The Annals of Statistics

Asymptotic equivalence for regression under fractional noise

Johannes Schmidt-Hieber

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Consider estimation of the regression function based on a model with equidistant design and measurement errors generated from a fractional Gaussian noise process. In previous literature, this model has been heuristically linked to an experiment, where the anti-derivative of the regression function is continuously observed under additive perturbation by a fractional Brownian motion. Based on a reformulation of the problem using reproducing kernel Hilbert spaces, we derive abstract approximation conditions on function spaces under which asymptotic equivalence between these models can be established and show that the conditions are satisfied for certain Sobolev balls exceeding some minimal smoothness. Furthermore, we construct a sequence space representation and provide necessary conditions for asymptotic equivalence to hold.

Article information

Ann. Statist., Volume 42, Number 6 (2014), 2557-2585.

First available in Project Euclid: 12 November 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression
Secondary: 62G20: Asymptotic properties

Asymptotic equivalence long memory fractional Brownian motion fractional Gaussian noise fractional calculus inverse problems nonharmonic Fourier series reproducing kernel Hilbert space (RKHS) stationarity


Schmidt-Hieber, Johannes. Asymptotic equivalence for regression under fractional noise. Ann. Statist. 42 (2014), no. 6, 2557--2585. doi:10.1214/14-AOS1262.

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Supplemental materials

  • Supplementary material: Asymptotic equivalence for regression under fractional noise. The supplement contains proofs for Section 4, some technical results and a brief summary of the Le Cam distance.