We make an observation that facilitates exact likelihood-based inference for the parameters of the popular ARFIMA model without requiring stationarity by allowing the upper bound $\bar{d}$ for the memory parameter $d$ to exceed $0.5$. We observe that estimating the parameters of a single non-stationary ARFIMA model is equivalent to estimating the parameters of a sequence of stationary ARFIMA models, which allows for the use of existing methods for evaluating the likelihood for an invertible and stationary ARFIMA model. This enables improved inference because many standard methods perform poorly when estimates are close to the boundary of the parameter space. It also allows us to leverage the wealth of likelihood approximations that have been introduced for estimating the parameters of a stationary process. We explore how estimation of the memory parameter $d$ depends on the upper bound $\bar{d}$ and introduce adaptive procedures for choosing $\bar{d}$. Via simulations, we examine the performance of our adaptive procedures for estimating the memory parameter when the true value is as large as $2.5$. Our adaptive procedures estimate the memory parameter well, can be used to obtain confidence intervals for the memory parameter that achieve nominal coverage rates, and perform favorably relative to existing alternatives.

翻译：我们提出一种观察结果，使得对流行ARFIMA模型参数进行精确似然推断成为可能，且无需满足平稳性条件——只需允许记忆参数$d$的上界$\bar{d}$超过0.5。我们发现，估计单个非平稳ARFIMA模型的参数等价于估计一系列平稳ARFIMA模型的参数，这使我们能够利用现有方法评估可逆平稳ARFIMA模型的似然函数。该方法可改进推断效果，因为许多标准方法在参数估计值接近参数空间边界时表现欠佳。同时，该方法还允许我们充分利用为平稳过程参数估计而提出的丰富似然近似方法。我们探究了记忆参数$d$的估计如何依赖于上界$\bar{d}$，并提出了选择$\bar{d}$的自适应程序。通过仿真实验，我们检验了当真实值高达2.5时，自适应程序对记忆参数的估计性能。结果表明，我们的自适应程序能够良好估计记忆参数，可构建达到名义覆盖率的记忆参数置信区间，且性能优于现有替代方法。