Two-criteria Estimation of Linear Regression Models Using Least Absolute Deviations and Squares

Abstract

This article is devoted to the problem of estimating multiple linear regression models. Today many different methods have been developed to solve this problem. The most popular of them in the scientific community is ordinary least squares, which consists in solving the optimization problem of minimizing the sum of squared errors. The advantage of this method is that the optimization problem solution is reduced to solving a system of linear algebraic equations, and also that within the framework of the method, a whole system of various adequacy criteria has been developed that are responsible for measuring a variety of qualitative characteristics of the regression model. But least squares estimates are very sensitive to outliers. Estimates less sensitive to outliers are provided the least absolute deviations, which consists of solving the optimization problem of minimizing the sum of absolute deviations. Its solution is equivalent to solving a specially formulated linear programming problem, which is somewhat slower than solving a linear system when implementing the least squares. In addition, when implementing the least absolute deviations, it is impossible to use such adequacy criteria as the coefficient of determination, the Durbin-Watson criterion, etc. The debate about which of the two methods is better continues to this day. In this work, with certain assumptions, the problem of two-criteria estimation of linear regressions is formulated using the least absolute deviations and squares. Using linear convolution, the two-criteria problem is reduced to a single-criteria linear programming problem. Using data from a chemical experiment as an example, both known and new linear regression estimates were obtained, the properties of which require further study.