The Annals of Statistics

Inference for single-index quantile regression models with profile optimization

Shujie Ma and Xuming He

Full-text: Open access


Single index models offer greater flexibility in data analysis than linear models but retain some of the desirable properties such as the interpretability of the coefficients. We consider a pseudo-profile likelihood approach to estimation and testing for single-index quantile regression models. We establish the asymptotic normality of the index coefficient estimator as well as the optimal convergence rate of the nonparametric function estimation. Moreover, we propose a score test for the index coefficient based on the gradient of the pseudo-profile likelihood, and employ a penalized procedure to perform consistent model selection and model estimation simultaneously. We also use Monte Carlo studies to support our asymptotic results, and use an empirical example to illustrate the proposed method.

Article information

Ann. Statist., Volume 44, Number 3 (2016), 1234-1268.

Received: April 2015
Revised: October 2015
First available in Project Euclid: 11 April 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G08: Nonparametric regression
Secondary: 62G20: Asymptotic properties

Quantile regression single-index profile principle polynomial spline model selection score test


Ma, Shujie; He, Xuming. Inference for single-index quantile regression models with profile optimization. Ann. Statist. 44 (2016), no. 3, 1234--1268. doi:10.1214/15-AOS1404.

Export citation


  • Andrews, D. W. K. (1987). Consistency in nonlinear econometric models: A generic uniform law of large numbers. Econometrica 55 1465–1471.
  • Belloni, A. and Chernozhukov, V. (2011). $\ell_{1}$-penalized quantile regression in high-dimensional sparse models. Ann. Statist. 39 82–130.
  • Bhattacharya, P. K. and Gangopadhyay, A. K. (1990). Kernel and nearest-neighbor estimation of a conditional quantile. Ann. Statist. 18 1400–1415.
  • Bosq, D. (1998). Nonparametric Statistics for Stochastic Processes, 2nd ed. Springer, New York.
  • Carroll, R. J., Fan, J., Gijbels, I. and Wand, M. P. (1997). Generalized partially linear single-index models. J. Amer. Statist. Assoc. 92 477–489.
  • Chaudhuri, P. (1991). Nonparametric estimates of regression quantiles and their local Bahadur representation. Ann. Statist. 19 760–777.
  • Chaudhuri, P., Doksum, K. and Samarov, A. (1997). On average derivative quantile regression. Ann. Statist. 25 715–744.
  • Cui, X., Härdle, W. K. and Zhu, L. (2011). The EFM approach for single-index models. Ann. Statist. 39 1658–1688.
  • de Boor, C. (2001). A Practical Guide to Splines, Revised ed. Applied Mathematical Sciences 27. Springer, New York.
  • De Gooijer, J. G. and Zerom, D. (2003). On additive conditional quantiles with high-dimensional covariates. J. Amer. Statist. Assoc. 98 135–146.
  • Fan, J., Hu, T. C. and Truong, Y. K. (1994). Robust non-parametric function estimation. Scand. J. Stat. 21 433–446.
  • Fan, J. and Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. J. Amer. Statist. Assoc. 96 1348–1360.
  • Geyer, C. J. (1994). On the asymptotics of constrained $M$-estimation. Ann. Statist. 22 1993–2010.
  • Hammer, S. M., Squires, K. E., Hughes, M. D., Grimes, J. M., Demeter, L. M., Currier, J. S., Eron, J. J., Feinberg, J. E., Balfour, H. H., Deyton, L. R., Chodakewitz, J. A. and Fischl, M. A. (1997). A controlled trial of two nucleoside analogues plus indinavir in persons with human immunodeficiency virus infection and CD4 cell counts of 200 per cubic millimeter or less. AIDS clinical trials group 320 study team. N. Engl. J. Med. 337 725–733.
  • He, X., Ng, P. and Portnoy, S. (1998). Bivariate quantile smoothing splines. J. R. Stat. Soc. Ser. B. Stat. Methodol. 60 537–550.
  • He, X. and Shi, P. D. (1994). Convergence rate of $B$-spline estimators of nonparametric conditional quantile functions. J. Nonparametr. Stat. 3 299–308.
  • He, X. and Shi, P. (1996). Bivariate tensor-product $B$-splines in a partly linear model. J. Multivariate Anal. 58 162–181.
