  We consider discrete linear Chebyshev approximation problems in which the
unknown parameters of linear function are fitted by minimizing the maximum
absolute deviation of errors. Such problems find application in the solution of
overdetermined systems of linear equations that appear in many practical
contexts. The least maximum absolute deviation estimator is used in regression
analysis in statistics when the distribution of errors has bounded support. To
derive a direct solution of the problem, we propose an algebraic approach based
on a parameter elimination technique. As a key component of the approach, an
elimination lemma is proved to handle the problem by reducing it to a problem
with one parameter eliminated, together with a box constraint imposed on this
parameter. We demonstrate the application of the lemma to the direct solution
of linear regression problems with one and two parameters. We develop a
procedure to solve multidimensional approximation (multiple linear regression)
problems in a finite number of steps. The procedure follows a method that
comprises two phases: backward elimination and forward substitution of
parameters. We describe the main components of the procedure and estimate its
computational complexity. We implement symbolic computations in MATLAB to
obtain exact solutions for two numerical examples.
