Limitations of the ARMA w-slice method using a linear combination of higher-order cumulants

The paper analyses autoregressive moving-average (ARMA) system identification method. This method belongs to higher-order statistical methods of a linear algebra type, showing a unique feature that the method works for any kind of model, i.e. MA, AR, or ARMA, and that the model's order (p, q) need not be known in advance. Our analyses of the ARMA approach proved that there is a class of systems not being identifiable. All these systems having poles s
i
; i = 1,…, p, and at least one zero of type of (s
i1
s
i2
… Si
k−1
)
⁻¹
; i
1
, i
2
,…, i
k−1
∊ (1,…, p) cannot be identified by ARMA w-slices using k
^th
-order cumulante, no matter whether with single cumulante, linear combination of cumulante, 1-D slices, or multidimensional slices. The analytical result is backed by simulations. Finally, we propose a procedure of verification of ARMA identifiability and an extension of ARMA w-slice in order to assure the identifiability.