ResearchTrend.AI
  • Papers
  • Communities
  • Events
  • Blog
  • Pricing
Papers
Communities
Social Events
Terms and Conditions
Pricing
Parameter LabParameter LabTwitterGitHubLinkedInBlueskyYoutube

© 2025 ResearchTrend.AI, All rights reserved.

  1. Home
  2. Papers
  3. 2405.16515
61
1

Adaptive estimation of L2\mathbb{L}_2L2​-norm of a probability density and related topics I. Lower bounds

26 May 2024
G. Cleanthous
A. G. Georgiadis
Oleg V. Lepski
ArXiv (abs)PDFHTML
Abstract

We deal with the problem of the adaptive estimation of the L2\mathbb{L}_2L2​-norm of a probability density on Rd\mathbb{R}^dRd, d≥1d\geq 1d≥1, from independent observations. The unknown density is assumed to be uniformly bounded and to belong to the union of balls in the isotropic/anisotropic Nikolskii's spaces. We will show that the optimally adaptive estimators over the collection of considered functional classes do no exist. Also, in the framework of an abstract density model we present several generic lower bounds related to the adaptive estimation of an arbitrary functional of a probability density. These results having independent interest have no analogue in the existing literature. In the companion paper Cleanthous et al (2024) we prove that established lower bounds are tight and provide with explicit construction of adaptive estimators of L2\mathbb{L}_2L2​-norm of the density.

View on arXiv
Comments on this paper