Cram\'er-type Moderate Deviation Theorems for Nonnormal Approximation

TL;DR

This paper extends Cramér-type moderate deviation theorems to nonnormal approximations using Stein's method, providing theoretical tools for tail probability estimation in complex models like Curie--Weiss.

Contribution

It develops a general framework for Cramér-type moderate deviations under nonnormal limits via Stein's identity, applicable to models such as Curie--Weiss.

Findings

Established Cramér-type moderate deviation theorems for nonnormal approximations.

Applied results to the Curie--Weiss model and monomer-dimer mean-field model.

Provided theoretical justification for tail probability estimates in complex models.

Abstract

A Cram\'er-type moderate deviation theorem quantifies the relative error of the tail probability approximation. It provides theoretical justification when the limiting tail probability can be used to estimate the tail probability under study. Chen Fang and Shao (2013) obtained a general Cram\'er-type moderate result using Stein's method when the limiting was a normal distribution. In this paper, Cram\'er-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 Cram\'er-type moderate deviation theorem is obtained for the general Curie--Weiss model and the imitative monomer-dimer mean-field model.