A Simple Iterative Solution for Mixed Additive and Multiplicative Random Error Model with Inequality Constraints

Objectives:  With the continuous development of modern observation techniques, the processing methods considering only additive errors cannot meet the requirements. Most of the existing methods for dealing with inequality constraints are based on additive error models, including Gaussian Markov models and errors-in-variables models, while there is less research on the processing methods for mixed additive and multiplicative (MAM) random error models.   Methods:  Based on the least squares principle and applying the ideas of zero and infinite weights, we construct a penalty function with the given inequality constraints. The simple iterative method (SIM) for the estimation of the MAM parameters under the inequality constraints is derived. Based on the original SIM, we add a penalty factor increasing with the number of iterations before the penalty term to address the defects of the original simple iterative method.   Results:  Two sets of cases show that the improved SIM can effectively solve the problem that the original method does not converge when used to deal with the MAM error model with inequality constraints. The structure of improved SIM is simple and easy to implement. In addition, it is shown that this method can obtain better parameter estimation by comparing other schemes.   Conclusions:  The feasibility and effectiveness of the improved SIM for parameter estimation of MAM error models with inequality constraints are verified, and it is verified that the method can be applied to the processing of large batches of data.