Electronic Journal of Statistics

Oracally efficient estimation for single-index link function with simultaneous confidence band

Lijie Gu and Lijian Yang

Full-text: Open access

Abstract

Over the last twenty-five years, various $\sqrt{n}$-consistent estimators have been devised for the coefficient vector in the popular semiparametric single-index model. In this paper, we prove under general assumptions that the kernel estimator of the link function by a univariate regression on the index variable is oracally efficient, namely, the estimator with the true single-index coefficient vector is asymptotically indistinguishable from that with any $\sqrt{n}$-consistent coefficient vector estimator. As a mathematical byproduct of the oracle efficiency, a simultaneous confidence band is constructed for the link function based on the oracally efficient kernel estimator. Simulation experiments corroborate the theoretical results. The proposed simultaneous confidence band is applied to analyze and test hypothesis about the Boston housing data.

Article information

Source
Electron. J. Statist. Volume 9, Number 1 (2015), 1540-1561.

Dates
Received: January 2015
First available in Project Euclid: 23 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1437658702

Digital Object Identifier
doi:10.1214/15-EJS1051

Mathematical Reviews number (MathSciNet)
MR3376116

Zentralblatt MATH identifier
1327.62254

Subjects
Primary: 62G08: Nonparametric regression
Secondary: 62G15: Tolerance and confidence regions

Keywords
Confidence band kernel link function oracle efficiency single-index

Citation

Gu, Lijie; Yang, Lijian. Oracally efficient estimation for single-index link function with simultaneous confidence band. Electron. J. Statist. 9 (2015), no. 1, 1540--1561. doi:10.1214/15-EJS1051. https://projecteuclid.org/euclid.ejs/1437658702


Export citation

References

  • [1] Bosq, D. (1998)., Nonparametric Statistics for Stochastic Processes. Springer-Verlag, New York.
  • [2] Breiman, L. and Friedman, J. H. (1985). Estimating optimal transformations of multiple regression and correlation., J. Amer. Statist. Assoc. 80 580–619.
  • [3] Carroll, R., Fan, J., Gijbels, I. and Wand, M. P. (1997). Generalized partially linear single-index models., J. Amer. Statist. Assoc. 92 477–489.
  • [4] Chen, H. (1991). Estimation of a projection-pursuit type regression model., Ann. Statist. 19 142–157.
  • [5] Fan, J. and Gijbels, I. (1996)., Local Polynomial Modelling and Its Applications. Chapman and Hall, London.
  • [6] Friedman, J. H. and Stuetzle, W. (1981). Projection pursuit regression., J. Amer. Statist. Assoc. 76 817–823.
  • [7] Gordon, J. J. (1982). Probabilities of maximal deviations for nonparametric regression function estimates., J. Multivariate Anal. 12 402–414.
  • [8] Harrison, D. and Rubinfeld, D. L. (1978). Hedonic housing prices and the demand for cleaning air., J. Environ. Econ. Manag. 5 81–102.
  • [9] Härdle, W. (1989). Asymptotic maximal deviation of M-smoothers., J. Multivariate Anal. 29 163–179.
  • [10] Härdle, W., Liang, H. and Gao, J. (2000)., Partially Linear Models. Physica-Verlag, Heidelberg.
  • [11] Härdle, W., Hall, P. and Ichimura, H. (1993). Optimal smoothing in single-index models., Ann. Statist. 21 157–178.
  • [12] