广义最小二乘估计(Generalized least squares estimation,GLSE)是最佳线性无偏估计,却有计算复杂高和依赖未知信息的局限性,使得普通最小二乘估计(Ordinary least squares estimation,OLSE)经常成为应用的无奈之选.本文探讨该现象背后的三个循序渐进的理论问题:第一,GLSE的退化问题,给出GLSE完全退化为OLSE的充要条件;第二,退化的分类问题,依据设计矩阵和误差协方差阵的结构把退化现象分为三类,并给出典型的退化特例;第三,不完全退化问题,研讨导致效率退化的因素,刻画效率曲线和效率曲面,最后给出效率不低于95％的退化边界.效率退化和边界分析的潜在应用价值主要包括两方面:第一,为进一步优化试验方案提供效率视角和反馈信息;第二,为设计更简洁更可靠的算法提供理论依据.
Efficiency degradation boundary analysis of parameter estimation for Generalized Gaussian-Markov model
The Generalized least squares estimation(GLSE)is the best linear unbiased estimation,but it has the limitations of high computational complexity and dependence on unknown information,making the Ordinary least squares estimation(OLSE)be the unbearable choice for applications.Three progressive theoretical issues behind this phenomenon are explored:firstly,the degradation problem of GLSE,provides a necessary and sufficient condition when GLSE completely degenerates into OLSE;Secondly,the degradation classification problem,classifies degradation phenomenon into three categories based on the structure of the design matrix and the error covariance matrix,and typical degradation exceptions are given;Thirdly,the problem of incomplete degradation,analyzes the factors that lead to efficiency degradation,draws up the efficiency curve and efficiency surface,and finally provides a degradation boundary with an efficiency of no less than 95％.The potential application value of efficiency degradation and boundary analysis mainly includes two aspects:firstly,providing an efficiency perspective and feedback information for further optimizing the experimental plan;Secondly,providing theoretical basis for designing simpler and more reliable algorithms.
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