# Optimal rate of convergence for nonparametric estimators in Sobolev space

Given a regression model on interval $[0,1]$ $$Y_{i}=f(x_{i})+\epsilon_{i},\ i=1,\ldots,N$$ with fixed design and standard error assumptions $E(\epsilon_{i})=0;\ E(\epsilon_{i}\epsilon_{j})=\delta_{i,j}\sigma^{2}$. The regression function $f$ is from Sobolev space $$f\in W^{q}\left[0,1\right]=\left[f,\ldots,f^{(q-1)} \text{ are absolutely continuous, } \int_{0}^{1}\left| f^{(q)}(x)\right| ^{2}<\infty\right]$$ Is the optimal convergence rate with respect to $L_{2}$-norm $N^{-q/(2q+1)}$ ? If yes, could you give a reference? If not, what additional assumptions are required ?

Yes, this is the classical Pinsker's theorem for Sobolev ellipsoids. Your rate is slightly off; also you need to restrict to $L^2$-balls. Part of the theorem says that, for $R > 0$,
$$\inf_{T_n} \sup_{f \in W^q[0,1], \|f\|_{L^2}<R} \| T_n f - f \|_{L^2} = O(n^{-\frac{2q}{2q+1}})$$
where $\inf_{T_n}$ denotes infimum over all estimators measurable with respect to data. Pinsker actually gave an linear shrinkage estimator that achieve the minimax risk exactly. Pinksker's estimator is, however, nonadaptive---it requires the smoothness parameter $q$ to be known. Later Stein gave a SURE block-threshold estimator that is adaptive over all $q>0$.