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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 ›› 2021, Vol. 38 ›› Issue (1): 63-72.doi: 10.3969/j.issn.1005-3085.2021.01.006

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m相依序列的样本分位数核估计的中偏差和大偏差

谢   超,   陈   夏,   闫   莉   

  1. 陕西师范大学数学与信息科学学院，西安  710119
  • 收稿日期:2018-06-21 接受日期:2018-11-29 出版日期:2021-02-15 发布日期:2021-04-15
  • 通讯作者: 闫 莉 E-mail: lyan@snnu.edu.cn
  • 基金资助:
    国家自然科学基金 (11801346)；教育部人文社会科学研究青年基金 (18YJC910014)；陕西省自然科学基础研究计划项目(2018JM1024; 2020JM-276).

The Moderate Deviation and Large Deviation for the Smooth Estimate on the m-dependent Sequences

XIE Chao,   CHEN Xia,   YAN Li   

  1. School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710119
  • Received:2018-06-21 Accepted:2018-11-29 Online:2021-02-15 Published:2021-04-15
  • Contact: L. Yan. E-mail address: lyan@snnu.edu.cn
  • Supported by:
    The National Natural Science Foundation of China (11801346); the Youth Fund for Humanities and Social Sciences Research of Ministry of Education (18YJC910014); the Natural Science Basis Research Plan in Shaanxi Province of China (2018JM1024; 2020JM-276).

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摘要： 分位数是统计学中的一个重要概念，它在可靠性统计分析以及经济、金融、生物信息、医学等领域都有非常广泛的应用．相依随机序列削弱了独立性的限制，得到了众多关注和研究．因此，本文基于 m 相依序列，研究了样本分位数核估计的大样本性质．首先，利用 m 相依序列的极限理论，通过计算Cramer函数，证明了样本分位数核估计的中偏差原理．其次，通过验证Cramer条件成立，得到了样本分位数核估计的大偏差原理．研究结果简化并推广了独立同分布样本情形下的证明方法及结果，为讨论其他类型相依序列的中偏差及大偏差性质提供了重要依据．

关键词: m相依序列, 样本分位数核估计, 中偏差, 大偏差

Abstract: The quantile is an important concept in statistics. It has been widely used in many fields such as reliability statistical analysis, economics, finance, bioinformatics and medicine. The study of the dependent random sequences has received a lot of attention since it weakens the limitation of independence. Therefore, based on the m-dependent sequences, this paper studies the large sample properties of the sample quantile kernel estimation. Firstly, using the limit theorem of m-dependent sequences, the Cramer function is calculated, and the moderate deviation principle of sample quantile kernel estimation is proved. Secondly, by verifying the Cramer condition, large deviation results of the sample quantile kernel estimation are obtained. The proof methods and results of the independent and identically distributed samples are simplified and generalized. Also the results provide an important basis for discussing the moderate deviation and large deviation of other types of dependent sequences.

Key words: m-dependent random sequence, the sample quantile kernel estimate, moderate deviation, large deviation

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