  • Horowitz, J. L. and Lee, S. (2005). Nonparametric estimation of an additive quantile regression model. J. Amer. Statist. Assoc. 100 1238–1249.
  • Ichimura, H. (1993). Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. J. Econometrics 58 71–120.
  • Kai, B., Li, R. and Zou, H. (2011). New efficient estimation and variable selection methods for semiparametric varying-coefficient partially linear models. Ann. Statist. 39 305–332.
  • Kim, M.-O. (2007). Quantile regression with varying coefficients. Ann. Statist. 35 92–108.
  • Knight, K. and Fu, W. (2000). Asymptotics for lasso-type estimators. Ann. Statist. 28 1356–1378.
  • Kocherginsky, M., He, X. and Mu, Y. (2005). Practical confidence intervals for regression quantiles. J. Comput. Graph. Statist. 14 41–55.
  • Koenker, R. (2005). Quantile Regression. Cambridge Univ. Press, Cambridge.
  • Koenker, R. (2011). Additive models for quantile regression: Model selection and confidence bandaids. Braz. J. Probab. Stat. 25 239–262.
  • Koenker, R. and Bassett, G. Jr. (1978). Regression quantiles. Econometrica 46 33–50.
  • Koenker, R., Ng, P. and Portnoy, S. (1994). Quantile smoothing splines. Biometrika 81 673–680.
  • Kong, E. and Xia, Y. (2012). A single-index quantile regression model and its estimation. Econometric Theory 28 730–768.
  • Liang, H., Liu, X., Li, R. and Tsai, C.-L. (2010). Estimation and testing for partially linear single-index models. Ann. Statist. 38 3811–3836.
  • Ma, S. and He, X. (2015). Supplement to “Inference for single-index quantile regression models with profile optimization.” DOI:10.1214/15-AOS1404SUPP.
  • Murphy, S. A. and van der Vaart, A. W. (2000). On profile likelihood. J. Amer. Statist. Assoc. 95 449–485.
  • Pollard, D. (1991). Asymptotics for least absolute deviation regression estimators. Econometric Theory 7 186–199.
  • Portnoy, S. (1997). Local asymptotics for quantile smoothing splines. Ann. Statist. 25 414–434.
  • Severini, T. A. and Staniswalis, J. G. (1994). Quasi-likelihood estimation in semiparametric models. J. Amer. Statist. Assoc. 89 501–511.
  • Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B 58 267–288.
  • Wang, H., Li, R. and Tsai, C.-L. (2007). Tuning parameter selectors for the smoothly clipped absolute deviation method. Biometrika 94 553–568.
  • Wang, J. and Yang, L. (2009). Polynomial spline confidence bands for regression curves. Statist. Sinica 19 325–342.
  • Wang, H. J., Zhu, Z. and Zhou, J. (2009). Quantile regression in partially linear varying coefficient models. Ann. Statist. 37 3841–3866.
  • Wu, Y. and Liu, Y. (2009). Variable selection in quantile regression. Statist. Sinica 19 801–817.
  • Wu, T. Z., Yu, K. and Yu, Y. (2010). Single-index quantile regression. J. Multivariate Anal. 101 1607–1621.
  • Xia, Y. and Härdle, W. (2006). Semi-parametric estimation of partially linear single-index models. J. Multivariate Anal. 97 1162–1184.
  • Xia, Y., Tong, H. and Li, W. K. (1999). On extended partially linear single-index models. Biometrika 86 831–842.
  • Yu, K. and Jones, M. C. (1998). Local linear quantile regression. J. Amer. Statist. Assoc. 93 228–237.
  • Yu, Y. and Ruppert, D. (2002). Penalized spline estimation for partially linear single-index models. J. Amer. Statist. Assoc. 97 1042–1054.
  • Zhu, L., Huang, M. and Li, R. (2012). Semiparametric quantile regression with high-dimensional covariates. Statist. Sinica 22 1379–1401.
  • Zou, H. and Li, R. (2008). One-step sparse estimates in nonconcave penalized likelihood models. Ann. Statist. 36 1509–1533.
  • Zou, Q. and Zhu, Z. (2014). M-estimators for single-index model using B-spline. Metrika 77 225–246.

Supplemental materials

  • Supplement to “Inference for single-index quantile regression models with profile optimization”. We present several lemmas that will be used in the proof of the main theorems, and the proof of equation (A.18). Then we present Example 2 for Case 1 and additional simulation results for Case 2 in the simulation studies.