Rates of convergence of powered order statistics from general error distribution

Abstract

Let $\{X_n:n\ge 1\}$ be a sequence of independent random variables with common general error distribution $\text{GED}\left(v\right)$ with shape parameter v>0, and let $M_{n,r}$ denote the r-th largest order statistics of $X_1,X_2,\dots ,X_n$. With different normalizing constants the distributional expansions and the uniform convergence rates of normalized powered order statistics $|M_{n,r}{|}^p$ are established. An alternative method is presented to estimate the probability of the r-th extremes. Numerical analyses are provided to support the main results.

Keywords:

• Distributional expansion
• uniform convergence rate
• general error distribution
• powered order statistic

Acknowledgements

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.

Disclosure statement

No potential conflict of interest was reported by the author(s).