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Objectives: For multiplicative error model, the parameters estimated by weighted least squares (WLS) are nonlinear functions of observations, and the weights of observations are nonlinear functions of the estimated parameters. The existing parameter estimation methods of multiplicative error model can theoretically achieve second‑order unbiased, but the precision of uncertainty can only achieve first‑order unbiased. Therefore, a new method is used to improve the accuracy of uncertainty. Methods: The effect of nonlinear iterative processes on WLS parameters is considered, and the relationship between the estimated parameters and the observations in iterative WLS process is regarded as a nonlinear nested function. The derivative‑free Sterling interpolation method with symmetric sampling is used to calculate the expectations of estimated parameters and the standard deviation. Results: From the analysis of two synthetic experiments, the following results are obtained: (1) Considering the impact of the randomness of each iteration on the parameter estimation, the proposed Sterling interpolation method can get better estimated parameters than the existing methods. (2) When the nonlinearity of model is high, the effectiveness of the Sterling interpolation method is more significant. (3) The uncertainty estimation method in this paper can achieve secondorder precision. Conclusions: The feasibility and effectiveness of the Sterling interpolation method for parameter estimation and precision estimation of multiplicative error model are verified. [ABSTRACT FROM AUTHOR]

Objectives: Partial errors⁃in⁃variables (PEIV) model is a more general form of errors in variables (EIV) model. It has been widely used in the data processing of global navigation satellite system (GNSS), coordinate conversion, deformation monitoring and data fitting in the past ten years.Methods: First, the development history of PEIV model is reviewed. Then, five core problems of PEIV model are summarized and analyzed, including parameter estimation algorithm, precision estimation, random model estimation, expansion algorithm and data processing application. Finally, the applications of PEIV model are outlined. Results: We point out three questions for further research, including how to process big data by PEIV parameter estimation method, how to correct the bias of weighted total least squares(WTLS) adjustment by sampling method, and how to derive the formula of higher order parameter estimation covariance matrix are.Conclusions: This review paper aims to further promote the development of surveying and mapping da⁃ ta processing and provide readers with suggestions and references. [ABSTRACT FROM AUTHOR]