Consistent \(G_ 8\)-estimators for solutions of systems of linear algebraic equations. I. (English. Russian original) Zbl 0748.62016

Girko, V. L.

Theory Probab. Math. Stat. 42, 13-22 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 42, 14-22 (1990).

Summary: The estimator \[ G_ 8=\text{Re}[I(\hat\theta+i\varepsilon)+\beta^{- 1}Z_ s'Z_ s]^{-1}Z_ s'\beta^{-1}b \] is proposed for regularized pseudo-solutions \[ x_ \alpha=[I\alpha+A'A\beta^{-1}]^{-1}\times A'b\beta^{-1} \] of systems of equations \(Ax=b\), where \(A\) is a matrix of order \(n\times m\), \(x\) and \(b\) are vectors, \(\alpha\geq 0\), and \(\beta\) is a sequence of numbers. Here \(\varepsilon\neq 0\); \(Z_ s=s^{- 1}\sum^ s_{i=1}X_ i\); \(X_ i\) are independent observations of the matrix \(A+\Xi\); \(\Xi\) is a random matrix; and \(\hat \theta\) is any measurable real solution of some nonlinear equation. It is proved under certain conditions that for any \(\gamma > 0\) \[ \lim_{\varepsilon \to 0}\lim_{n\to\infty}P\{\| G_ 8-x_ \alpha\| > \gamma\}=0. \]

MSC:

62F12	Asymptotic properties of parametric estimators
15B52	Random matrices (algebraic aspects)
15A06	Linear equations (linear algebraic aspects)

Keywords:

consistency; \(G(8)\)-estimators; systems of linear algebraic equations; regularized pseudo-solutions; measurable real solution; nonlinear equation