基于M残差的方差分量估计

The Estimate of Variance Components Based on M-Estimate Residuals

根据M估计的线性表达式原理,导出了不同类观测M估计的线性表达式、多余参数以及观测量和参数估计量的方差协方差矩阵。M残差的二次型的无偏估计是方差分量和多余参数的函数。当误差密度已知时,多余参数的显式可以由方差分量表达,此时二次型是方差分量的显线性函数,由此构成了基于M残差的方差分量无偏估计公式。对Lp估计和正态分布,导出了方差分量估计的实用公式,在边角网中进行了应用。与赫尔默特方法进行比较,结果表明,有粗差时,方差分量估计和参数估计结果随着Lp估计的p的变化相差显著,无粗差时(或粗差被剔除时),不同的方差分量估计方法的结果相差甚微。该方法可以对赫尔默特方法进行有效的检查。

Based on the principle of the linear representation of the M-estimate, this paper derives the linear representation and nuisances of M-estimate for Heteroscedastic observations and the asymptotic variance-covariance matrix of the observations and the estimator of the unknown parameters; The unbiased estimate of the weighted quadric type of M-estimate residuals is derived from the asymptotic variance-covariance matrixes, it is the implicit function of heteroscedastic variances （or variance components） and nuisances. For the known error density, nuisances have their explicit representation only relative Heteroscedastic variances or their square roots, which constructs unbiased estimate form of heteroscedastic variance for computation. For Lp estimate and normal errors, the practical form estimating heteroscedastic variances is derived and applied in side-angle network. As compared with Helmert method, it shows that the estimate result of variance components and parameters varies significantly with p of Lp-estimate as gross errors occur; if there exist no gross errors or they are rejected right, there is a little gap in the estimate result. The estimate method derived in this paper can be used in adjustment of heteroscedastic model and for a good checkout of Helmert method.