The element-wise weighted total least-squares problem

A new technique is considered for parameter estimation in a linear measurement error model AX≈BAX≈B, A=A0+A˜, B=B0+B˜, A0X0=B0A0X0=B0 with row-wise independent and non-identically distributed measurement errors A˜, B˜. Here, A0A0 and B0B0 are the true values of the measurements AA and BB, and X0X0 is the true value of the parameter XX. 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 X^ converges to the true value X0X0 as the number of measurements increases. The new estimator is a solution of an optimization problem with the parameter estimate X^ and the correction ΔD=[ΔAΔB], applied to the measured data D=[AB], as decision variables. An equivalent unconstrained problem is derived by minimizing analytically over the correction ΔDΔD, 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.