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Estimation of general semi-parametric quantile regression. (English) Zbl 1428.62154
Summary: Quantile regression introduced by R. Koenker and G. Bassett jun. [Econometrica 46, 33–50 (1978; Zbl 0373.62038)] produces a comprehensive picture of a response variable on predictors. In this paper, we propose a general semi-parametric model of which part of predictors are presented with a single-index, to model the relationship of conditional quantiles of the response on predictors. Special cases are single-index models, partially linear single-index models and varying coefficient single-index models. We propose the qOPG, a quantile regression version of outer-product gradient estimation method (OPG, [Y. Xia et al., J. R. Stat. Soc., Ser. B, Stat. Methodol. 64, No. 3, 363–410 (2002; Zbl 1091.62028)]) to estimate the single-index. Large-sample properties, simulation results and a real-data analysis are provided to examine the performance of the qOPG.

MSC:
62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
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[1] Cai, Z.; Xu, X., Nonparametric quantile estimations for dynamic smooth coefficient models, Journal of the American Statistical Association, 104, 371-383, (2009) · Zbl 1375.62003
[2] Chaudhuri, P., Nonparametric estimates of regression quantiles and their local bahadur representation, Annals of Statistics, 19, 760-777, (1991) · Zbl 0728.62042
[3] Chaudhuri, P.; Doksum, K.; Samarov, A., On average derivative quantile regression, Annals of Statistics, 25, 715-744, (1997) · Zbl 0898.62082
[4] Fan, J., Design-adaptive nonparametric regression, Journal of the American Statistical Association, 87, 998-1004, (1992) · Zbl 0850.62354
[5] Fan, J., Local linear regression smoothers and their minimax efficiency, Annals of Statistics, 21, 196-216, (1993) · Zbl 0773.62029
[6] Gooijer, J. G.; Zerom, D., On additive conditional quantiles with high-dimensional covariates, Journal of the American Statistical Association, 98, 135-146, (2003) · Zbl 1047.62027
[7] He, X.; Shi, P., Bivariate tensor-product B-splines in a partly linear model, Journal of Multivariate Analysis, 58, 162-181, (1996) · Zbl 0865.62027
[8] Honda, T., Quantile regression in varying coefficient models, Journal of Statistical Planning and Inference, 121, 113-125, (2004) · Zbl 1038.62041
[9] Horowitz, J. L.; Lee, S., Nonparametric estimation of an additive quantile regression model, Journal of the American Statistical Association, 100, 1238-1249, (2005) · Zbl 1117.62355
[10] Hjort, N.L., Pollard, D., 1993. Asymptotics for minimisers of convex process. Technical Report, Yale University.
[11] Kai, B.; Li, R.; Zou, H., New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models, Annals of Statistics, 39, 305-332, (2011) · Zbl 1209.62074
[12] Kim, M., Quantile regression with varying coefficients, Annals of Statistics, 35, 92-108, (2007) · Zbl 1114.62051
[13] Knight, K., Limiting distributions for L1 regression estimates under general conditions, Annals of Statistics, 26, 755-770, (1998) · Zbl 0929.62021
[14] Koenker, R.; Bassett, G., Regression quantiles, Econometrica, 46, 33-50, (1978) · Zbl 0373.62038
[15] Koenker, R.; Ng, P.; Portnoy, S., Quantile smoothing splines, Biometrika, 81, 673-680, (1994) · Zbl 0810.62040
[16] Koenker, R., Quantile regression, (2005), Cambridge University Press New York · Zbl 1111.62037
[17] Kong, E., Xia, Y., 2011. A single-Index quantile regression model and its estimation, Econometric Theory, preprint. · Zbl 1419.62090
[18] Lee, S., Efficient semiparametric estimation of a partially linear quantile regression model, Econometric Theory, 19, 1-31, (2003) · Zbl 1031.62034
[19] Serfling, R. J., Approximation theorems of mathematical statistics, (1980), Wiley New York · Zbl 0538.62002
[20] Stone, C. J., Consistent nonparametric regression, with discussion, Annals of Statistics, 5, 595-645, (1977) · Zbl 0366.62051
[21] Wang, H. J.; Zhu, Z. Y.; Zhou, J. H., Quantile regression in partially linear varying coefficient models, Annals of Statistics, 37, 3841-3866, (2009) · Zbl 1191.62077
[22] Wu, T. Z.; Yu, K.; Yu, Y., Single-index quantile regression, Journal of Multivariate Analysis, 101, 1607-1621, (2010) · Zbl 1189.62075
[23] Xia, Y.; Tong, H.; Li, W. K.; Zhu, L., An adaptive estimation of dimension reduction space, Journal of the Royal Statistical Society Series B, 64, 363-410, (2002) · Zbl 1091.62028
[24] Xia, Y.; Hardle, W., Semi-parametric estimation of partially linear single-index models, Journal of Multivariate Analysis, 97, 1162-1184, (2006) · Zbl 1089.62050
[25] Yu, K.; Jones, M. C., A comparison of local constant and local linear regression quantile estimation, Computational Statistics and Data Analysis, 25, 159-166, (1997) · Zbl 0900.62182
[26] Yu, K.; Jones, M. C., Local linear quantile regression, Journal of the American Statistical Association, 93, 228-237, (1998) · Zbl 0906.62038
[27] Zhu, L. P.; Huang, M.; Li, R. Z., Semiparametric quantile regression with high-dimensional covariates, Statistica Sinica, 22, 1379-1401, (2012) · Zbl 1253.62032
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