Autoregressive moving average (ARMA) models are widely used for analyzing time series data. However, standard likelihood-based inference methodology for ARMA models has avoidable limitations. We show that currently accepted standards for ARMA likelihood maximization frequently lead to sub-optimal parameter estimates. Existing algorithms have theoretical support, but can result in parameter estimates that correspond to a local optimum. While this possibility has been previously identified, it remains unknown to most users, and no routinely applicable algorithm has been developed to resolve the issue. We introduce a novel random initialization algorithm, designed to take advantage of the structure of the ARMA likelihood function, which overcomes these optimization problems. Additionally, we show that profile likelihoods provide superior confidence intervals to those based on the Fisher information matrix. The efficacy of the proposed methodology is demonstrated through a data analysis example and a series of simulation studies. This work makes a significant contribution to statistical practice by identifying and resolving under-recognized shortcomings of existing procedures that frequently arise in scientific and industrial applications.

翻译：自回归滑动平均（ARMA）模型被广泛应用于时间序列数据分析。然而，针对ARMA模型的标准基于似然的推断方法存在可避免的局限性。我们证明，当前被广泛接受的ARMA似然最大化标准常导致次优的参数估计。现有算法虽具有理论支持，但可能产生对应于局部最优的参数估计值。尽管这一可能性先前已被指出，但多数用户仍未知晓，且尚未开发出可常规应用的算法来解决此问题。我们提出一种新颖的随机初始化算法，该算法旨在利用ARMA似然函数的结构，从而克服这些优化问题。此外，我们证明轮廓似然所提供的置信区间优于基于费舍尔信息矩阵的置信区间。通过一个数据分析实例和一系列模拟研究，验证了所提方法的有效性。本研究通过识别并解决现有方法在科学与工业应用中频繁出现但未被充分认识的缺陷，为统计实践作出了重要贡献。