Asymptotic equivalence for nonparametric generalized linear models
- Grama, Ion G.
- Nussbaum, Michael
2010 Mathematics Subject Classification
- 62B15 62G07 62G20
- Nonparametric regression, statistical experiment, deficiency distance, global white noise approximation, exponential family, variance stabilizing transform
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 ƒ(ti) of a regression function ƒ at a grid point ti (nonparametric GLM). When ƒ is in a Hölder ball with exponent β > 1⁄2, 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.
- Probab. Theory Related Fields, 111 (1998), pp. 167-214