On Lasso and Slope drift estimators for L\'evy-driven Ornstein--Uhlenbeck processes

TL;DR

This paper demonstrates that Lasso and Slope estimators can optimally estimate the drift in high-dimensional Le9vy-driven Ornstein-Uhlenbeck processes under sparsity, with nonasymptotic guarantees.

Contribution

It establishes minimax optimal convergence rates for Lasso and Slope estimators in this setting, improving upon previous results for standard Ornstein-Uhlenbeck processes.

Findings

Lasso and Slope estimators achieve minimax optimal rates

Results are nonasymptotic and hold in probability and expectation

Estimates are valid under sparsity constraints

Abstract

We investigate the problem of estimating the drift parameter of a high-dimensional L\'evy-driven Ornstein--Uhlenbeck process under sparsity constraints. It is shown that both Lasso and Slope estimators achieve the minimax optimal rate of convergence (up to numerical constants), for tuning parameters chosen independently of the confidence level, which improves the previously obtained results for standard Ornstein--Uhlenbeck processes. The results are nonasymptotic and hold both in probability and conditional expectation with respect to an event resembling the restricted eigenvalue condition.