Quasi-estimation as a Basis for Two-stage Solving of Regression Problem

TL;DR

This paper introduces a two-stage estimation method for linear regression using a quasi-estimator that produces two estimates, selecting the better one with minimal additional information, resulting in lower quadratic risk and high robustness.

Contribution

The paper proposes a novel quasi-estimator for linear regression that offers two estimates and a minimal-information-based selection process, improving accuracy and robustness over traditional methods.

Findings

The quasi-estimator has significantly lower quadratic risk than least squares.

The method is robust to violations of initial assumptions.

It effectively incorporates minimal additional information for estimate selection.

Abstract

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.