Multiple L${}_{v}$-estimation of linear regression models

There are many methods for estimating multiple linear regression models: ordinary least squares, least absolute deviations, anti-robust estimation, L${}_{v}$-estimation, and multiple estimations. The purpose of this work is to generalize these methods by a loss function. First, an estimation problem was formulated where the minimization criteria are the anti-robust and L${}_{v}$-estimations. The disadvantage of this problem statement is that it is difficult to determine the initial values of the parameters for a numerical solution, since the variables may have different scales. Besides, the loss function is non-uniform, which also complicates the estimation. To solve these problems, we introduced a new criterion, equal to the anti-robust estimation criterion raised to the power v. We stated the problem of multiple L${}_{v}$-estimation using the new criterion and the loss function. The functional of this problem is homogeneous, therefore, for multiple L${}_{v}$-estimations, it is advisable to normalize the initial variables and then apply the standardized linear regression estimates. We also developed an algorithm for multiple L${}_{v}$-estimations. A result of such estimations is a set containing linear regression estimates obtained both by the existing and new methods. The optimal choice of the best estimates from the set of estimates remains an open problem. We successfully simulated the passenger railway traffic in the Irkutsk region with the proposed multiple L${}_{v}$-estimations.