On estimation of the \(L_r\) norm of a regression function
Let \(X(t)\), \(t\in[0,1]\), be a process satisfying the stochastic differential equation \[ dX(t)=f(t)dt+n^{-1/2}dW(t), \] where \(f(t)\) is an unknown function, and \(W(t)\), \(t\in[0,1]\), is the standard Wiener process. The authors' concern is to estimate the functional \[ \| f\|_r=\left[ \int^1_0\bigl| f(t)\bigr|^r dt\right]^{1/r}\quad \text{with given }r\geq 1. \] The asymptotic behavior of \[ {\mathcal B}(\widehat f_n)=\sup_{f\in\Sigma (\beta,L)}E\bigl| \widehat f_n-\| f\|_r \bigr|\text{ and }{\mathcal B}^*(n)= \inf_{\widehat f_n}\sup_{f\in\Sigma (\beta,L)}E\bigr| \widehat f_n-\| f\|_r\bigr | \] is studied. In particular, the main result shows that the optimal rate of convergence of the estimator of the functional \(\| f\|_r\) is better than the rate of convergence of nonparametric estimates of \(f(t)\).
scientific article; zbMATH DE number 1271431