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. Furthermore, we derive a bias formula for exact IVQR estimators up to order O\left(1{n}\right).
After 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 with nearly optimal rates. Second, we study the higher-order bias of exact quantile estimators up to order O\left(1{n}\right).