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Asymptotic equivalence for nonparametric generalized linear models
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  • Published: June 1998

Asymptotic equivalence for nonparametric generalized linear models

  • Ion Grama1 &
  • Michael Nussbaum2 

Probability Theory and Related Fields volume 111, pages 167–214 (1998)Cite this article

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  • 40 Citations

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Summary.

We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance Δ; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value f(t i ) of a regression function f at a grid point t i (nonparametric GLM). When f is in a Hölder ball with exponent we establish global asymptotic equivalence to observations of a signal Γ(f(t)) in Gaussian white noise, where Γ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process.

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Authors and Affiliations

  1. Institute of Mathematics, Academy of Sciences, Academiei Str. 5, Chişinău 277028, Moldova e-mail: 16grama@mathem.moldova.su, , , , , , MD

    Ion Grama

  2. Weierstrass Institute, Mohrenstr. 39, D-10117 Berlin, Germany e-mail: nussbaum@wias-berlin.de, , , , , , DE

    Michael Nussbaum

Authors
  1. Ion Grama
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  2. Michael Nussbaum
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Received: 4 February 1997

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Grama, I., Nussbaum, M. Asymptotic equivalence for nonparametric generalized linear models. Probab Theory Relat Fields 111, 167–214 (1998). https://doi.org/10.1007/s004400050166

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  • Issue Date: June 1998

  • DOI: https://doi.org/10.1007/s004400050166

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  • Mathematics Subject Classification (1991): Primary 62B15; Secondary 62G07
  • 62G20
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