Berry-Esseen bounds for estimating undirected graphs

Abstract

We consider the problem of providing nonparametric confidence guarantees — with finite sample Berry-Esseen bounds — for undirected graphs under weak assumptions. We do not assume sparsity or incoherence. 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 weak assumptions then $D$ must be less than $n$. In that case, we show that methods based on Normal approximations and on the bootstrap lead to valid inferences and we provide new Berry-Esseen bounds on the accuracy of the Normal approximation and the bootstrap. When the dimension is large relative to sample size, accurate inferences for graphs under weak 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.