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On nonparametric estimation of the interaction function in particle system models

Denis Belomestny, Mark Podolskij, Shi-Yuan Zhou

TL;DR

There is a natural metric under which the corresponding minimax estimation error of the interaction function converges to zero with parametric rate, which is rather suprising given complexity of the underlying estimation problem and rather large classes of interaction functions for which the above parametric rate holds.

Abstract

This paper delves into a nonparametric estimation approach for the interaction function within diffusion-type particle system models. We introduce two estimation methods based upon an empirical risk minimization. Our study encompasses an analysis of the stochastic and approximation errors associated with both procedures, along with an examination of certain minimax lower bounds. In particular, we show that there is a natural metric under which the corresponding minimax estimation error of the interaction function converges to zero with parametric rate. This result is rather suprising given complexity of the underlying estimation problem and rather large classes of interaction functions for which the above parametric rate holds.

On nonparametric estimation of the interaction function in particle system models

TL;DR

There is a natural metric under which the corresponding minimax estimation error of the interaction function converges to zero with parametric rate, which is rather suprising given complexity of the underlying estimation problem and rather large classes of interaction functions for which the above parametric rate holds.

Abstract

This paper delves into a nonparametric estimation approach for the interaction function within diffusion-type particle system models. We introduce two estimation methods based upon an empirical risk minimization. Our study encompasses an analysis of the stochastic and approximation errors associated with both procedures, along with an examination of certain minimax lower bounds. In particular, we show that there is a natural metric under which the corresponding minimax estimation error of the interaction function converges to zero with parametric rate. This result is rather suprising given complexity of the underlying estimation problem and rather large classes of interaction functions for which the above parametric rate holds.
Paper Structure (20 sections, 14 theorems, 208 equations)

This paper contains 20 sections, 14 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. Risk minimization over compact functional classes
  3. Construction of the estimator and main results
  4. Approximation error
  5. Risk minimization over vector spaces
  6. Setting and construction of the estimator
  7. Main results
  8. Checking Assumption \ref{['assm']} and bounding the norm $\|\varphi_{\star} - \varphi\|_{\star \star}$
  9. Lower bounds
  10. Proofs
  11. Proof of Theorem \ref{['Th2.3']}
  12. Proof of Theorem \ref{['approxerr']}
  13. Proof of Proposition \ref{['prop1']}
  14. Some properties of $\Psi$
  15. Matrix concentration inequality for independent observations
  16. ...and 5 more sections

Key Result

Theorem 2.3

It holds under Assumption assLip, for any $p>2,$$2\leq q\leq p,$ where $C>0$ is an absolute constant and $\varphi^\star$ stands for the best approximation of $\varphi$ as defined in defphiN.

Theorems & Definitions (27)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Proposition 3.1
  • Theorem 3.2
  • Remark 3.3
  • ...and 17 more