Higher-order expansions of sample range from general error distribution

Abstract

Let $\{X_n,n\ge 1\}$ be a sequence of independent random variables with common general error distribution $\text{GED}(v)$ with shape parameter $v>0$, and denote $M_n$ and $m_n$ the partial maximum and minimum of $\{X_n,n\ge 1\}$. With different normalizing constants, the distributional expansions of normalized sample range $M_n-m_n$ 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 $1/\hspace{0.17em}\log \hspace{0.17em}n$ as $v\in (0,1)\cup (1,\infty )$ contrary to the case of $v=1$, which is proportional to $1/n$. Furthermore, numerical analysis is provided to illustrate the theoretical findings.

Keywords:

• Sample range
• distributional expansion
• general error distribution

Acknowledgments

The authors would like to thank the Editor-in-Chief, the Associated Editor and the two referees for careful reading and comments which greatly improved the paper.

Additional information

Funding

This work was supported by the Natural Science Foundation of Sichuan Province (No. 2022NSFSC1838) and the Project for Humanities and Social Sciences of USST (No. 21SKPY03).