Autoregressive moving average (ARMA) models are frequently used to analyze time series data. Despite the popularity of these models, algorithms for fitting ARMA models have weaknesses that are not well known. We provide a summary of parameter estimation via maximum likelihood and discuss common pitfalls that may lead to sub-optimal parameter estimates. We propose a random restart algorithm for parameter estimation that frequently yields higher likelihoods than traditional maximum likelihood estimation procedures. We then investigate the parameter uncertainty of maximum likelihood estimates, and propose the use of profile confidence intervals as a superior alternative to intervals derived from the Fisher's information matrix. Through a series of simulation studies, we demonstrate the efficacy of our proposed algorithm and the improved nominal coverage of profile confidence intervals compared to the normal approximation based on Fisher's Information.

翻译：自回归移动平均（ARMA）模型常用于时间序列数据分析。尽管这些模型应用广泛，但拟合ARMA模型的算法存在一些鲜为人知的缺陷。我们概述了通过极大似然法进行参数估计的方法，并讨论了可能导致参数估计次优的常见陷阱。我们提出了一种随机重启算法进行参数估计，该算法通常能比传统极大似然估计方法获得更高的似然值。随后，我们研究了极大似然估计的参数不确定性，并提出使用轮廓置信区间作为基于Fisher信息矩阵导出的区间的更优替代方案。通过一系列模拟研究，我们证明了所提算法的有效性，并表明与基于Fisher信息的正态近似相比，轮廓置信区间具有更优的名义覆盖水平。