Consistency in least-squares estimation: A Bayesian approach
type: Item
instance of: scholarly article
title: Consistency in least-squares estimation: A Bayesian approach
zbMATH Open document ID: 0547.93062
DOI: 10.1016/0005-1098(84)90105-5
author: Holger Rootzén
author: Jan Sternby
published in: Automatica
publication date: 1984-01-01T00:00:00Z
review text: 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 interprete…
Mathematics Subject Classification ID: 93E10
Mathematics Subject Classification ID: 62A01
Mathematics Subject Classification ID: 62F35
Mathematics Subject Classification ID: 62F12
Mathematics Subject Classification ID: 93C40
zbMATH DE Number: 3873202
zbMATH Keywords: consistency
zbMATH Keywords: recursive least squares estimate
MaRDI profile type: Publication
full work available at URL: 0005-1098(84)90105-5
OpenAlex ID: W1606299524
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description: scientific article; zbMATH DE number 3873202