Asymptotic expansions of powered order statistics for generalized Maxwell distribution

Let $M_{n,r}$ denote the $r$th largest-order statistics of a sequence of independent random variables with common generalized Maxwell distribution with positive parameter $k$. This paper mainly considers the higher-order asymptotic expansions of the distribution of normalized powered order statistics $|M_{n,r}{|}^t$ and the corresponding convergence rates under different norming constants. The results show that when $t=2k$ and the optimal norming constants are selected, and the convergence speed of the distribution of $|M_{n,r}{|}^t$ toward its extreme limit is proportional to $1/(\log n{)}^2$, and for the case of $t\ne 2k$, the convergence rate is proportional to $1/\log n$. The main results are confirmed by some numerical analysis.