Autoregressive moving average (ARMA) models are frequently used to analyze time series data. Despite the popularity of these models, likelihood-based inference for ARMA models has subtleties that have been previously identified but continue to cause difficulties in widely used data analysis strategies. 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 initialization 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）模型常用于分析时间序列数据。尽管这些模型应用广泛，但基于似然的ARMA模型推断存在一些先前已被指出、却仍在广泛使用的数据分析策略中持续引发困难的微妙问题。本文总结了通过最大似然法进行参数估计的方法，并讨论了可能导致次优参数估计的常见缺陷。我们提出了一种用于参数估计的随机初始化算法，该算法通常能获得比传统最大似然估计程序更高的似然值。随后，我们研究了最大似然估计的参数不确定性，并提出使用剖面置信区间作为基于费希尔信息矩阵所得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，并展示了相较于基于费希尔信息的正态近似法，剖面置信区间在名义覆盖率上的显著提升。