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

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.

References in corpus (3)

• Multifractal stationary random measures and multifractal random walks with log-infinitely divisible scaling laws
• Martingale approximations for sums of stationary processes
• Stochastic equations generating continuous multiplicative cascades

Cited by in corpus (5)

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• New procedures controlling the false discovery proportion via Romano-Wolf's heuristic
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• Asymptotics for Duration-Driven Long Range Dependent Processes
• Shift-invariant homogeneous classes of random fields