On Estimation of $L_{r}$-Norms in Gaussian White Noise Models

TL;DR

This paper characterizes the asymptotic minimax estimation of $L_r$-norms in Gaussian white noise models over Nikolskii-Besov spaces, extending previous results and exploring adaptive estimation differences for even and non-even $r$.

Contribution

It completes the understanding of minimax $L_r$-norm estimation in Gaussian models for all $r \\ge 1$, including adaptive estimation and the distinction between even and non-even $r$.

Findings

Established asymptotic minimax bounds for all $r \\ge 1$.

Demonstrated differences in adaptive estimation capabilities between even and non-even $r$.

Extended previous work from H"{o}lder spaces to Nikolskii-Besov spaces.

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.