Minimum Mismatch Modeling (3M) Hyperparameter Selection in Autoregressive Moving Average (ARMA) Modeling

Hyperparameters of Autoregressive Moving Average (ARMA) modeling are the number of AR coefficients and the number of MA coefficients. The hyperparameter selection (HS) in ARMA modeling plays a critical role and can dominate the coefficient (parameter) estimation process. This work provides a novel method of HS estimation that works with the Conditional Least Square Estimator (CLSE), which is the most efficient ARMA parameter estimator. The proposed HS method focuses on a rational cost function in the form of mismatch modeling error. The error aims to capture the estimation difference between the true and unknown HS parameters and the competing hyperparameters. This error can be calculated using the available mean square error (MSE) in the parameter estimation step. The proposed method, denoted by the minimum mismatch modeling (3M) approach, has already shown superiority over other HS approaches in AR modeling. In AR modeling, the parameter estimator is based on the Yule-Walker method, which is a linear estimator, and the 3M calculation process using the available MSE has been provided for this modeling. However, in ARMA modeling the CLSE estimator is a nonlinear estimator, and one main challenge is to solve for calculation of the 3M using the MSE of CLSE. The method proposed here, denoted by 3M-CLSE, provides the steps to get to the desired 3M from the available CLSE MSE. It can be shown that the criteria of most used HS methods Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) are special cases of 3M-CLSE for particular choices of confidence and validation probabilities. The simulation results confirm the superiority of 3M-CLSE over the existing HS approaches in terms of HS accuracy, as well as in terms of modeling MSE error.