The estimation of the long memory parameter d is a widely discussed issue in the literature. The harmonically weighted (HW) process was recently introduced for long memory time series with an unbounded spectral density at the origin. In contrast to the most famous fractionally integrated process, the HW approach does not require the estimation of the d parameter, but it may be just as able to capture long memory as the fractionally integrated model, if the sample size is not too large. Our contribution is a generalization of the HW model, denominated the Generalized harmonically weighted (GHW) process, which allows for an unbounded spectral density at \(k \ge 1\) frequencies away from the origin. The convergence in probability of the Whittle estimator is provided for the GHW process, along with a discussion on simulation methods. Fit and forecast performances are evaluated via an empirical application on paleoclimatic data. Our main conclusion is that the above generalization is able to model long memory, as well as its classical competitor, the fractionally differenced Gegenbauer process, does. In addition, the GHW process does not require the estimation of the memory parameter, simplifying the issue of how to disentangle long memory from a (moderately persistent) short memory component. This leads to a clear advantage of our formulation over the fractional long memory approach.

长记忆参数d的估计是文献中广泛讨论的问题。最近针对长时间存储时间序列引入了谐波加权（HW）过程，其起始处具有无穷大的频谱密度。与最著名的分数积分过程相比，硬件方法不需要估计d参数，但是，如果样本大小不太大，它可能与分数积分模型一样能够捕获较长的内存。我们的贡献是HW模型的广义化，命名为广义谐波加权（GHW）过程，该过程允许\（k \ ge 1 \）处的无穷频谱密度远离原点的频率。GHW过程提供了Whittle估计量的概率收敛性，并提供了关于仿真方法的讨论。通过对古气候数据的经验应用来评估拟合和预测性能。我们的主要结论是，上述概括能够建模长记忆，以及它的经典竞争者（分数差分Gegenbauer过程）能够做到。另外，GHW过程不需要估计内存参数，从而简化了如何使长内存与（中等持久性）短内存组件分离的问题。这导致我们的配方相对于分数长记忆方法具有明显的优势。