Testing degree heterogeneity in directed networks
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914218101440512 |
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| author | Pan, Lu Wang, Qiuping Yan, Ting |
| author_facet | Pan, Lu Wang, Qiuping Yan, Ting |
| contents | In this study, we focus on the likelihood ratio tests in the $p_0$ model for testing degree heterogeneity in directed networks, which is an exponential family distribution on directed graphs with the bi-degree sequence as the naturally sufficient statistic. For testing the homogeneous null hypotheses $H_0: α_1 = \cdots = α_r$, we establish Wilks-type results in both increasing-dimensional and fixed-dimensional settings. For increasing dimensions, the normalized log-likelihood ratio statistic $[2\{\ell(\widehat{\mathbfθ})-\ell(\widehat{\mathbfθ}^0)\}-r]/(2r)^{1/2}$ converges in distribution to a standard normal distribution. For fixed dimensions, $2\{\ell(\widehat{\mathbfθ})-\ell(\widehat{\mathbfθ}^0)\}$ converges in distribution to a chi-square distribution with $r-1$ degrees of freedom as $n\rightarrow \infty$, independent of the nuisance parameters. Additionally, we present a Wilks-type theorem for the specified null $H_0: α_i=α_i^0$, $i=1,\ldots, r$ in high-dimensional settings, where the normalized log-likelihood ratio statistic also converges in distribution to a standard normal distribution. These results extend the work of \cite{yan2025likelihood} to directed graphs in a highly non-trivial way, where we need to analyze much more expansion terms in the fourth-order asymptotic expansions of the likelihood function and develop new approximate inverse matrices under the null restricted parameter spaces for approximating the inverse of the Fisher information matrices in the $p_0$ model. Simulation studies and real data analyses are presented to verify our theoretical results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_09865 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Testing degree heterogeneity in directed networks Pan, Lu Wang, Qiuping Yan, Ting Statistics Theory In this study, we focus on the likelihood ratio tests in the $p_0$ model for testing degree heterogeneity in directed networks, which is an exponential family distribution on directed graphs with the bi-degree sequence as the naturally sufficient statistic. For testing the homogeneous null hypotheses $H_0: α_1 = \cdots = α_r$, we establish Wilks-type results in both increasing-dimensional and fixed-dimensional settings. For increasing dimensions, the normalized log-likelihood ratio statistic $[2\{\ell(\widehat{\mathbfθ})-\ell(\widehat{\mathbfθ}^0)\}-r]/(2r)^{1/2}$ converges in distribution to a standard normal distribution. For fixed dimensions, $2\{\ell(\widehat{\mathbfθ})-\ell(\widehat{\mathbfθ}^0)\}$ converges in distribution to a chi-square distribution with $r-1$ degrees of freedom as $n\rightarrow \infty$, independent of the nuisance parameters. Additionally, we present a Wilks-type theorem for the specified null $H_0: α_i=α_i^0$, $i=1,\ldots, r$ in high-dimensional settings, where the normalized log-likelihood ratio statistic also converges in distribution to a standard normal distribution. These results extend the work of \cite{yan2025likelihood} to directed graphs in a highly non-trivial way, where we need to analyze much more expansion terms in the fourth-order asymptotic expansions of the likelihood function and develop new approximate inverse matrices under the null restricted parameter spaces for approximating the inverse of the Fisher information matrices in the $p_0$ model. Simulation studies and real data analyses are presented to verify our theoretical results. |
| title | Testing degree heterogeneity in directed networks |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2502.09865 |