Asymptotic equivalence for nonparametric generalized linear models |
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Authors: | Ion Grama Michael Nussbaum |
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Institution: | (1) Institute of Mathematics, Academy of Sciences, Academiei Str. 5, Chişinău 277028, Moldova e-mail: 16grama@mathem.moldova.su, MD;(2) Weierstrass Institute, Mohrenstr. 39, D-10117 Berlin, Germany e-mail: nussbaum@wias-berlin.de, DE |
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Abstract: | 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 H?lder 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.
Received: 4 February 1997 |
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Keywords: | Mathematics Subject Classification (1991): Primary 62B15 Secondary 62G07 62G20 |
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