L_q估计的渐近方差-协方差矩阵及其特点

The Asymptotic Variance-covariance Matrix in L_q-norm Estimate and Its Characteristic

针对由独立同分布误差膨胀而成的独立不等精度误差,根据未知参数的M估计的Ba hadur型线性表达式,本文导出了由观测量、残差向量、参数估计量和观测量平差向量组成的基本向量的Bahadur型表达式。进一步地,根据方差传播定律导出了M估计的基本向量的渐近方差 协方差矩阵,该矩阵由3个多余参数决定,第三多余参数由本文定义。对Lq范估计, 收稿日期:2003 06 03;修回日期:2004 05 12基金项目:国家973项目(G1998040703);国家自然科学基金重点项目(19833030);国家自然科学基金项目(NSFC 10073017);广西青年自然科学基金资助(9912008)作者简介:彭军还(1964 ),男,重庆人,博士,教授,从事量数据处理理论及其在大地测量、天体测量以及空间飞行器精密定轨中的应用研究。分别计算了误差分别为正态分布和q范分布时的3个多余参数,以及相应的基本向量的方差协方差矩阵。对最小二乘估计,残差向量与参数估计量和观测量的平差向量统计独立,相应的协方差矩阵为零,这一性质与误差分布无关。对正态分布的Lq估计,残差向量与参数估计量和观测量平差向量的协方差不为零;而对q范分布的Lq估计,即是相应的极大似然估计,残差向量与参数估计量和观测量平差向量的协方差为零。文中所得公式和结论可用于统计分析。

For independent and heteroscedastic errors generated by increasing from the independent, identically distributed errors, according to the Bahadur-type linear representation of M-estimate of unknowns, this paper derives the Bahadur-type linear representation of the basic vector including the observational vector, the residual vector, the estimated vector of the unknowns and the adjusted observational vector. The asymptotic variance-covariance matrix of the basic vector for statistical analysis is further derived from the law of variance propagation and determined by the three nuisance parameters. The third nuisance parameter is defined first in this paper. For L_q-norm estimate, the three nuisance parameters and the corresponding variance-covariance matrix are derived respectively from errors being normally distributed and errors being distributed in L_q-norm function. For the Least Squares estimate or L_2-norm estimate, residuals are respectively independent of the estimator of the unknown parameters and the adjusted observations, statistically; the property is irrelative to the error distribution. For L_q-norm estimate with errors being normally distributed, the covariance matrices between the residual vector and the estimated vector of the unknown parameters, as well as the adjusted observational vector are not zero. However, for L_q-norm estimate with errors being distributed in q-norm function, it is the corresponding maximal likelihood estimate, the covariance matrices between the residual vector and the estimated vector of the unknown parameters, as well as the adjusted observational vector are zero. The derived forms and conclusions can be used in statistical analysis.