不完全二元样本空间的分层Aitchison-Silvey模型
Hierarchical Aitchison-Silvey models for incomplete binary sample spaces
摘要：
多元样本空间可以是不完全的笛卡尔积，当变量的类别的某些组合是不可能的。传统的对数线性模型，概括独立性和条件独立性，不适用于这种情况下，因为它们可能与不存在的细胞的正概率。为了描述不完备样本空间中的关联结构，本文发展了一类分层乘法模型，它是通过使某些非齐次广义优势比等于1来定义的，并以Aitchison和Silvey的名字命名.这些模型是弯曲的指数族，不包含整体效应，从代数的角度来看，是非齐次复曲面理想。从统计学和代数几何学的角度研究了这类模型与对数线性模型和拟对数线性模型的关系。讨论了最大似然估计的存在性及其性质，并给出了相应的算法。
英文摘要：
Multivariate sample spaces may be incomplete Cartesian products, when certain combinations of the categories of the variables are not possible. Traditional log-linear models, which generalize independence and conditional independence, do not apply in such cases, as they may associate positive probabilities with the non-existing cells. To describe the association structure in incomplete sample spaces, this paper develops a class of hierarchical multiplicative models which are defined by setting certain non-homogeneous generalized odds ratios equal to one and are named after Aitchison and Silvey who were among the first to consider such ratios. These models are curved exponential families that do not contain an overall effect and, from an algebraic perspective, are non-homogeneous toric ideals. The relationship of this model class with log-linear models and quasi log-linear models is studied in detail in terms of both statistics and algebraic geometry. The existence of maximum likelihood estimates and their properties, as well as the relevant algorithms are also discussed.
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