Higher-order expansions of sample range from general error distribution

Abstract

Let be a sequence of independent random variables with common general error distribution with shape parameter , and denote and the partial maximum and minimum of . With different normalizing constants, the distributional expansions of normalized sample range are established in this article. A byproduct is to deduce the convergence rates of distributions of normalized sample range to their limits, which shows that the optimal convergence rate is proportional to as contrary to the case of , which is proportional to . Furthermore, numerical analysis is provided to illustrate the theoretical findings.