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On a Strictly Decreasing Nonparametric Estimator of the Drift Function for Recurrent Diffusion Processes

Nicolas Marie

Abstract

This paper deals with a copies-based continuously differentiable and strictly decreasing estimator of the drift function for stochastic differential equations defining recurrent diffusion processes. The first part of our paper deals with non-asymptotic $\mathbb L^1$-risk bounds and a bandwidths selection procedure for a universal monotone estimator. These results are tailor-made to our framework, and then applied to the estimation of the drift function of recurrent diffusion processes in the second part of the paper.

On a Strictly Decreasing Nonparametric Estimator of the Drift Function for Recurrent Diffusion Processes

Abstract

This paper deals with a copies-based continuously differentiable and strictly decreasing estimator of the drift function for stochastic differential equations defining recurrent diffusion processes. The first part of our paper deals with non-asymptotic -risk bounds and a bandwidths selection procedure for a universal monotone estimator. These results are tailor-made to our framework, and then applied to the estimation of the drift function of recurrent diffusion processes in the second part of the paper.
Paper Structure (14 sections, 9 theorems, 98 equations, 2 figures, 1 table)

This paper contains 14 sections, 9 theorems, 98 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Non-asymptotic $\mathbb L^1$-risk bounds on some universal monotone estimators
  3. Back to the estimation of $a$ in Model (\ref{['main_equation']})
  4. Numerical experiments
  5. Proofs
  6. Proof of Proposition \ref{['risk_bound_monotone_estimator_inverse_b']}
  7. Proof of Lemma \ref{['error_bound_inverse_b']}
  8. Proof of Theorem \ref{['risk_bound_monotone_estimator_b']}
  9. Proof of Lemma \ref{['error_bound_b']}
  10. Proof of Proposition \ref{['risk_bound_practical_estimator_b']}
  11. Proof of Proposition \ref{['risk_bound_adaptive_monotone_estimator_b']}
  12. Proof of Theorem \ref{['risk_bound_monotone_estimator_a']}
  13. Proof of Lemma \ref{['risk_bound_NW_estimator_a']}
  14. Proof of Proposition \ref{['risk_bound_practical_estimator_a']}

Key Result

Proposition 2.2

Under Assumption assumption_b, if $h\in (0,\mathfrak m_b\varepsilon)$, then

Figures (2)

  • Figure 1: Plots of 5 adaptive strictly decreasing estimations (black dashed lines) of $a$ (red line) for Model (A).
  • Figure 2: Plots of 5 adaptive strictly decreasing estimations (black dashed lines) of $a$ (red line) for Model (B).

Theorems & Definitions (14)

  • Proposition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Proposition 2.5
  • Remark 2.6
  • Proposition 2.7
  • Remark 2.8
  • Theorem 3.1
  • Remark 3.2
  • Proposition 3.3
  • ...and 4 more