偏正态纵向数据混合效应模型的统计推断及应用

The mixed effects model (MEM) is an important statistical model with broad application backgrounds. However, the classical theories and methods, mostly based on the assumption of normal distribution, can not be directly applied to statistical modeling with skew-normal longitudinal data which are common in reality, and therefore are inadequate for practical data analysis and problem-solving. This research is aimed at MEMs with skew-normal longitudinal data, and will be conducted as follows. Firstly, via combining moment method, maximum likelihood method with fine matrix techniques, we will establish the feasible estimates of interest parameters and prove their statistical excellent properties. Secondly, setting up the noncentral skew F distribution and using the concepts of generalized p-value and generalized confidence interval, we will construct the test statistics of interest parameters, which are then proved to be exact. Thirdly, the above-mentioned theories and methods are applied to analyze practical data regarding several economic and social hot issues. This research is of scientific value, since it helps further explore statistical inference theories and establish several new efficient methods for MEMs with skew-normal longitudinal data. In addition, it will provide efficient statistical methods and tools for skew-normal longitudinal data analysis in various fields such as economics, finance, biology and medical science, which also features it by practical significance.

混合效应模型是一类具有广泛应用背景的重要统计模型，但其经典理论和方法大多基于正态分布假设，难以直接应用于现实中大量存在的偏正态纵向数据的统计建模，不利于实际数据分析及问题解决。本项目拟针对偏正态纵向数据混合效应模型，开展以下研究工作：其一，将矩法、极大似然法与矩阵技术中的高度技巧相结合，构造感兴趣参数的可行估计，并证明其统计优良性。其二，建立非中心偏F分布并利用广义p-值、广义置信区间的概念，构造感兴趣参数的检验统计量，进而证明其为精确检验。其三，将上述理论和方法应用于当前经济社会领域若干重点、热点问题的实际数据分析。本项目预期可对偏正态纵向数据混合效应模型的统计推断理论作出创新性探索，建立若干新的有效的统计推断方法，具有科学价值；亦可为经济、金融、生物、医学等领域的偏正态纵向数据分析提供更为有效的统计方法和工具，具有现实意义。

混合效应模型是一类具有广泛应用背景的重要统计模型，但其现有基本理论和传统方法大多基于正态分布假设，难以直接应用于现实中大量存在的偏正态纵向数据的统计建模，不利于实际数据分析及问题解决。本项目针对偏正态纵向数据混合效应模型，综合运用矩阵技术、非中心偏F分布、EM算法、Monte Carlo模拟等多种方法和工具，给出其密度函数、矩生成函数、均值向量、协方差矩阵、独立性条件等统计性质。在此基础上，建立该模型下Cochran定理，构造回归系数和方差分量的可行估计以及具有优良性质的精确检验。进而，将上述理论与方法推广到含k+1个随机效应的偏正态混合效应模型和偏正态随机矩阵。最后，将上述理论研究成果应用于能耗强度、碳排放强度、绿色经济效率等当前经济领域十分重要的若干问题，作出精度更高的统计推断，改善实际数据分析效果，从而更为有效地促进实际经济问题的解决。本项目对偏正态纵向数据混合效应模型的统计推断理论作出创新性探索，建立若干新的有效的统计推断方法，具有科学价值；亦可为经济、金融、生物、医学等实际领域的偏正态纵向数据分析提供更为有效的统计方法和工具，具有现实意义。