High dimensional Wilks phenomena in random graph models

Details High dimensional Wilks phenomena in random graph models High dimensional Wilks phenomena in random graph models

2011-12-30

arxiv.org/abs/1201.0058v2

Mathematics

Statistics Theory

22 pages, 1 figure, two tables

Scientific paper

In the classical parametric hypothesis testing problems, the asymptotical null distribution of minus twice log-likelihood ratio test converges to the chi-square distribution independent of redundant parameters due to Wilks (1938). This phenomena is not monopolized by fixed dimension problems. Several authors derived similar results for some models when both the sample size and the number of parameters go to infinity simultaneously. In this paper, we show Wilks type of theorems in simple random graph models, which are known as the $\beta$-model in the undirected case and the Bradley-Terry model in the directed case, when the number of graphic vertices goes to infinity and the number of statistical experiments for each edge is a fixed constant. Numerical studies and a data application are carried out to demonstrate the theoretical results.

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