Bounded Influence Propagation $\tau$ -Estimation: A New Robust Method for ARMA Model Estimation

A new robust and statistically efficient estimator for ARMA models called the bounded influence propagation <inline-formula><tex-math notation="LaTeX">$\tau$</tex-math></inline-formula>-estimator is proposed. The estimator incorporates an auxiliary model, which prevents the propagation of outliers. Strong consistency and asymptotic normality of the estimator for ARMA models that are driven by independently and identically distributed (iid) innovations with symmetric distributions are established. To analyze the infinitesimal effect of outliers on the estimator, the influence function is derived and computed explicitly for an AR(1) model with additive outliers. To obtain estimates for the AR(<inline-formula><tex-math notation="LaTeX">$p$</tex-math></inline-formula>) model, a robust Durbin–Levinson type and a forward–backward algorithm are proposed. An iterative algorithm to robustly obtain ARMA(<inline-formula><tex-math notation="LaTeX">$p,q$</tex-math></inline-formula>) parameter estimates is also presented. The problem of finding a robust initialization is addressed, which for orders <inline-formula> <tex-math notation="LaTeX">$p+q > 2$</tex-math></inline-formula> is a nontrivial matter. Numerical experiments are conducted to compare the finite sample performance of the proposed estimator to existing robust methodologies for different types of outliers both in terms of average and of worst case performance, as measured by the maximum bias curve. To illustrate the practical applicability of the proposed estimator, a real-data example of outlier cleaning for R–R interval plots derived from electrocardiographic data is considered. The proposed estimator is not limited to biomedical applications, but is also useful in any real-world problem whose observations can be modeled as an ARMA process disturbed by outliers or impulsive noise.