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

Philippe Soulier (1)

1 MODAL'X - Modélisation aléatoire de Paris X

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.

Mots clés: Long memory time series; memory parameter estimation; minimax rates of convergence

Domaines: Statistiques [math.ST]

Philippe Soulier. Best attainable rates of convergence for the estimation of the memory parameter. Paul Doukhan, Gabriel Lang, Donatas Surgailis and Gilles Teyssière. Dependence in Probability and Statistics, Springer, pp.45-37, 2010, Lecture Notes in Statistics, Volume 200, ⟨10.1007/978-3-642-14104-1⟩. ⟨hal-00390120⟩