CramÉr-type moderate deviation theorems for nonnormal approximation

A Cramér-type moderate deviation theorem quantifies the relative error of the tail probability approximation. It provides a criterion whether the limiting tail probability can be used to estimate the tail probability under study. Chen, Fang and Shao (2013) obtained a general Cramér-type moderate result using Stein’s method when the limiting was a normal distribution. In this paper, Cramér-type moderate deviation theorems are established for nonnormal approximation under a general Stein identity, which is satisfied via the exchangeable pair approach and Stein’s coupling. In particular, a Cramér-type moderate deviation theorem is obtained for the general Curie–Weiss model and the imitative monomer-dimer mean-field model.