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### 基于改进Huber损失的部分线性模型稳健经验似然推断

1. 1. 首都经济贸易大学统计学院, 北京 100070;
2. 盐城师范学院 数学与统计学院, 盐城 224002
• 收稿日期:2021-06-29 修回日期:2021-12-27 出版日期:2022-05-25 发布日期:2022-07-23
• 通讯作者: 刘强,Email:cuebliuqiang0910@126.com.
• 基金资助:
国家自然科学基金青年项目(11901508)资助课题.

SUN Huihui, LIU Qiang. Robust Orthogonal Empirical Likelihood for Partial Linear Models Based on Modified Huber's Loss Function[J]. Journal of Systems Science and Mathematical Sciences, 2022, 42(5): 1330-1343.

### Robust Orthogonal Empirical Likelihood for Partial Linear Models Based on Modified Huber's Loss Function

SUN Huihui1,2, LIU Qiang1

1. 1. School of Statistics, Capital University of Economics and Business, Beijing 100070;
2. School of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224002
• Received:2021-06-29 Revised:2021-12-27 Online:2022-05-25 Published:2022-07-23

This paper discusses an effective and robust empirical likelihood inference for partial linear models. The procedure applies a modified Huber's function with tail function replaced by the exponential squared loss (ESL) to achieve robustness and effectiveness. Combined with the modified Huber's loss function and QR decomposition technique, an orthogonal empirical likelihood based on Huber-ESL is proposed by improving the estimation equation in the constraint condition of empirical likelihood method to suppress the influence of outliers. Meanwhile, the parametric and nonparametric part of the models are estimated separately to avoid the mutual influence and improve the effectiveness of the estimation. Under some mild conditions, the asymptotic behavior of the robust empirical likelihood approach is established. The finite sample performance of our proposed method is studied through simulations and the proposed method is applied to the Boston house price data. The results show that the performance of our Huber-ESL based empirical likelihood method is competitive with Huber-based procedure and much better than nonrobust empirical likelihood method when the data are contaminated.

MR(2010)主题分类:

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 [1] Owen A. Empirical Likelihood. New York:Chapman and Hall, 2001.[2] 刘强,薛留根,陈放.删失数据下部分线性EV模型中参数的经验似然置信域.数学学报, 2009, 52(3):135-146.(Liu Q, Xue L G, Chen F. Empirical likelihood confidence regions of parameters in a censored partially linear EV model. Acta Mathematica Sinica, 2009, 52(3):135-146.)[3] Xue L G, Dong X. Empirical likelihood for semiparametric regression model with missing response data. Mathematica Applicata, 2008, 102(4):723-740.[4] Tang N S, Zhao P X. Empirical likelihood semiparametric nonlinear regression analysis for longitudinal data with responses missing at random. Annals of the Institute of Statistical Mathematics, 2013, 65:639-665.[5] 丁先文,徐亮,林金官.非线性回归模型的经验似然诊断.应用数学学报, 2012, 35(4):693-702.(Ding X W, Xu L, Lin J G. Diagnostic measures for nonlinear regression models based on empirical likelihood method. Acta Mathematica Applicatae Sinica, 2012, 35(4):693-702.)[6] 徐亮,丁先文,林金官.基于经验似然的部分线性模型的统计诊断.应用概率统计, 2011, 27(1):91-102.(Xu L, Ding X W, Lin J G. Diagnostic measures for partial linear models based on empirical likelihood method. Chinese Journal of Applied Probability and Statistics, 2011, 27(1):91-102.)[7] Molanes Lopez E M, Keilegom I V, Veraverbeke N. Empirical likelihood for non-smooth criterion functions. Scandinavian Journal of Statistics, 2009, 36(3):413-432.[8] Lazar N A. A review of empirical likelihood. Annual Review of Statistics and Its Application, 2021, 8(1):329-344.[9] Cheng W L, Tang N S. Smoothed empirical likelihood inference for ROC curve in the presence of missing biomarker values. Biometrical Journal, 2020, DOI:10.1002/bimj.201900121.[10] Huber P. Robust Estimation. New York:John Wiley&Sons, 1981.[11] Qin G Y, Bai Y, Zhu Z Y. Robust empirical likelihood inference for generalized partial linear models with longitudinal data. Journal of Multivariate Analysis, 2012, 105:32-44.[12] Qin G Y, Zhu Z Y. Robust estimation in generalized semiparametric mixed models for longitudinal data. Journal of Multivariate Analysis, 2007, 98:1658-1683.[13] Maronna R A, Martin R D, Yohai V J, et al. Robust Statistics:Theory and Methods (with R). New York:Wiley, 2018.[14] Wang X Q, Jiang Y L, Huang M, et al. Robust variable selection with exponential squared loss. Journal of the American Statistical Association, 2013, 108(502):632-643.[15] Jiang Y L, Ji Q H, Xie B J. Robust estimation for the varying coefficient partially nonlinear models. Journal of Computational and Applied Mathematics, 2017, 326:31-43.[16] Jiang Y L, Tian G L, Fei Y. A robust and efficient estimation method for partially nonlinear models via a new MM algorithm. Statistical Papers, 2019, 60(6):2063-2085.[17] Song Y Q, Liang X J, Zhu Y J, et al. Robust variable selection with exponential squared loss for the spatial autoregressive model. Computational Statistics&Data Analysis, 2020, DOI:10.1016/j.csda.2020.107094.[18] Yu P, Zhu Z Y, Zhang Z Z. Robust exponential squared loss-based estimation in semi-functional linear regression models. Computational Statistics, 2018, 34(2):503-525.[19] Li S M, Wang K N, Ren Y Y. Robust estimation and empirical likelihood inference with exponential squared loss for panel data models. Economics Letters, 2018, 164:19-23.[20] Jiang Y L, Wang Y G, Fu L Y, et al. Robust estimation using modifed Huber's functions with new tails. Technometrics, 2019, 61(1):111-122.[21] Cai X, Xue L G, Wang Z L. Robust estimation with modified Huber's function for functional linear models. Statistics:A Journal of Theoretical and Applied Statistics, 2020, 54(6):1276-1286.[22] Cai X, Xue L G, Lu F. Robust estimation with a modified Huber's loss for partial functional linear models based on splines. Journal of the Korean Statistical Society, 2020, 49(3):1214-1237.[23] Ma S J. Estimation and inference in functional single-index models. Annals of the Institute of Statistical Mathematics, 2016, 68(1):181-208.[24] Huang J Z. Local asymptotics for polynomial spline regression. The Annals of Statistics, 2003, 31(5):1600-1635.[25] Yao W X, Lindsay B G, Li R Z. Local modal regression. Journal of Nonparametric Statistics, 2012, 24(3):647-663.[26] He X M, Fung W K, Zhu Z Y. Robust estimation in generalized partial linear models for clustered data. Journal of the American Statistical Association, 2005, 100:1176-1184.[27] Sinha S K. Robust analysis of generalized linear mixed models. Journal of American Statistical Association, 2004, 99(466):451-460.[28] Hampel F R, Ronchetti E M, Rousseeuw P J, et al. Robust Statistics:The Approach Based on Influence Functions. New York:Wiley, 1986.[29] Zhao P X, Xue L G. Empirical likelihood inferences for semiparametric varying-coefficient partially linear models with longitudinal data. Communications in Statistics-Theory and Methods, 2010, 39:1898-1914.[30] Schumaker L L. Spline Functions. New York:Wiley, 1981.[31] Xue L G, Zhu L X. Empirical likelihood semiparametric regression analysis for longitudinal data. Biometrika, 2007, 94:921-937.
 [1] 王诗芸，赵培信，杨宜平. 基于模态回归的纵向部分线性模型的有效稳健估计[J]. 系统科学与数学, 2020, 40(12): 2459-2473. [2] 袁晖坪. 关于酉对称矩阵的QR分解及其算法[J]. 系统科学与数学, 2012, 32(2): 172-180. [3] 周兴才;胡舒合. NA样本部分线性模型估计的强相合性[J]. 系统科学与数学, 2010, 30(1): 60-071. [4] 于卓熙;王德辉;史宁中. NA误差下部分线性模型的经验似然推断[J]. 系统科学与数学, 2009, 29(4): 490-501. [5] 许冰. $NA$ 相依样本部分线性模型估计理论[J]. 系统科学与数学, 2004, 24(2): 232-242.