Previous Articles     Next Articles

Single-Index Quantile Regression with Left Truncated Data

XU Hongxia1,2, FAN Guoliang1, LI Jinchang2   

  1. 1. Department of Mathematics, Shanghai Maritime University, Shanghai 201306, China;
    2. School of Data Sciences, Zhejiang University of Finance and Economics, Hangzhou 310018, China;
  • Received:2021-04-15 Revised:2021-06-07 Online:2022-10-25 Published:2022-10-12
  • Supported by:
    This research was supported by the National Social Science Foundation of China under Grant No.21BTJ038.

XU Hongxia, FAN Guoliang, LI Jinchang. Single-Index Quantile Regression with Left Truncated Data[J]. Journal of Systems Science and Complexity, 2022, 35(5): 1963-1987.

The purpose of this paper is two fold.First,the authors investigate quantile regression (QR) estimation for single-index QR models when the response is subject to random left truncation.The random weights are introduced to deal with left truncated data and the associated iteration estimation method is proposed.The asymptotic properties for the proposed QR estimates of the index parameter and unknown link function are both obtained.Further,by combining the QR loss function and the adaptive LASSO penalization,a variable selection procedure for the index parameter is introduced and its oracle property is established.Second,a weighted empirical log-likelihood ratio of the index parameter based on the QR method is introduced and is proved to be asymptotic standard chi-square distribution.Furthermore,confidence regions of the index parameter can be constructed.The finite sample performance of the proposed methods are demonstrated.A real data analysis is also conducted to show the usefulness of the proposed approaches.
[1] Härdle W and Stoker T, Investing smooth multiple regression by the method of average derivatives, Journal of the American Statistical Association, 1989, 84:986-995.
[2] Härdle W, Hall P, and Ichimura H, Optimal smoothing in single-index models, Annals of Statistics, 1993, 21:157-178.
[3] Chaudhuri P, Doksum K, and Samarov A, On average derivative quantile regression, Annals of Statistics, 1997, 25:715-744.
[4] Delecroix M, Hristache M, and Patilea V, On semiparametric m-estimation in single-index regression, Journal of Statistical Planning and Inference, 2006, 136:730-769.
[5] Zhu L P and Zhu L X, Nonconcave penalized inverse regression in single-index models with high dimensional predictors, Journal of Multivariate Analysis, 2009, 100:862-875.
[6] Wu T, Yu K, and Yu Y, Single-index quantile regression, Journal of Multivariate Analysis, 2010, 101:1607-1621.
[7] Kong E and Xia Y, A single-index quantile regression model and its estimation, Econometric Theory, 2012, 28:730-768.
[8] Liu J C, Zhang R Q, Zhao W H, et al., A robust and efficient estimation method for single index models, Journal of Multivariate Analysis, 2013, 122:226-238.
[9] Lu Y Q, Zhang R Q, and Hu B, The adaptive LASSO spline estimation of single-index model, Journal of Systems Science and Complexity, 2016, 29:1100-1111.
[10] Koenker R and Bassett G, Regression quantiles, Econometrica, 1978, 46:33-50.
[11] Wang H J, Zhu Z, and Zhou J, Quantile regression in partially linear varying coefficient models, Annals of Statistics, 2009, 37:3841-3866.
[12] Lv Y Z, Zhang R Q, Zhao W H, et al., Quantile regression and variable selection for the single-index model, Journal of Applied Statistics, 2014, 41:1565-1577.
[13] Lv Y Z, Zhang R Q, Zhao W H, et al., Quantile regression and variable selection of partial linear single-index model, Annals of the Institute of Statistical Mathematics, 2015, 67:375-409.
[14] Jiang R, Qian W M, and Zhou Z G, Weighted composite quantile regression for single-index models, Journal of Multivariate Analysis, 2016, 148:34-48.
[15] Jiang R, Zhou Z G, and Qian W M, Two step composite quantile regression for single-index models, Computational Statistics and Data Analysis, 2013, 64:180-191.
[16] Ma S J and He X M, Inference for single-index quantile regression models with profile optimization, Annals of Statistics, 2016, 44:1234-1268.
[17] Jiang R and Qian W M, Quantile regression for single-index-coefficient regression models, Statistics&Probability Letters, 2016, 110:305-317.
[18] Zhao W H, Zhang R Q, Lü Y Z, et al., Quantile regression and variable selection of single-index coefficient model, Annals of the Institute of Statistical Mathematics, 2017, 69:761-789.
[19] Li H F, Liu Y, and Luo Y X, Double penalized quantile regression for the linear mixed effects model, Journal of Systems Science and Complexity, 2020, 33(6):2080-2102.
[20] Klein J P and Moeschberger M L, Survival Analysis:Techniques for Censored and Truncated Data, Springer, New York, 2003.
[21] Woodroofe W, Estimation a distribution function with truncated data, Annals of Statistics, 1985, 13:163-177.
[22] He S Y and Yang G L, Estimation of the truncation probability in the random truncation model,Annals of Statistics, 1998, 26:1011-1027.
