Likelihood Ratio Tests in Random Graph Models with Increasing Dimensions

• 成果类型: Article; Early Access
• 署名作者: Yan, Ting; Li, Yuanzhang; Xu, Jinfeng; Yang, Yaning; Zhu, Ji
• 署名单位: Central China Normal University; George Washington University; City University of Hong Kong; Chinese Academy of Sciences; University of Science & Technology of China, CAS; University of Michigan System; University of Michigan
• 刊物名称: JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
• ISSN/ISSBN: 0162-1459
• DOI: 10.1080/01621459.2025.2488477
• 发表日期: 2025
• 关键词: CENTRAL-LIMIT-THEOREM; PROBABILITY-DISTRIBUTIONS; parameters tends; diverging number; beta-model; asymptotics; statistics; FAMILY

摘要

We explore the Wilks phenomena in two random graph models: the beta-model and the Bradley-Terry model. For two increasing dimensional null hypotheses, including a specified null H-0:beta(i)=beta(0)(i) for i=1,...,r and a homogenous null H-0:beta(1)=...=beta(r), we reveal high dimensional Wilks' phenomena that the normalized log-likelihood ratio statistic, [2{l(beta)-l(beta(0))}-r]/(2r)(1/2), converges in distribution to the standard normal distribution as r goes to infinity. Here, l(beta) is the log-likelihood function on the model parameter beta=(beta(1),...,beta(n))(inverted perpendicular), beta is its maximum likelihood estimator (MLE) under the full parameter space, and beta(0) is the restricted MLE under the null parameter space. For the homogenous null with a fixed r, we establish Wilks-type theorems that 2{l(beta)-l(beta(0))} converges in distribution to a chi-square distribution with r-1 degrees of freedom, as the total number of parameters, n, goes to infinity. When testing the fixed dimensional specified null, we find that its asymptotic null distribution is a chi-square distribution in the beta-model. However, unexpectedly, this is not true in the Bradley-Terry model. By developing several novel technical methods for asymptotic expansion, we explore Wilks-type results in a principled manner; these principled methods should be applicable to a class of random graph models beyond the beta-model and the Bradley-Terry model. Simulation studies and real network data applications further demonstrate the theoretical results. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.