Before revision:  We propose a theoretical framework for analyzing small sample properties based on a novel approximation of the discontinuous sample moments with a H\"older continuous process . Using this approximation , we derive remainder bounds with nearly-optimal rates for the asymptotic linear expansions of exact and k-step estimators of IVQR models.
After revision:  We propose a theoretical framework for analyzing small sample properties based on a novel approximation of the discontinuous sample moments by a H\"older-continuous process with a negligible error. For any consistent estimator, this approximation leads to asymptotic linear expansions of exact and k-step estimators of IVQR models.