Consistency in least-squares estimation: A Bayesian approach
The author removes the restrictions imposed on either the feedback structure or on the initial values for the estimates and presents a (necessary and) sufficient condition for consistency in linear discrete- time system with Gaussian equation noise. The system model described in this paper is \((1)\quad Y(t)=\phi (t)^ Tx+e(t),\) where \(\phi\) (t) is an n-dimensional column vector of inputs and outputs, e(t) is some unobservable one-dimensional disturbance and x is a random n-dimensional column vector of parameters which are to be estimated from observations of the y(t) values. Assumptions: (a) \(\{e(t)\}\) are independent Gaussian random variables with mean zero and variances \(\sigma^ 2>0\); (b) \(\{e(t)\}_ 1^{\infty}\) is independent of x and \({\mathcal F}_ 0\), \(\phi (t)\in {\mathcal F}_{t-1},\) and \({\mathcal F}_ t\) is a \(\sigma\) -algebra generated by \({\mathcal F}\) and \(y(1),...,y(t)\) for \(t\geq 1\). (c) The conditional distribution of the parameter vector x, given \({\mathcal F}_ 0\), is n-dimensional normal with mean m and covariance R. The main result is: Theorem 3. Let y(t) be the output of system (1) which satisfies (a) and (b) above, and let \({\hat\xi }{}_ t\) be the recursive least squares estimate with initial values \({\hat\xi }{}_ 0\), \(S_ 0\). (i) If the true conditional distribution of the parameter vector x given \({\mathcal F}^ a.\)e. is absolutely continuous with respect to the Gaussian distribution with mean \({\hat\xi }{}_ 0\) and covariance matrix \(S_ 0\), then \({\hat\xi }{}_ t\) converges a.e. and if \(a\in {\mathcal F}_{\infty},\) then \(a^ T{\hat\xi }_ t\to a^ Tx, a.e.\quad on\quad\{\omega| S_{\infty}a=0\}.\) (ii) If \(S_ 0\) is strictly positive definite and \(S_ t\to 0\quad a.e.,\) then there is a set \(A\in R^ n\) such that the complement of A has measure zero and \({\hat\xi }_ t\to x\quad a.e.\) for each fixed \(x\in A\) (i.e. with respect to the conditional distribution, given x). The consistency condition is interpreted in terms of conditions on the feedback and adaptive control laws.
scientific article; zbMATH DE number 3873202
scientific article