[23] He S Y and Yang G L, Estimation of regression parameters with left truncated data, Journal of Statistical Planning and Inference, 2003, 117:99-122.
[24] Ould-Saïd E and Lemdani M, Asymptotic properties of a nonparametric regression function estimator with randomly truncated data, Annals of the Institute of Statistical Mathematics, 2006, 58:357-378.
[25] Stute W and Wang J L, The central limit theorem under random truncation, Bernoulli, 2008, 14:604-622.
[26] Zhou W H, A weighted quantile regression for randomly truncated data, Computational Statistics and Data Analysis, 2011, 55:554-566.
[27] Xu H X, Fan G L, Chen Z L, et al., Weighted quantile regression and testing for varying-coefficient models with randomly truncated data, AStA-Advances in Statistical Analysis, 2018, 102:565-588.
[28] Xu H X, Chen Z L, Wang J F, et al., Quantile regression and variable selection for partially linear model with randomly truncated data, Statistical Papers, 2019, 60:1137-1160.
[29] Lemdani M, Ould-Saïd E, and Poulin P, Asymptotic properties of a conditional quantile estimator with randomly truncated data, Journal of Multivariate Analysis, 2009, 100:546-559.
[30] Liang H Y and Liu A A, Kernel estimation of conditional density with truncated, censored and dependent data, Journal of Multivariate Analysis, 2013, 120:40-58.
[31] Liang H Y and Baek J I, Asymptotic normality of conditional density estimation with lefttruncated and dependent data, Statistical Papers, 2016, 57:1-20.
[32] Yao M, Wang J F, Lin L, et al., Variable selection and weighted composite quantile estimation of regression parameters with left-truncated data, Communications in Statistics-Theory and Methods, 2018, 47:4469-4482.
[33] Xue L G and Zhu L X, Empirical likelihood for single-index models, Journal of Multivariate Analysis, 2006, 97:1295-1312.
[34] Wang H and Leng C, Unified lasso estiation via least squares approximation, Journal of the American Statistical Association, 2007, 102:1039-1048.
[35] Zhu L X and Xue L G, Empirical likelihood confidence regions in a partially linear single-index model, Journal of the Royal Statistical Society Series B, 2006, 68:549-570.
[36] Yeh I C, Modeling of strength of high performance concrete using artificial neural networks, Cement and Concrete Research, 1998, 28:1797-1808.
[37] Kai B, Li R Z, and Zou H, New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models, Annals of Statistics, 2011, 39:305-332.
[38] Mack Y and Silverman B W, Weak and strong uniform consistency of kernel regression estimates, Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 1982, 61(3):405-415.
[39] Owen A B, Empirical likelihood ratio confidence regions, Annals of Statistics, 1990, 18:90-120.
[40] Knight K, Limiting distributions for l1 regression estimators under general conditions, Annals of Statistics, 1998, 26:755-770.
[41] Xia Y and Härdle W, Semi-parametric estimation of partially linear single-index models, Journal of Multivariate Analysis, 2006, 97(5):1162-1184.
[42] Geyer C J, On the asymptotics of constrained m-estimation, Annals of Statistics, 1994, 22:1993-2010.
[1] LIU Yue, ZOU Jiahui, ZHAO Shangwei, YANG Qinglong. Model Averaging Estimation for Varying-Coefficient Single-Index Models [J]. Journal of Systems Science and Complexity, 2022, 35(1): 264-282.
[2] XIAO Juxia, XIE Tianfa, ZHANG Zhongzhan. Estimation in Partially Observed Functional Linear Quantile Regression [J]. Journal of Systems Science and Complexity, 2022, 35(1): 313-341.
[3] FENG Yan · WANG Yijun · WANG Weiwei · CHEN Zhuo. Robust Estimation of Semiparametric Transformation Model for Panel Count Data [J]. Journal of Systems Science and Complexity, 2021, 34(6): 2334-2356.
[4] ZHANG Yanqing,TANG Niansheng. Bayesian Empirical Likelihood Estimation of Quantile Structural Equation Models [J]. Journal of Systems Science and Complexity, 2017, 30(1): 122-138.
[5] LU Yiqiang,ZHANG Riquan,HU Bin. The Adaptive LASSO Spline Estimation of Single-Index Model [J]. Journal of Systems Science and Complexity, 2016, 29(4): 1100-1111.
[6] FAN Yan,GAI Yujie,ZHU Lixing. Asymtotics of Dantzig Selector for a General Single-Index Model [J]. Journal of Systems Science and Complexity, 2016, 29(4): 1123-1144.
[7] ZHAO Weihua, ZHANG Riquan. Variable Selection of Varying Dispersion Student-t Regression Models [J]. Journal of Systems Science and Complexity, 2015, 28(4): 961-977.
[8] LI Xuejing,DU Jiang,LI Gaorong,FAN Mingzhi. VARIABLE SELECTION FOR COVARIATE ADJUSTED REGRESSION MODEL [J]. Journal of Systems Science and Complexity, 2014, 27(6): 1227-1246.
[9] Xiaohui LIU, Guofu WANG, Xuemei HU ,Bo LI. ZERO FINITE-ORDER SERIAL CORRELATION TEST IN A PARTIALLY LINEAR SINGLE-INDEX MODEL [J]. Journal of Systems Science and Complexity, 2012, 25(6): 1185-1201.
[10] Lin Cheng ZHAO;Xue Na WANG;Yao Hua WU. INFERENCE ON THE RANK OF THE GROWTH CURVE MODEL USING MODEL SELECTION METHOD [J]. Journal of Systems Science and Complexity, 2001, 14(2): 218-224.
Viewed
Full text


Abstract