🤖 AI Summary
Standard likelihood-based inference for ARMA models frequently converges to local optima, resulting in biased parameter estimates and undercoverage of confidence intervals; existing approaches lack a general-purpose remedy. This paper proposes a structural-aware stochastic initialization optimization algorithm and a profile likelihood confidence interval method. We introduce the first initialization strategy explicitly tailored to the geometric structure of the ARMA likelihood surface. Furthermore, we rigorously establish and generalize the superiority of the profile likelihood approach—demonstrating its substantially improved coverage accuracy relative to conventional Fisher information–based methods. Through extensive Monte Carlo simulations and empirical analyses, our methods markedly enhance estimation consistency and restore confidence interval coverage toward nominal levels. The proposed framework effectively mitigates long-standing inferential biases in both industrial applications and scientific research.
📝 Abstract
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 common ARMA likelihood maximization strategies often lead to sub-optimal parameter estimates. While this possibility has been previously identified, 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 confidence intervals 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.