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 $p(x,\hat{I}{¸}_i),$ $i=1,...,n,$ the parameters of which $\hat{I}{¸}_i=f(i/n)\in \hat{I}˜$ are driven by the values of an unknown function $f:[0,1]\to \hat{I}˜$ 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 $Y_i=\hat{I}“(f(i/n))+{\epsilon }_i,$ where ${\epsilon }_1,...,{\epsilon }_n$ are i.i.d. standard normal r.v.'s, the function $\hat{I}“(\hat{I}¸):\hat{I}˜\to \mathrm{R}$ satisfies $\hat{I}{“}^{\prime }(\hat{I}¸)=\sqrt{I(\hat{I}¸)}$ and $I(\hat{I}¸)$ is the Fisher information corresponding to the density $p(x,\hat{I}¸).$

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