This paper presents a comparative study of two Bayesian approaches - Markov Chain Monte Carlo (MCMC) and Approximate Bayesian Computation (ABC) - for estimating the parameters of autoregressive fractionally-integrated moving average (ARFIMA) models, which are widely used to capture long-memory in time series data. We propose a novel MCMC algorithm that filters the time series into distinct long-memory and ARMA components, and benchmarked it against standard approaches. Additionally, a new ABC method is proposed, using three different summary statistics used for posterior estimation. The methods are implemented and evaluated through an extensive simulation study, as well as applied to a real-world financial dataset, specifically the quarterly U.S. Gross National Product (GNP) series. The results demonstrate the effectiveness of the Bayesian methods in estimating long-memory and short-memory parameters, with the filtered MCMC showing superior performance in various metrics. This study enhances our understanding of Bayesian techniques in ARFIMA modeling, providing insights into their advantages and limitations when applied to complex time series data.

翻译：本文对两种贝叶斯方法——马尔可夫链蒙特卡洛（MCMC）与近似贝叶斯计算（ABC）——在估计自回归分数阶积分移动平均（ARFIMA）模型参数方面进行了比较研究，该模型被广泛用于捕捉时间序列数据中的长记忆性。我们提出了一种新颖的MCMC算法，该算法将时间序列滤波为独立的长记忆分量与ARMA分量，并与标准方法进行了基准测试。此外，本文还提出了一种新的ABC方法，该方法采用了三种不同的汇总统计量进行后验估计。这些方法通过广泛的模拟研究进行了实现与评估，并应用于真实世界的金融数据集，特别是美国季度国民生产总值（GNP）序列。结果表明，贝叶斯方法在估计长记忆与短记忆参数方面具有有效性，其中滤波MCMC方法在多项指标上均表现出更优的性能。本研究深化了我们对ARFIMA建模中贝叶斯技术的理解，为将其应用于复杂时间序列数据时的优势与局限性提供了见解。