Empirical detection of long range dependence (LRD) of a time series often consists of deciding whether an estimate of the memory parameter $d$ corresponds to LRD. Surprisingly, the literature offers numerous spectral domain estimators for $d$ but there are only a few estimators in the time domain. Moreover, the latter estimators are criticized for relying on visual inspection to determine an observation window $[n_1, n_2]$ for a linear regression to run on. Theoretically motivated choices of $n_1$ and $n_2$ are often missing for many time series models. In this paper, we take the well-known variance plot estimator and provide rigorous asymptotic conditions on $[n_1, n_2]$ to ensure the estimator's consistency under LRD. We establish these conditions for a large class of square-integrable time series models. This large class enables one to use the variance plot estimator to detect LRD for infinite-variance time series (after suitable transformation). Thus, detection of LRD for infinite-variance time series is another novelty of our paper. A simulation study indicates that the variance plot estimator can detect LRD better than the popular spectral domain GPH estimator.

翻译：时间序列长程依赖性的实证检测通常涉及判断记忆参数$d$的估计值是否对应长程依赖性。令人惊讶的是，文献中提供了大量$d$的频域估计方法，但时域估计方法却寥寥无几。此外，后者常被批评为依赖于目视检查来确定线性回归的运行观测窗口$[n_1, n_2]$。对于许多时间序列模型而言，$n_1$和$n_2$的理论驱动选择往往缺失。本文采用著名的方差图估计方法，并严格推导了$[n_1, n_2]$的渐近条件，以确保该估计量在长程依赖性下的一致性。我们针对一大类平方可积时间序列模型建立了这些条件。该大类模型使得方差图估计量能够（通过适当变换）用于检测无限方差时间序列的长程依赖性。因此，对无限方差时间序列长程依赖性的检测是本文的另一创新。模拟研究表明，方差图估计量在检测长程依赖性方面优于流行的频域GPH估计量。