On Estimation of \(L_{r}\)-Norms in Gaussian White Noise Models

Abstract

We provide a complete picture of asymptotically minimax estimation of \(L_r\)-norms (for any \(r\ge 1\)) of the mean in Gaussian white noise model over Nikolskii-Besov spaces. In this regard, we complement the work of Lepski, Nemirovski and Spokoiny (1999), who considered the cases of \(r=1\) (with poly-logarithmic gap between upper and lower bounds) and \(r\) even (with asymptotically sharp upper and lower bounds) over H\"{o}lder spaces. We additionally consider the case of asymptotically adaptive minimax estimation and demonstrate a difference between even and non-even \(r\) in terms of an investigator's ability to produce asymptotically adaptive minimax estimators without paying a penalty.