On Estimation of Lr-Norms in Gaussian White Noise Models

We provide a complete picture of asymptotically minimax estimation of Lr-norms (for any r≥ 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"older 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.