误差服从多元t分布的一类线性模型下参数估计的若干注记

SOME NOTES ON PARAMETER ESTIMATION IN A CLASS OF LINEAR MODELS WITH AN ERROR VECTOR HAVING MULTIVARIATE t DISTRIBUTION

研究一类线性模型下参数估计的若干问题．这类模型包含了多个因变量线性模型、增长曲线模型、扩充的增长曲线模型、似乎不相关回归方程组、方差分量模型等常用模型．在这类线性模型下,证明了当误差服从多元t分布时与误差服从多元正态分布时,具有相同的完全统计量和无偏估计,且在后一种情况下的充分统计量必为前一种情况下的充分统计量．对于带有多种协方差结构的前述几种模型,把在误差服从多元正态分布下,相应的协方差阵及有关参数的一致最小风险无偏(UMRU)估计存在性的结论推广到了相应的误差服从多元t分布情形．此外,对于误差服从多元t分布的这类统一的线性模型,给出了回归系数的线性可估函数的无偏估计的协方差阵的C-R下界．

This paper studies some problems of parameter estimation in a class of linear models, which include the following common models： the multivariate linear model, the growth curve model, the extended growth curve model, the seemingly unrelated regression equations, the variance components model, and so on. It is proved that the class of linear models with multivariate t error terms and the one with multivariate normal error terms possess the same complete statistic and the same unbiased estimator, and that a sufficient statistic for the later models is also one for the former models. For the above several special models with the various covariance structures, the conclusions on the existence of the uniformly minimum risk unbi- ased（UMRU） estimators of the covariance matrix and concerned parameters under the error vector having multivariate normal distribution are extended to the case of the error vector having multivariate t distribution. Moreover, the C-R lower bound of the covariance matrix of the unbiased estimator for the estimable linear function of regression coefficients in the class of linear models with multivariate t error terms is given.