带传播噪声的递归max-线性模型
Recursive max-linear models with propagating noise
摘要：
递归最大线性向量通过结构方程模型对节点变量之间的因果关系进行建模，将每个节点变量表示为有向无圈图（DAG）中其父节点的最大线性函数和一些外生创新。对于这样的模型，存在唯一的最小DAG，由其边权重矩阵的Kleene星矩阵表示，该最小DAG识别模型并且可以被估计。为了更真实的统计建模，我们引入了一些随机的观测噪声。这个新的噪声模型的概率分析表明，唯一的最小DAG代表的非噪声模型的分布保持不变，可识别的。此外，模型参数在其左极限的最小比估计量的分布完全由噪声变量的分布决定，直到一个正的常数。在噪声变量的一个规则的变化条件下，我们证明了估计的Kleene星矩阵收敛到一个矩阵的独立的Weibull条目后，适当的中心和缩放。
英文摘要：
Recursive max-linear vectors model causal dependence between node variables by a structural equation model, expressing each node variable as a max-linear function of its parental nodes in a directed acyclic graph (DAG) and some exogenous innovation. For such a model, there exists a unique minimum DAG, represented by the Kleene star matrix of its edge weight matrix, which identifies the model and can be estimated. For a more realistic statistical modeling we introduce some random observational noise. A probabilistic analysis of this new noisy model reveals that the unique minimum DAG representing the distribution of the non-noisy model remains unchanged and identifiable. Moreover, the distribution of the minimum ratio estimators of the model parameters at their left limits are completely determined by the distribution of the noise variables up to a positive constant. Under a regular variation condition on the noise variables we prove that the estimated Kleene star matrix converges to a matrix of independent Weibull entries after proper centering and scaling.
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