Robust estimations from distribution structures: V. Non-asymptotic

Abstract: Due to the complexity of order statistics, the finite sample behaviour of robust statistics is generally not analytically solvable. While the Monte Carlo method can provide approximate solutions, its convergence rate is typically very slow, making the computational cost to achieve the desired accuracy unaffordable for ordinary users. In this paper, we propose an approach analogous to the Fourier transformation to decompose the finite sample structure of the uniform distribution. By obtaining sets of sequences that are consistent with parametric distributions for the first four sample moments, we can approximate the finite sample behavior of other estimators with significantly reduced computational costs. This article reveals the underlying structure of randomness and presents a novel approach to integrate multiple assumptions.