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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 $$(\delta_i,\varepsilon_i)$$.
##### MSC:
 62J05 Linear regression; mixed models 62F12 Asymptotic properties of parametric estimators