记忆参数估计的最佳可达收敛速度
Best attainable rates of convergence for the estimation of the memory parameter
摘要：
本文证明了平稳长记忆过程记忆参数估计的一个下界。记忆参数在这里被定义为在0处的谱密度的规则变化的指数。在文献中得到的收敛速度假设二阶正则变化的谱密度在零。在本文中，我们不做这个假设，并表明在这种情况下的收敛速度可能非常慢。我们证明了对数周期图回归（GPH）估计达到高斯长记忆过程的最佳收敛速率。
英文摘要：
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.
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