On the estimation of the derivatives of a function with the derivatives of an estimate
Let \({\tilde \theta}{}_ n(x)\) be an estimator of a smooth function \(\theta\) (x). It is proved that \(\theta\) (x) can be estimated easier than its derivative \(\theta^{(s)}(x)\), providing for \(\| {\hat \theta}_ n^{(s)}-\theta^{(s)}\|_ q\) an upper bound that depends on \(\| {\hat \theta}_ n-\theta \|_ q\). The same bound can be used as a tool to derive automatically rates of convergence when we are estimating derivatives of densities or regression functions.
bounds on norm differences of derivatives of smooth functions
scientific article; zbMATH DE number 4088733