Likelihood ratio tests in random graph models with increasing dimensions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908422702628864 |
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| author | Yan, Ting Li, Yuanzhang Xu, Jinfeng Yang, Yaning Zhu, Ji |
| author_facet | Yan, Ting Li, Yuanzhang Xu, Jinfeng Yang, Yaning Zhu, Ji |
| contents | We explore the Wilks phenomena in two random graph models: the $β$-model and the Bradley-Terry model. For two increasing dimensional null hypotheses, including a specified null $H_0: β_i=β_i^0$ for $i=1,\ldots, r$ and a homogenous null $H_0: β_1=\cdots=β_r$, we reveal high dimensional Wilks' phenomena that the normalized log-likelihood ratio statistic, $[2\{\ell(\widehat{\mathbfβ}) - \ell(\widehat{\mathbfβ}^0)\} - r]/(2r)^{1/2}$, converges in distribution to the standard normal distribution as $r$ goes to infinity. Here, $\ell( \mathbfβ)$ is the log-likelihood function on the model parameter $\mathbfβ=(β_1, \ldots, β_n)^\top$, $\widehat{\mathbfβ}$ is its maximum likelihood estimator (MLE) under the full parameter space, and $\widehat{\mathbfβ}^0$ is the restricted MLE under the null parameter space. For the homogenous null with a fixed $r$, we establish Wilks-type theorems that $2\{\ell(\widehat{\mathbfβ}) - \ell(\widehat{\mathbfβ}^0)\}$ 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 $β$-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 $β$-model and the Bradley-Terry model. Simulation studies and real network data applications further demonstrate the theoretical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_05806 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Likelihood ratio tests in random graph models with increasing dimensions Yan, Ting Li, Yuanzhang Xu, Jinfeng Yang, Yaning Zhu, Ji Statistics Theory Methodology We explore the Wilks phenomena in two random graph models: the $β$-model and the Bradley-Terry model. For two increasing dimensional null hypotheses, including a specified null $H_0: β_i=β_i^0$ for $i=1,\ldots, r$ and a homogenous null $H_0: β_1=\cdots=β_r$, we reveal high dimensional Wilks' phenomena that the normalized log-likelihood ratio statistic, $[2\{\ell(\widehat{\mathbfβ}) - \ell(\widehat{\mathbfβ}^0)\} - r]/(2r)^{1/2}$, converges in distribution to the standard normal distribution as $r$ goes to infinity. Here, $\ell( \mathbfβ)$ is the log-likelihood function on the model parameter $\mathbfβ=(β_1, \ldots, β_n)^\top$, $\widehat{\mathbfβ}$ is its maximum likelihood estimator (MLE) under the full parameter space, and $\widehat{\mathbfβ}^0$ is the restricted MLE under the null parameter space. For the homogenous null with a fixed $r$, we establish Wilks-type theorems that $2\{\ell(\widehat{\mathbfβ}) - \ell(\widehat{\mathbfβ}^0)\}$ 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 $β$-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 $β$-model and the Bradley-Terry model. Simulation studies and real network data applications further demonstrate the theoretical results. |
| title | Likelihood ratio tests in random graph models with increasing dimensions |
| topic | Statistics Theory Methodology |
| url | https://arxiv.org/abs/2311.05806 |