publication . Preprint . 2017

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

Han, Yanjun; Jiao, Jiantao; Mukherjee, Rajarshi;
Open Access English
  • Published: 10 Oct 2017
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 with...
Subjects
arXiv: Statistics::Theory
free text keywords: Mathematics - Statistics Theory, Computer Science - Machine Learning
Funded by
NSF| Emerging Frontiers of Science of Information
Project
  • Funder: National Science Foundation (NSF)
  • Project Code: 0939370
  • Funding stream: Directorate for Computer & Information Science & Engineering | Division of Computing and Communication Foundations
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Lemma 5.2. Under the conditions of Theorem 3.4, the following hold for all x ∈ [0, 1], k ≥ 2, c1 > √8k, and constants C1 (depending on (c1, c2, ǫ, σ, KM )) and C2 (depending on (c1, c2, ǫ, σ, KM , k)).

= P Lemma 6.3. (Boucheron, Lugosi and Massart, 2013) For X ∼ N (µ, σ 2), we have Lemma 6.12. Let X, Y ∼ N (µ, λ 2h) be independent, k ≥ 2 be any integer, 4c21 ≥ c2, c2 ln n ≥ 1, and c1 > √8k, 7c2 ln 2 < ǫ ∈ (0, 1), nh ≥ 1. Then there exists constants C1 (depending on (c1, c2, ǫ, KM )) and C2 (depending on (c1, c2, ǫ, KM , k)) such that the following hold.

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