# Distributional expansions of powered order statistics from general error distribution

## TL;DR

This paper derives distributional expansions and convergence rates for normalized powered order statistics from independent variables with general error distribution, extending known results from normal sequences.

## Contribution

It generalizes Hall's results on powered-extremes to variables with general error distribution, providing new distributional expansions and convergence rates.

## Key findings

- Distributional expansions for normalized powered order statistics.
- Convergence rates of powered order statistics to their limits.
- Extension of Hall's results to general error distributions.

## Abstract

Let $\{X_{n}, n\ge 1\}$ be a sequence of independent random variables with common general error distribution $GED(v)$ with shape parameter $v>0$, and let $M_{n,r}$ denote the $r$th largest order statistics of $X_{1}, X_{2}, \cdots, X_{n}$. With different normalizing constants the distributional expansions of normalized powered order statistics $|M_{n,r}|^{p}$ are established, from which the convergence rates of powered order statistics to their limits are derived. This paper generalized Hall's results on powered-extremes of normal sequence.