删失数据下单指标模型的经验似然推断

数学物理学报 ›› 2012, Vol. 32 ›› Issue (2): 297-311.

删失数据下单指标模型的经验似然推断

杨宜平¹|薛留根²|程维虎²

1.重庆工商大学数学与统计学院重庆 400067;
2.北京工业大学应用数理学院北京 100124

收稿日期:2009-09-28 修回日期:2010-09-16 出版日期:2012-04-25 发布日期:2012-04-25
基金资助:
国家社科基金(11CTJ004)、国家自然科学基金(11126332, 11101452)、重庆市自然科学基金(CSTC2011JJA00014)、重庆市教委科学技术研究项目(KJ110720)、北京市自然科学基金(1102008)和广西自然科学基金(2010GXNSFB013051)资助

Inferences in a Censored Single-index Model with Empirical Likelihood

YANG Yi-Ping¹, XUE Liu-Gen², CHENG Wei-Hu²

1.College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067;
2.College of Applied Sciences, Beijing University of Technology, Beijing 100124

Received:2009-09-28 Revised:2010-09-16 Online:2012-04-25 Published:2012-04-25
Supported by:
国家社科基金(11CTJ004)、国家自然科学基金(11126332, 11101452)、重庆市自然科学基金(CSTC2011JJA00014)、重庆市教委科学技术研究项目(KJ110720)、北京市自然科学基金(1102008)和广西自然科学基金(2010GXNSFB013051)资助

摘要/Abstract

摘要：

考虑删失数据下单指标模型, 研究了模型中参数的经验似然推断, 证明了所提出的调整的经验对数似然比渐近于卡方分布, 由此构造相应兴趣参数的置信域. 进一步, 由于模型中参数向量的范数等于1, 利用该约束条件来降低参数的维数, 从而增加置信域的精度. 模拟研究比较了经验似然方法和正态逼近方法的有限样本性质, 从置信域的面积和覆盖概率两方面进行了比较, 模拟结果表明经验似然方法优于正态逼近方法.

关键词: 删失数据, 单指标模型, 经验似然, χ² 分布

Abstract:

Empirical likelihood method for single-index models with randomly censored data is investigated. An estimated empirical likelihood for the parameters by using a synthetic data approach is developed. An adjusted empirical log-likelihood ratio function is proven to be asymptotically standard chi-squared, and the corresponding confidence regions of the parameters are then constructed. Furthermore,
since the index is of norm 1, this constraint is used to reduce the dimension of parameters, which increases the accuracy of the confidence regions. A simulation study indicates that compared with a normal approximation method, the proposed method works better in terms of coverage probabilities and areas of confidence regions.

Key words: Censored data, Single-index model, Empirical likelihood, , χ²-distribution

中图分类号:

62G05

杨宜平, 薛留根, 程维虎. 删失数据下单指标模型的经验似然推断[J]. 数学物理学报, 2012, 32(2): 297-311.

YANG Yi-Ping, XUE Liu-Gen, CHENG Wei-Hu. Inferences in a Censored Single-index Model with Empirical Likelihood[J]. Acta mathematica scientia,Series A, 2012, 32(2): 297-311.

参考文献

[1] Ichimura H. Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J Econometrics, 1993, 58: 71--120

[2] Härdle W, Tsybakov A B. How sensitive are average derivative. J Econometrics, 1993, 58: 31--48

[3] Xue L G, Zhu L X. Empirical likelihood for single-index models. J Multi Anal, 2006, 97: 1295--1312

[4] Fan J, Gijbels I. Censored regression: local linear approximations and their applications. J Amer Statist Assoc, 1994, 89: 560--570

[5] Wang Q H, Li G. Empirical likelihood semiparametric regression analysis under random censorship. J Multi Anal, 2002, 83: 469--486

[6] Lu X, Chen G, Song X K, Singh R S. A class of partially linear single-index survival models. Canad J Statist, 2006, 34: 99--116

[7] Lu X, Cheng T L. Randomly censored partially linear single-index models. J Multi Anal, 2007, 98: 1895--1922

[8] Owen A B. Empirical likelihood ratio confidence intervals for a single function. Biometrika, 1988, 75: 237--325

[9] Owen A B. Empirical likelihood ratio confidence regions. Ann Statist, 1990, 18: 90--120

[10] 冯三营, 薛留根, 范承华. 非线性半参数回归模型中参数的经验似然置信域. 数学物理学报, 2009, 29(5): 1338--1349

[11] Zhu L X, Xue L G. Empirical likelihood confidence regions in a partially linear single-index model. J Roy Statist Soc (Ser B), 2006, 68: 549--570

[12] Fan J, Gijbels I. Local Polynomial Modeling and Its Applications. London: Chapman and Hall, 1996

[13] Rao J, Scott A. The analysis of categorical data from complex sample surveys: Chi-squared tests for goodness-of-fit and
independence in two-way tables. J Amer Statist Assoc, 1981, 76: 221--230

[14] Chen M, Yuen K C, Zhu L X. Asymptotics of the goodness-of-fit test for a partial linear model with randomly censored data. Sci China (Ser A), 2003, 46(2): 145--158

[15] Lai T L, Ying Z, Zheng Z. Asymptotic normality of a class of adaptive statistics with applications of synthetic data methods for censored
regression. J Multi Anal, 1995, 52: 259--279

[16] Zhou M. Asymptotic normality of the synthetic estimator for censored survival data. Ann Statist, 1992, 20: 1002--1021