Asymptotic Equivalence for Nonparametric Regression

Abstract

We consider a nonparametric model ${\mathcal{E}}^n,$ generated by independent observations $X_i,$ $i=1,...,n,$ with densities $p(x,{\theta }_i),$ $i=1,...,n,$ the parameters of which ${\theta }_i=f(i/n)\in \Theta$ are driven by the values of an unknown function $f:[0,1]\to \Theta$ in a smoothness class. The main result of the paper is that, under regularity assumptions, this model can be approximated, in the sense of the Le Cam deficiency pseudodistance, by a nonparametric Gaussian shift model $Y_i=\Gamma (f(i/n))+{\epsilon }_i,$ where ${\epsilon }_1,...,{\epsilon }_n$ are i.i.d. standard normal r.v.'s, the function $\Gamma (\theta ):\Theta \to \mathrm{R}$ satisfies ${\Gamma }^{\prime }(\theta )=\sqrt{I(\theta )}$ and $I(\theta )$ is the Fisher information corresponding to the density $p(x,\theta ).$

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