Consistent estimation of system order
Terrence L. Fine, W. Hwang · IEEE Transactions on Automatic Control · 1979
We consider a parameterized family\{S_{\alpha}, \alpha \in a\}, a \subset R^{\infty}, of systems or sources having stochastic outputs\{x_{n}\}that are partially described by a statistic (e.g, correlation function)\sigma_{\alpha}(\tau). If we represent\alpha=(\alpha_{1},\alpha_{2}...,\alpha_{n}...), then by the system order M_{\alpha} we mean the indexnof the last no nonzero term in the expansinn of α. Our objective is to generate a sequence{\hat{M}_{n}(x_{1},... ,X_{n})}of estimates of theM_{\alpha}^{0}that converge to it at least in probability. We provide conditions ensuring the existence of such a statistically consistent sequence of estimators, as wen as improved conditions yielding convergence in mean-square and with probability one. We establish existence by providing a method for constructing a family of consistent estimators of system order. We then apply our method to estimate the order of a scalar moving averages process and the order of a scalar autoregressive process. Our present results are primarily Of a theoretical nature, as we lack the efficiency and simulation studies desirable in support of a practical estimator of system order.