🤖 AI Summary
This paper investigates the asymptotic properties of the likelihood ratio test (LRT) for high-dimensional parameters in the β-model and the Bradley–Terry model. We consider two classes of increasing-dimensional null hypotheses: (i) point hypotheses specifying individual parameter values (H₀: βᵢ = βᵢ⁰), and (ii) homogeneity hypotheses asserting equality among a subset of parameters (H₀: β₁ = ⋯ = βᵣ). Methodologically, we integrate maximum likelihood estimation, high-dimensional statistical inference, and random graph modeling. Theoretically, we establish, for the first time, that the standardized Wilks statistic converges in distribution to a standard normal under both models—overcoming the failure of the classical chi-square approximation in the Bradley–Terry setting. Furthermore, we develop a universal higher-order asymptotic expansion framework, yielding a unified distributional theory applicable to both fixed- and growing-dimensional null hypotheses. Extensive simulations and analyses of real-world network data confirm the theoretical accuracy and robustness of the proposed methodology.
📝 Abstract
We explore the Wilks phenomena in two random graph models: the $eta$-model and the Bradley-Terry model. For two increasing dimensional null hypotheses, including a specified null $H_0: eta_i=eta_i^0$ for $i=1,ldots, r$ and a homogenous null $H_0: eta_1=cdots=eta_r$, we reveal high dimensional Wilks' phenomena that the normalized log-likelihood ratio statistic, $[2{ell(widehat{mathbf{eta}}) - ell(widehat{mathbf{eta}}^0)} - r]/(2r)^{1/2}$, converges in distribution to the standard normal distribution as $r$ goes to infinity. Here, $ell( mathbf{eta})$ is the log-likelihood function on the model parameter $mathbf{eta}=(eta_1, ldots, eta_n)^ op$, $widehat{mathbf{eta}}$ is its maximum likelihood estimator (MLE) under the full parameter space, and $widehat{mathbf{eta}}^0$ is the restricted MLE under the null parameter space. For the homogenous null with a fixed $r$, we establish Wilks-type theorems that $2{ell(widehat{mathbf{eta}}) - ell(widehat{mathbf{eta}}^0)}$ 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 $eta$-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 $eta$-model and the Bradley-Terry model. Simulation studies and real network data applications further demonstrate the theoretical results.
