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 $ \hbox{GED} (v) $ with shape parameter v>0, and let $ M_{n,r} $ denote the r-th largest order statistics of $ X_{1}, X_{2}, \ldots, 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