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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (4): 524-532.doi: 10.3969/j.issn.1005-3085.2015.04.006

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正相协样本分位数估计的Bahadur表示

李永明1,   张文婷2,   李乃医3,   姚   竟4   

  1. 1- 上饶师范学院数学与计算机科学学院，上饶 334001
    2- 桂林航天工业学院来宾校区，来宾 546100
    3- 广东海洋大学理学院，湛江 524088
    4- 广西师范学院数学科学学院，南宁 530023
  • 收稿日期:2014-01-09 接受日期:2014-10-09 出版日期:2015-08-15 发布日期:2015-10-15
  • 基金资助:
    国家自然科学基金 (11461057)；江西省自然科学基金 (20122BAB201007)；江西省教育厅科技项目 (GJJ12604).

The Bahadur Representation for the Estimator of Sample Quantiles under Positive Associated Samples

LI Yong-ming1,   ZHANG Wen-ting2,   LI Nai-yi3,   YAO Jing4   

  1. 1- School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001
    2- Laibin Campus, Guilin University of Aerospace Technology, Laibin 546100
    3- College of Science, Guangdong Ocean University, Zhanjiang 524088
    4- College of Mathematical Science, Guangxi Teachers Education University, Nanning 530023
  • Received:2014-01-09 Accepted:2014-10-09 Online:2015-08-15 Published:2015-10-15
  • Supported by:
    The National Natural Science Foundation of China (11461057); the Natural Science Foundation of Jiangxi Province (20122BAB201007); the Natural Science Foundation of the Education Department of Jiangxi Province (GJJ12604).

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摘要： 作为一类常见的随机变量序列，正相协随机变量序列在可靠性理论和多元统计分析中有着广泛应用．本文的主要目的是研究一类严平稳正相协随机样本分位数的估计问题．首先，利用正相协随机序列的性质，我们获得了一个有关正相协随机变量的协方差不等式．然后，利用正相协序列的指数不等式获得了一个有关经验分布函数的不等式．最后，我们利用所得不等式，在适当的条件下，进一步讨论了样本分位数估计的强相合性，并给出了其Bahadur表示及其收敛速度．

关键词: 正相协序列, 样本分位数, Bahadur表示

Abstract:

The positively associated sequence is a general class of random variables, and has been widely utilized in multivariate statistical analysis and system reliability. The purpose of this paper is to estimate sample quantiles based on a stationary and positively associated sequ-ence. By applying the property of a positively associated sequence, we establish a covariance inequality for the positively associated variables. And then, by using the exponential inequality of a positively associated sequence, we obtain an inequality for the empirical distribution function. Furthermore, under certain conditions, by virtue of the obtained inequality, we discuss the consistency of the sample quantile estimator for positively associated sequence, and derive the Bahadur representation together with its convergence rate.

Key words: positive associated sequence, sample quantiles, Bahadur representation

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