We consider the problem of using an autoregressive (AR) approximation to estimate the spectral density function and the n × n autocovariance matrix based on stationary data X₁, … , X_n. The consistency of the autoregressive spectral density estimator has been proven since the 1970s under a linearity assumption. We extend these ideas to the nonlinear setting, and give an application to estimating the n × n autocovariance matrix. Under mild assumptions on the underlying dependence structure and the order p of the fitted AR(p) model, we are able to show that the autoregressive spectral estimate and the associated AR-based autocovariance matrix estimator are consistent. We are also able to establish an explicit bound on the rate of convergence of the proposed estimators.

中文翻译：

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一致的自回归谱估计：非线性时间序列和大型自协方差矩阵

我们考虑使用自回归 (AR) 近似来估计谱密度函数和基于平稳数据X ₁ , … ,  X _n的n  ×  n自协方差矩阵的问题。自 1970 年代以来，在线性假设下已经证明了自回归谱密度估计量的一致性。我们将这些想法扩展到非线性设置，并给出了估计n  ×  n自协方差矩阵的应用。下对底层的依赖结构和顺序温和假设p拟合的AR（p) 模型，我们能够证明自回归谱估计和相关的基于 AR 的自协方差矩阵估计是一致的。我们还能够对所提议的估计器的收敛速度建立一个明确的界限。