Best attainable rates of convergence for the estimation of the memory parameter

TL;DR

This paper establishes the fundamental limits on estimating the memory parameter in long memory processes, showing that without second order regular variation assumptions, convergence can be very slow, but the GPH estimator is optimal for Gaussian cases.

Contribution

It proves a lower bound for the convergence rate of memory parameter estimation without second order regular variation assumptions and confirms the optimality of the GPH estimator in Gaussian scenarios.

Findings

Convergence rates can be extremely slow without second order regular variation.

The GPH estimator achieves the optimal rate for Gaussian long memory processes.

Lower bounds on estimation accuracy are established under minimal assumptions.

Abstract

The purpose of this note is to prove a lower bound for the estimation of the memory parameter of a stationary long memory process. The memory parameter is defined here as the index of regular variation of the spectral density at 0. The rates of convergence obtained in the literature assume second order regular variation of the spectral density at zero. In this note, we do not make this assumption, and show that the rates of convergence in this case can be extremely slow. We prove that the log-periodogram regression (GPH) estimator achieves the optimal rate of convergence for Gaussian long memory processes.