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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 ›› 2017, Vol. 34 ›› Issue (2): 171-181.doi: 10.3969/j.issn.1005-3085.2017.02.006

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$\phi$-混合样本下缺失数据情形线性模型回归系数的经验似然比统计量的渐近分布

郑李玲1,   秦永松2,   李英华2   

  1. 1- 钦州学院理学院，广西 钦州 535011 
    2- 广西师范大学数学与统计学院，广西 桂林 541004
  • 收稿日期:2015-07-22 接受日期:2016-10-12 出版日期:2017-04-15 发布日期:2017-06-15
  • 基金资助:
    国家自然科学基金(11671102; 11271088)；广西自然科学基金(2013GXNSFBA019001; 2016GXNSFAA3800163; 2016GXNSFAA380102; 2016GXNSFAA380317).

Asymptotic Distributions of Empirical Likelihood Ratio Statistics for Regression Coefficients in a Linear Model under $\phi$-mixing Samples with Missing Data

ZHENG Li-ling1,   QIN Yong-song2,   LI Ying-hua2   

  1. 1- School of Science, Qinzhou University, Qinzhou, Guangxi 535011
    2- School of Mathematics and Statistics, Guangxi Normal University, Guilin, Guangxi 541004
  • Received:2015-07-22 Accepted:2016-10-12 Online:2017-04-15 Published:2017-06-15
  • Supported by:
    The National Natural Science Foundation of China (11671102; 11271088); the Natural Science Foundation of Guangxi (2013GXNSFBA019001; 2016GXNSFAA3800163; 2016GXNSFAA380102; 2016GXNSFAA380317).

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摘要： $\phi$-混合的概念作为弱相关的衡量尺度在实际中被广泛应用，且缺失数据现象在各领域常有发生，已有文献对相依和缺失数据两种情形的统计推断分别进行了深入研究，但对同时存在相依和缺失数据情形的研究较少 .本文研究既有相依又有缺失情形的统计推断，即研究$\phi$-混合样本下缺失数据情形线性模型回归系数的经验似然比统计量的渐近分布．我们采取回归填补方法对响应变量的缺失值进行补足，得到线性模型回归系数的“完全”样本数据．在此基础上利用记分函数构造线性模型回归系数的经验似然比统计量，在一定条件下证明经验似然比统计量渐近服从卡方分布，这一结论为构造$\phi$-混合样本下缺失数据情形线性模型回归系数的置信域提供了理论依据．

关键词: $\phi$-混合样本, 缺失数据, 经验似然, 卡方分布

Abstract: The concept of $\phi$-mixing has been used extensively as measures of the weak depen-dence, and the phenomenon of missing data often occurs in various application fields. In existing literatures, the statistical inference under the dependence and missing data, has been deeply studied. However, there are few studies on the case of the dependent and missing data simultaneously. This paper is concerned with the statistical inference simultaneously under the dependence and missing data. In other words, this paper discusses the asymptotic distributions of empirical likelihood ratio statistics for regression coefficients in a linear model under $\phi$-mixing samples with missing data. The regression imputation method is applied to impute the missing data of the response variables, and thus `complete' data for regression coefficients in the linear model are obtained. Furthermore, we employ the score functions to establish the empirical likelihood ratio statistics for the regression vector in the linear model. Under some conditions, it is proved that the empirical likelihood ratio statistics are asymptotically Chi square distributed. This conclusion provides a theoretical basis for the confidence region of the regression coefficients of a linear model under $\phi$-mixing samples with missing data.

Key words: $\phi$-mixing sample, missing data, empirical likelihood, $\chi^2$ distribution

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