Fitting Data with Different Error Models

A maximum likelihood estimator has been applied to find regression parameters of a straight line in case of different error models. Assuming Gaussian-type noise for the measurement errors, explicit results for the parameters can be given employing Mathematica. In the case of the ordinary least squares (OLSy), total least squares (TLS), and least geometric mean deviation (LGMD) approaches, as well as the error model of combining ordinary least squares (OLSx and OLSy) in the Pareto sense, simple formulas are given to compute the parameters via a reduced Gröbner basis. Numerical examples illustrate the methods, and the results are checked via direct global minimization of the residuals.