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 et al. (Probab Theory Relat Fields 113(2):221–253, 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 Holder 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.

中文翻译：

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高斯白噪声模型中$$L_{r}$$-范数的估计

我们提供了在 Nikolskii-Besov 空间上的高斯白噪声模型中均值的 $$L_r$$ -范数（对于任何 $$r\ge 1$$ ）的渐近极大极小估计的完整图片。在这方面，我们补充了 Lepski 等人的工作。（Probab Theory Relat Fields 113(2):221–253, 1999），他考虑了 $$r=1$$（上下界之间的多对数差距）和 r even（渐近尖锐的上界和下界）的情况下限）在持有人空间。我们还考虑了渐近自适应极小极大估计的情况，并证明了偶数和非偶数 r 之间的差异，就研究人员生成渐近自适应极小极大估计器而无需支付惩罚的能力而言。