In this work, we will investigate a Bayesian approach to estimating the parameters of long memory models. Long memory, characterized by the phenomenon of hyperbolic autocorrelation decay in time series, has garnered significant attention. This is because, in many situations, the assumption of short memory, such as the Markovianity assumption, can be deemed too restrictive. Applications for long memory models can be readily found in fields such as astronomy, finance, and environmental sciences. However, current parametric and semiparametric approaches to modeling long memory present challenges, particularly in the estimation process. In this study, we will introduce various methods applied to this problem from a Bayesian perspective, along with a novel semiparametric approach for deriving the posterior distribution of the long memory parameter. Additionally, we will establish the asymptotic properties of the model. An advantage of this approach is that it allows to implement state-of-the-art efficient algorithms for nonparametric Bayesian models.

翻译：本文研究长记忆模型参数估计的贝叶斯方法。长记忆以时间序列中双曲线型自相关衰减为特征，已受到广泛关注。这是因为在许多情况下，短记忆假设（如马尔可夫性假设）可能过于严格。长记忆模型在天文学、金融学、环境科学等领域具有广泛应用。然而，当前建模长记忆的参数化与半参数化方法在估计过程中存在挑战。本研究将从贝叶斯视角介绍该问题的多种解决方法，并提出一种推导长记忆参数后验分布的新型半参数方法。此外，我们将建立该模型的渐近性质。此方法的优势在于能够为非参数贝叶斯模型实现最先进的高效算法。