The TSCR method for precision estimation of ill-posed mixed additive and multiplicative random error model

Estimating the precision information of parameter estimation can fully reflect the quality of parameter estimation. In this paper, we first derive the weighted least-square regularization iterative (WLSRI) solution and mean square error (MSE) matrix of the ill-posed mixed additive and multiplicative random error (MAMRE) model. Then, considering that the gradual iterative process of the WLSRI solution will affect the final parameter estimation and precision information and further lead to a complex nonlinear function relationship, the traditional Taylor expansion approximate function method cannot be used to solve. Therefore, this paper introduces the derivative-free third-degree spherical-radial cubature rule (TSCR) method for precision estimation of the ill-posed MAMRE model, which generates a series of samples with the same weight by the fixed sampling strategy and further uses the WLSRI method to calculate. Finally, the experiment research and analysis results illustrate that compared with the existing solutions without considering the ill-posed problem, the WLSRI method is applicable and can obtain reasonable parameter estimation and precision information in solving the ill-posed MAMRE model; while the TSCR method can obtain more accurate parameter estimation and precision information than the WLSRI method, which enriches the theoretical research on the precision estimation problem of ill-posed MAMRE model.