Yuan K H, Bentler P M
Department of Psychology and Center for Statistics, University of California, Los Angeles, CA 90095-1563, USA.
Br J Math Stat Psychol. 1998 May;51 ( Pt 1):63-88. doi: 10.1111/j.2044-8317.1998.tb00667.x.
Covariance structure analysis is used to evaluate hypothesized influences among unmeasured latent and observed variables. As implemented, it is not robust to outliers and bad data. Several robust methods in model fitting and testing are proposed. These include direct estimation of M-estimators of structured parameters and a two-stage procedure based on robust M- and S-estimators of population covariances. The large sample properties of these estimators are obtained. The equivalence between a direct M-estimator and a two-stage estimator based on an M-estimator of population covariance is established when sampling from an elliptical distribution. Two test statistics are presented in judging the adequacy of a hypothesized model; both are asymptotically distribution free if using distribution free weight matrices. So these test statistics possess both finite sample and large sample robustness. The two-stage procedures can be easily adapted into standard software packages by modifying existing asymptotically distribution free procedures. To demonstrate the two-stage procedure, S-estimator and M-estimators under different weight functions are calculated for some real data sets.
协方差结构分析用于评估未测量的潜在变量和观测变量之间的假设影响。在实际应用中，它对异常值和不良数据不具有稳健性。本文提出了几种在模型拟合和检验中的稳健方法。这些方法包括直接估计结构化参数的M估计量，以及基于总体协方差的稳健M估计量和S估计量的两阶段程序。得到了这些估计量的大样本性质。当从椭圆分布中抽样时，建立了直接M估计量与基于总体协方差M估计量的两阶段估计量之间的等价关系。文中给出了两个用于判断假设模型是否合适的检验统计量；如果使用无分布权重矩阵，这两个统计量在渐近意义下都是无分布的。因此，这些检验统计量同时具有有限样本和大样本稳健性。通过修改现有的渐近无分布程序，两阶段程序可以很容易地应用到标准软件包中。为了演示两阶段程序，针对一些实际数据集计算了不同权重函数下的S估计量和M估计量。
