引用本文: 王乐洋, 邹传义. 乘性误差模型参数估计及精度评定的Sterling插值方法[J]. 武汉大学学报 ( 信息科学版), 2022, 47(2): 219-225.
WANG Leyang, ZOU Chuanyi. Sterling Interpolation Method for Parameter Estimation and Precision Estimation in Multiplicative Error Model[J]. Geomatics and Information Science of Wuhan University, 2022, 47(2): 219-225.
 Citation: WANG Leyang, ZOU Chuanyi. Sterling Interpolation Method for Parameter Estimation and Precision Estimation in Multiplicative Error Model[J]. Geomatics and Information Science of Wuhan University, 2022, 47(2): 219-225.

## Sterling Interpolation Method for Parameter Estimation and Precision Estimation in Multiplicative Error Model

• 摘要: 乘性误差模型加权最小二乘参数估值是观测值的非线性函数，观测值的权是加权最小二乘参数估值的非线性函数。已有的乘性误差模型参数估计方法理论上可以达到二阶无偏，但精度评定方法只能达到一阶精度，并且参数估计逐步的迭代过程使得参数及改正数的每一步估值都具有随机性，使得最终的参数估值与观测值为复杂的非线性关系。考虑到非线性迭代过程对加权最小二乘参数带来的影响，使用一种无需求导的Sterling插值方法求解参数估值的均值和标准差。模拟实验表明，当模型非线性较高时，考虑每次迭代的随机性对参数估值的影响可以得到更接近真值的参数估值，并且所提方法的精度评定可以达到二阶精度，验证了Sterling插值方法用于乘性误差模型参数估计及其精度评定的适用性和有效性。

Abstract:
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 second-order precision.
Conclusions  The feasibility and effectiveness of the Sterling interpolation method for parameter estimation and precision estimation of multiplicative error model are verified.

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