Minimax convergence rates of a binary plug-in type classification procedure for time-homogeneous SDE paths under low-noise conditions

TL;DR

This paper considers a more complex diffusion model with space-dependent drift and diffusion coefficients where the drift depends on the class and the diffusion coefficient is common to all classes, and establishes a faster convergence rate over a Holder space.

Abstract

The study of minimax convergence rates for classification procedures adapted to SDE paths is rarely addressed in the literature. Only one paper established optimal convergence rates for a binary classifier for SDE paths constructed from the white noise model. In this paper, we consider a more complex diffusion model with space-dependent drift and diffusion coefficients where the drift depends on the class and the diffusion coefficient is common to all classes. We establish, under the low-noise condition, a faster convergence rate over a Holder space. This result will require the establishment of an exponential inequality, which is essential to obtain the expected rate. We then study the lower bound on the excess risk of the empirical classifier.

Theorems & Definitions (17)

• Lemma 2.3
• Proposition 2.4
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5
• Lemma 5.1
• Lemma 5.2
• proof : Proof of Theorem \ref{['thm:exp-bound-drift']}
• proof : Proof of Theorem \ref{['thm:upper-bound']}
• proof