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The element-wise weighted total least-squares problem. (English) Zbl 1429.62298
Summary: A new technique is considered for parameter estimation in a linear measurement error model $$AX\approx B, A = A_{0}+ \tilde A B = B_0+ \tilde B, A_0 X_{0}=B_{0}$$ with row-wise independent and non-identically distributed measurement errors $$\tilde A, \tilde B$$. Here, $$A_{0}$$ and $$B_{0}$$ are the true values of the measurements $$A$$ and $$B$$, and $$X_{0}$$ is the true value of the parameter $$X$$. The total least-squares method yields an inconsistent estimate of the parameter in this case. Modified total least-squares problem, called element-wise weighted total least-squares, is formulated so that it provides a consistent estimator, i.e., the estimate $$\hat X$$ converges to the true value $$X_{0}$$ as the number of measurements increases. The new estimator is a solution of an optimization problem with the parameter estimate $$\hat X$$ and the correction $$\Delta D = [\Delta A\Delta B]$$, applied to the measured data $$D=[AB]$$, as decision variables. An equivalent unconstrained problem is derived by minimizing analytically over the correction $$\Delta D$$, and an iterative algorithm for its solution, based on the first order optimality condition, is proposed. The algorithm is locally convergent with linear convergence rate. For large sample size the convergence rate tends to quadratic.

##### MSC:
 62J05 Linear regression; mixed models 62-08 Computational methods for problems pertaining to statistics 65F20 Numerical solutions to overdetermined systems, pseudoinverses
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