ARMA spectral estimation based on partial autocorrelations. II: Statistical analysis (Q1076661)

The authors present an analysis of the ARMA estimation method introduced by the second author in the first part of the paper [ibid. 2, 341-360 (1983; Zbl 0535.93060)]. Within the method analysed, estimates of the ARMA parameters are obtained by a nonlinear least-squares fit of the sample partial autocorrelations to the ARMA partial autocorrelations. The asymptotic accuracy properties of the method are established. A procedure for computing the covariance matrix (say P) of the estimation errors is presented. Furthermore, a lower bound (say \(P_{LB})\) on the covariance matrix of any consistent estimator based on a given number of sample partial autocorrelation, is derived. Numerical comparisons between P, \(P_{LB}\) and the corresponding Cramér-Rao lower bound (say \(P_{CR})\) are presented. In the examples presented \(P\simeq P_{LB}\) converges to \(P_{CR}\) as the number of partial correlations used in the estimation tends to infinity. In the opinion of this reviewer the approximate equality \(P\simeq P_{LB}\) is due to the almost diagonal form of the covariance matrix of the sample partial autocorrelations, and the convergence of \(P_{LB}\) to \(P_{CR}\) could be established in the manner of the analysis of the second author and \textit{B. Friedlander} [IEEE Trans. Autom. Control AC-31, 579-582 (1986; Zbl 0589.93061)]. The shape of the least-squares performance criterion (questions of unimodality etc.) would also be worth investigating.