mathematical Statistics

Citation

Cai, T. Toni; Zhou, Harrison H. Nonparametric regression in natural exponential families. Borrowing Strength: Theory Powering Applications – A Festschrift for Lawrence D. Brown, 199--215, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2010. doi:10.1214/10-IMSCOLL614. https://projecteuclid.org/euclid.imsc/1288099021


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