RECORD
Adaptive estimation of the $\mathbb{L}_2$-norm of a probability density and related topics II. Upper bounds via the oracle approach
Statistics Theory · Submitted 26 May 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this part we show that these bounds are tight and present the adaptive estimator which is obtained by a data-driven selection from a family of kernel-based estimators.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
This is the second part of the research project initiated in Cleanthous et al (2024). We deal with the problem of the adaptive estimation of the $\mathbb{L}_2$-norm of a probability density on $\mathbb{R}^d$, $d\geq 1$, from independent observations. The unknown density is assumed to be uniformly bounded by unknown constant and to belong to the union of balls in the isotropic/anisotropic Nikolskii's spaces. In Cleanthous et al (2024) we have proved that the optimally adaptive estimators do no exist in the considered problem and provided with several lower bounds for the adaptive risk. In this part we show that these bounds are tight and present the adaptive estimator which is obtained by a data-driven selection from a family of kernel-based estimators. The proposed estimation procedure as well as the computation of its risk are heavily based on new concentration inequalities for decoupled $U$-statistics of order two established in Section 4. It is also worth noting that all our results are derived from the unique oracle inequality which may be of independent interest.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | This is the second part of the research project initiated in Cleanthous et al (2024). | background 1.00 | 0.11 | 0.02 | asserted 1.00 |
| s1 | We deal with the problem of the adaptive estimation of the $\mathbb{L}_2$-norm of a probability density on $\mathbb{R}^d$, $d\geq 1$, from independent observations. | background 0.84 | 0.67 | 0.04 | asserted 0.87 |
| s2 | The unknown density is assumed to be uniformly bounded by unknown constant and to belong to the union of balls in the isotropic/anisotropic Nikolskii's spaces. | background 0.96 | 0.14 | 0.02 | asserted 1.00 |
| s3 | In Cleanthous et al (2024) we have proved that the optimally adaptive estimators do no exist in the considered problem and provided with several lower bounds for the adaptive risk. | background 0.94 | 0.64 | 0.07 | asserted 1.00 |
| s4 | In this part we show that these bounds are tight and present the adaptive estimator which is obtained by a data-driven selection from a family of kernel-based estimators. | contribution 0.64 | 0.96 | 0.77 | asserted 1.00 |
| s5 | The proposed estimation procedure as well as the computation of its risk are heavily based on new concentration inequalities for decoupled $U$-statistics of order two established in Section 4. | method 0.97 | 0.84 | 0.06 | asserted 1.00 |
| s6 | It is also worth noting that all our results are derived from the unique oracle inequality which may be of independent interest. | interpretation 0.46 | 0.92 | 0.09 | hedged possibility 0.66 |
What was examined, and what was not
- arXiv listing abstract v1
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
This is the second part of the research project initiated in Cleanthous et al (2024). We deal with the problem of the adaptive estimation of the $\mathbb{L}_2$-norm of a probability density on $\mathbb{R}^d$, $d\geq 1$, from independent observations. The unknown density is assumed to be uniformly bounded by unknown constant and to belong to the union of balls in the isotropic/anisotropic Nikolskii's spaces. In Cleanthous et al (2024) we have proved that the optimally adaptive estimators do no exist in the considered problem and provided with several lower bounds for the adaptive risk. In this part we show that these bounds are tight and present the adaptive estimator which is obtained by a data-driven selection from a family of kernel-based estimators. The proposed estimation procedure as well as the computation of its risk are heavily based on new concentration inequalities for decoupled $U$-statistics of order two established in Section 4. It is also worth noting that all our results are derived from the unique oracle inequality which may be of independent interest.
Human review
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.