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Bivariate gamma model
Journal of Multivariate Analysis ( IF 1.4 ) Pub Date : 2020-11-01 , DOI: 10.1016/j.jmva.2020.104666
Ruijian Han , Kani Chen , Chunxi Tan

Among undirected graph models, the β-model plays a fundamental role and is widely applied to analyze network data. It assumes the edge probability is linked with the sum of the strength parameters of the two vertices through a sigmoid function. Because of the univariate nature of the link function, this formulation, despite its popularity, can be too restrictive for practical applications, even with a straightforward extension of the link function. For example, it is possible that vertices with similar strength parameters are more likely to be connected, in which case the edge probability depends on the distance of the strength parameters. Such common cases are not included in the β-model or its immediate extensions. In this paper, we propose a bivariate gamma model that links the edge probability with the two strength parameters by a symmetric bivariate function. The proposed model is more flexible than the β-model and its existing variants. It is also applicable to mirror various undirected networks. We show some special but representative cases of the bivariate gamma model by considering sparsity, mixture and other modifications, which cannot be properly handled by the β-model. Asymptotic theory is established to justify the consistency and asymptotic normality of the moment estimators. Numerical studies present evidence in support of the theory and an example involving real data further illustrates the applications.

中文翻译:

双变量伽马模型

在无向图模型中,β-模型起着基础性的作用,被广泛应用于分析网络数据。它假设边缘概率通过 sigmoid 函数与两个顶点的强度参数之和相关联。由于链接函数的单变量性质,这种公式尽管很受欢迎,但对于实际应用来说可能过于严格,即使是链接函数的直接扩展也是如此。例如,具有相似强度参数的顶点更有可能被连接,在这种情况下,边缘概率取决于强度参数的距离。这种常见情况不包括在 β 模型或其直接扩展中。在本文中,我们提出了一个双变量伽玛模型,该模型通过对称双变量函数将边缘概率与两个强度参数联系起来。所提出的模型比 β 模型及其现有变体更灵活。也适用于镜像各种无向网络。通过考虑 β 模型无法正确处理的稀疏性、混合性和其他修改,我们展示了双变量 gamma 模型的一些特殊但具有代表性的案例。渐近理论的建立是为了证明矩估计量的一致性和渐近正态性。数值研究提供了支持该理论的证据,一个涉及真实数据的例子进一步说明了这些应用。通过考虑 β 模型无法正确处理的稀疏性、混合性和其他修改,我们展示了双变量 gamma 模型的一些特殊但具有代表性的情况。渐近理论的建立是为了证明矩估计量的一致性和渐近正态性。数值研究提供了支持该理论的证据,一个涉及真实数据的例子进一步说明了这些应用。通过考虑 β 模型无法正确处理的稀疏性、混合性和其他修改,我们展示了双变量 gamma 模型的一些特殊但具有代表性的情况。渐近理论的建立是为了证明矩估计量的一致性和渐近正态性。数值研究提供了支持该理论的证据,一个涉及真实数据的例子进一步说明了这些应用。
更新日期:2020-11-01
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