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\neq 2k $, the convergence rate is proportional to $ 1/\log n $. The main results are confirmed by some numerical analysis.
Citation: Jianwen Huang, Xinling Liu, Jinping Jia, Runke Wang. Asymptotic expansions of powered order statistics for generalized Maxwell distribution[J]. AIMS Mathematics, 2026, 11(3): 8842-8858. doi: 10.3934/math.2026364
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\neq 2k $, the convergence rate is proportional to $ 1/\log n $. The main results are confirmed by some numerical analysis.
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Jianwen Huang, Xinling Liu, Jinping Jia, Runke Wang. Asymptotic expansions of powered order statistics for generalized Maxwell distribution[J]. AIMS Mathematics, 2026, 11(3): 8842-8858. doi: 10.3934/math.2026364
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