Optimal Rates of Convergence for Nonparametric Statistical Inverse Problems

Abstract

Consider an unknown regression function $f$ of the response $Y$ on a $d$-dimensional measurement variable $X$. It is assumed that $f$ belongs to a class of functions having a smoothness measure $p$. Let $T$ denote a known linear operator of order $q$ which maps $f$ to another function $T(f)$ in a space $G$. Let $\hat{T}_n$ denote an estimator of $T(f)$ based on a random sample of size $n$ from the distribution of $(X, Y)$, and let $\|\hat{T}_n - T(f)\|_G$ be a norm of $\hat{T}_n - T(f)$. Under appropriate regularity conditions, it is shown that the optimal rate of convergence for $\|\hat{T}_n - T(f)\|_G$ is $n^{-(p - q)/(2p + d)}$. The result is applied to differentiation, fractional differentiation and deconvolution.