Simultaneous sharp estimation of functions and their derivatives
There are a number of reasons for wishing to estimate both a function and its derivatives; for instance, first and second derivatives may be of intrinsic interest as measures of slope and curvature. So far the attention has focused on estimation where typically derivatives of a rate optimal estimate are rate optimal estimates of the corresponding derivatives. We consider the filtering statistical model where on the interval \([0,1]\) we observe random processes \(Y_n(t)\) having stochastic differential \[ dY_n(t)= f(t)dt+ n^{-1/2} dW(t), \quad 0\leq t\leq 1, \] and \(W(t)\) is a standard Brownian motion. We assume that \(f\) is one-periodic, including its \(\alpha\) derivatives, and belongs to the Sobolev class \(W^\alpha_2 Q=\{f: \int^1_0 [f^({(\alpha)} (t)]^2 dt\leq Q\), \(\alpha \geq 2\), \(0<Q< \infty\}\) where \(f^{(k)}\) denotes the \(k\) th derivative and \(f^{(0)} =f\). A data-driven estimate is given that, over the Sobolev space, is simultaneously asymptotically sharp minimax for estimating both the function and its derivatives under integrated squared error loss. It is also shown that linear estimates cannot be simultaneously asymptotically sharp minimax over a given Sobolev space.
scientific article; zbMATH DE number 1359517