Asymptotic equivalence for nonparametric generalized linear models

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 Delta; 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 ti (nonparametric GLM). When f is in a Holder ball with exponent beta>1/2, we establish global asymptotic equivalence to observations of a signal Gamma(f(t)) in Gaussian white noise, where Gamma 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.