Estimating Undirected Graphs Under Weak Assumptions

We consider the problem of providing nonparametric confidence guarantees for undirected graphs under weak assumptions. In particular, we do not assume sparsity, incoherence or Normality. We allow the dimension D to increase with the sample size n. First, we prove lower bounds that show that if we want accurate inferences with low assumptions then there are limitations on the dimension as a function of sample size. When the dimension increases slowly with sample size, we show that methods based on Normal approximations and on the bootstrap lead to valid inferences and we provide Berry-Esseen bounds on the accuracy of the Normal approximation. When the dimension is large relative to sample size, accurate inferences for graphs under low assumptions are not possible. Instead we propose to estimate something less demanding than the entire partial correlation graph. In particular, we consider: cluster graphs, restricted partial correlation graphs and correlation graphs.