ARMA parameter estimation

A number of approaches are available to estimate ARMA parameters from the data [31]. The approach we prefer operates using a recursion. We first assume the process is AR only and estimate ${\bf a}= [a_1, \ldots a_P]$ and $\sigma^2$. We then estimate ${\bf b}= [b_1, \ldots b_P]$ from the sequence

$$u_n = \sum_{i=0}^P \; a_i \; x_{n-i}.$$ (10.20)

Next, ${\bf a}$ is re-estimated from

$$v_n = -\sum_{i=1}^P \; b_i \;v_{n-i} + x_n.$$ (10.21)

The algorithm works by alternately estimating ${\bf b}$ from the sequence that has the AR component removed, then ${\bf a}$ from the sequence that has the MA component removed. See software/get_arma.m. As a last step, refinement of the estimate by maximizing the exact PDF (10.6) or the approximate frequency-domain PDF (10.18) is possible using the approach in (section 17.5). For this we need the CR bound analysis.