论文标题
Wilks的定理在$β$ - 模型中
Wilks' theorems in the $β$-model
论文作者
论文摘要
可能性比测试和Wilks定理在统计数据中至关重要，但在维度越来越高的网络模型中很少探索。我们在这里关心的是在$β$模型中用于无向图的可能性比测试。对于两个增长的维零假设，包括指定的null $ h_0：β_i=β_i^0 $ for $ i = 1，\ ldots，r $和一个同质的null $ h_0：β_1= \ cd_1 = \ cdots =β_r$ $ [2 \ {\ ell（\ wideHat {\boldsymbolβ}） - \ ell（\ wideHat {\boldsymbolβ}^0）\}^0）\} - r]/（2r）^{1/2} $，以$ r $ $ r $的标准正态分布为$ r $。在这里，$ \ ell（\boldsymbolβ）$是对矢量参数$ \boldsymbolβ=（β_1，\ ldots，β_n）^\ top $，$ \ wideHat {\boldsymbolβ} $的最大可能估计值（MELE（mle），以及$ \ wideHat {\boldsymbolβ}^0 $是null参数空间下的受限mle。对于相应的固定尺寸为null $ h_0：β_i=β_i^0 $ for $ i = 1，\ ldots，r $和均匀的null $ h_0：β_1= \ cdots =β_r$ a $ $ r $，我们建立了$ 2 \ ell（\ ell）的结果， \ ell（\ wideHat {\boldsymbolβ}^0）\} $在分布中收敛到带有各个$ r $和$ r-1 $ $ r $的卡方分布，作为参数的总数，$ n $，转到Infinity。威尔克斯的类型结果进一步扩展到了与比较的密切相关的布拉德利模型中，我们发现在固定尺寸指定的渐近渐近上既不遵循Chi-Square也不遵循重新校正的Chi-square-square-square-square-square-square分布。模拟研究和NBA数据的应用说明了理论结果。
Likelihood ratio tests and the Wilks theorems have been pivotal in statistics but have rarely been explored in network models with an increasing dimension. We are concerned here with likelihood ratio tests in the $β$-model for undirected graphs. For two growing 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, $[2\{\ell(\widehat{\boldsymbolβ}) - \ell(\widehat{\boldsymbolβ}^0)\} - r]/(2r)^{1/2}$, converges in distribution to the standard normal distribution as $r$ goes to infinity. Here, $\ell( \boldsymbolβ)$ is the log-likelihood function on the vector parameter $\boldsymbolβ=(β_1, \ldots, β_n)^\top$, $\widehat{\boldsymbolβ}$ is its maximum likelihood estimator (MLE) under the full parameter space, and $\widehat{\boldsymbolβ}^0$ is the restricted MLE under the null parameter space. For the corresponding fixed dimensional null $H_0: β_i=β_i^0$ for $i=1,\ldots, r$ and the homogenous null $H_0: β_1=\cdots=β_r$ with a fixed $r$, we establish Wilks type of results that $2\{\ell(\widehat{\boldsymbolβ}) - \ell(\widehat{\boldsymbolβ}^0)\}$ converges in distribution to a Chi-square distribution with respective $r$ and $r-1$ degrees of freedom, as the total number of parameters, $n$, goes to infinity. The Wilks type of results are further extended into a closely related Bradley--Terry model for paired comparisons, where we discover a different phenomenon that the log-likelihood ratio statistic under the fixed dimensional specified null asymptotically follows neither a Chi-square nor a rescaled Chi-square distribution. Simulation studies and an application to NBA data illustrate the theoretical results.
