  An effective two-stage method for an estimation of parameters of the linear
regression is considered. For this purpose we introduce a certain
quasi-estimator that, in contrast to usual estimator, produces two alternative
estimates. It is proved that, in comparison to the least squares estimate, one
alternative has a significantly smaller quadratic risk, retaining at the same
time unbiasedness and consistency. These properties hold true for
one-dimensional, multi-dimensional, orthogonal and non-orthogonal problems.
Moreover, a Monte-Carlo simulation confirms high robustness of the
quasi-estimator to violations of the initial assumptions. Therefore, at the
first stage of the estimation we calculate mentioned two alternative estimates.
At the second stage we choose the better estimate out of these alternatives. In
order to do so we use additional information, among it but not exclusively of a
priori nature. In case of two alternatives the volume of such information
should be minimal. Furthermore, the additional information is not built-in into
the quasi-estimator structure, so that any kind of information, even intuitive
one, can be used. These features, in combination with decrease of the quadratic
risk, provide a great advantage of our method. A variety of types of the
additional information for choosing the better estimate is considered. One
example is the successful processing of the famous experiment conducted by
astronomers in 1919 to verify the General The-ory of Relativity of A. Einstein.
