This paper introduces a new kind of seasonal fractional autoregressive process (SFAR) driven by fractional Gaussian noise (fGn). The new model includes a standard seasonal AR model and fGn. {The estimation of the parameters of this new model has to solve two problems: nonstationarity from the seasonal structure and long memory from fGn. We innovatively solve these by getting a stationary subsequence, making a stationary additive sequence, and then obtaining their spectral density. Then, we use one-step procedure for Generalized Least Squares Estimator (GLSE) and the Geweke Porter-Hudak (GPH) method to get better results. We prove that both the initial and one-step estimators are consistent and asymptotically normal. Finally, we use Monte Carlo simulations with finite-sized samples to demonstrate the performance of these estimators. Moreover, through empirical analysis, it is shown that the SFAR model can simulate some real world phenomena better than general models.

翻译：本文引入了一种由分数高斯噪声(fGn)驱动的新型季节分数自回归过程(SFAR)。该新模型包含标准季节AR模型与fGn。对此新模型的参数估计需解决两个问题：季节结构带来的非平稳性与fGn带来的长记忆性。我们通过获取平稳子序列、构造平稳可加序列，进而得到其谱密度，创新性地解决了这些问题。随后，我们采用广义最小二乘估计量(GLSE)的一步程序与Geweke Porter-Hudak (GPH)方法以获得更优结果。我们证明了初始估计量与一步估计量均具有一致性且渐近正态。最后，我们通过有限样本的蒙特卡洛模拟展示了这些估计量的性能。此外，实证分析表明，SFAR模型比通用模型能更好地模拟某些现实世界现象。