Msplit Estimation Based on L1 Norm Condition

The variants of M-split estimation proposed so far are based on the objective function related to the least-squares method and hence they are known as the squared M-split estimation. We propose to design a new variant whose objective function is based on the L-1 norm condition. The main objective is to investigate the properties of the new solution, especially in the context of robustness. Thus, the respective influence functions and the weight functions are also derived. The influence functions provide us with information about general properties of the method, whereas the weight functions are the basis for the modified iterative process of M-split estimation proposed in this paper. Theoretical considerations are supplemented with empirical analyses that concern M-split estimation of the competitive parameters of the univariate functional model or the linear regression, and finally the multivariate functional model (the latter case is related to the deformation analysis). The results show that the new variant is generally less sensitive to outliers or wrong assignment of a particular observation to its functional model than the squared M-split estimates. The solution proposed in this paper is also more stable, that is, it is independent of the chosen starting point of the iterative process or local fluctuations of random observation errors. That seems to be an interesting practical feature. Both variants have similar accuracy; however, the accuracy of the new variant is higher in many presented cases.