Abstract
In this paper, by relaxing the mixing coefficients to α(n) = O(n −β), β > 3, we investigate the Bahadur representation of sample quantiles under α-mixing sequence and obtain the rate as \({O(n^{-\frac{1}{2}}(\log\log n\cdot\log n)^{\frac{1}{2}})}\). Meanwhile, for any δ > 0, by strengthening the mixing coefficients to α(n) = O(n −β), \({\beta > \max\{3+\frac{5}{1+\delta},1+\frac{2}{\delta}\}}\), we have the rate as \({O(n^{-\frac{3}{4}+\frac{\delta}{4(2+\delta)}}(\log\log n\cdot \log n)^{\frac{1}{2}})}\). Specifically, if \({\delta=\frac{\sqrt{41}-5}{4}}\) and \({\beta > \frac{\sqrt{41}+7}{2}}\), then the rate is presented as \({O(n^{-\frac{\sqrt{41}+5}{16}}(\log\log n\cdot \log n)^{\frac{1}{2}})}\).
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Zhang, Q., Yang, W. & Hu, S. On Bahadur representation for sample quantiles under α-mixing sequence. Stat Papers 55, 285–299 (2014). https://doi.org/10.1007/s00362-012-0472-z
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DOI: https://doi.org/10.1007/s00362-012-0472-z