On Estimation of $L_{r}$-Norms in Gaussian White Noise Models
This work addresses a theoretical gap in statistical estimation for researchers in mathematical statistics, offering incremental extensions to prior results.
The paper tackles the problem of asymptotically minimax estimation of L_r-norms of the mean in Gaussian white noise models over Nikolskii-Besov spaces, providing a complete picture for all r ≥ 1 and demonstrating differences in adaptive estimation between even and non-even r.
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 Hölder 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.