Single-Index Quantile Regression with Left Truncated Data

XU Hongxia, FAN Guoliang, LI Jinchang

系统科学与复杂性(英文) ›› 2022, Vol. 35 ›› Issue (5) : 1963-1987.

PDF(636 KB)
中国科学院数学与系统科学研究院期刊网
JSSC
PDF(636 KB)
系统科学与复杂性(英文) ›› 2022, Vol. 35 ›› Issue (5) : 1963-1987. DOI: 10.1007/s11424-022-1118-4

Single-Index Quantile Regression with Left Truncated Data

    XU Hongxia1,2, FAN Guoliang1, LI Jinchang2
作者信息 +

Single-Index Quantile Regression with Left Truncated Data

    XU Hongxia1,2, FAN Guoliang1, LI Jinchang2
Author information +
文章历史 +

摘要

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.

Abstract

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.

关键词

Adaptive LASSO penalty / left truncated data / quantile regression / single-index model / variable selection

Key words

Adaptive LASSO penalty / left truncated data / quantile regression / single-index model / variable selection

引用本文

EndNote

Ris (Procite)

Bibtex

导出引用
XU Hongxia , FAN Guoliang , LI Jinchang. Single-Index Quantile Regression with Left Truncated Data. 系统科学与复杂性(英文), 2022, 35(5): 1963-1987 https://doi.org/10.1007/s11424-022-1118-4
XU Hongxia , FAN Guoliang , LI Jinchang. Single-Index Quantile Regression with Left Truncated Data. Journal of Systems Science and Complexity, 2022, 35(5): 1963-1987 https://doi.org/10.1007/s11424-022-1118-4

参考文献

[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.

基金

This research was supported by the National Social Science Foundation of China under Grant No.21BTJ038.
PDF(636 KB)

211

Accesses

0

Citation

Detail
段落导航
相关文章

/

〈 〉