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A criterion for the consistency of the least squares estimator for a functional linear model with errors in variables. (English. Ukrainian original) Zbl 0947.62044
Theory Probab. Math. Stat. 60, 105-112 (2000); translation from Teor. Jmovirn. Mat. Stat. 60, 95-101 (1999).
The following linear regression model with errors in the variables is considered: \[ y_i=\xi_i\beta+\delta_i,\quad x_i=\xi_i+\varepsilon_i,\qquad i=1,\dots,n, \] where \(x_i\), \(y_i\) are vectors of observations, \(\beta_0\) is an unknown (matrix) parameter to be estimated, \(\xi_i\) are nuisance parameters, \(\delta_i\), \(\varepsilon_i\) are zero-mean random errors (stationary and weakly dependent for different \(i\)). The weighted least squares estimator \(\hat\beta\) is considered which minimizes \[ F_n(\beta)=n^{-1}\sum_{i=1}^n\min_\xi \|(y_i-\xi\beta;x_i-\xi)W^{-1/2}\|^2. \] It is shown that consistency of the estimate \(\hat\beta\) is equivalent to the condition \(W=c\Gamma,\) where \(\Gamma\) is the covariance matrix of \((\delta_i,\varepsilon_i)\).
62J05 Linear regression; mixed models
62F12 Asymptotic properties of parametric estimators