Efficient pointwise estimation based on discrete data in ergodic nonparametric diffusions

TL;DR

This paper introduces a new sequential estimation method for the drift coefficient at a specific point in ergodic diffusion processes using discrete data, achieving optimal convergence rates and efficiency.

Contribution

It develops a truncated sequential procedure with nonasymptotic error bounds and establishes its optimality and efficiency in pointwise estimation for ergodic diffusions.

Findings

Nonasymptotic upper bounds for estimation error.

Optimal convergence rate with sharp constants.

Proven efficiency of the proposed method.

Abstract

A truncated sequential procedure is constructed for estimating the drift coefficient at a given state point based on discrete data of ergodic diffusion process. A nonasymptotic upper bound is obtained for a pointwise absolute error risk. The optimal convergence rate and a sharp constant in the bounds are found for the asymptotic pointwise minimax risk. As a consequence, the efficiency is obtained of the proposed sequential procedure.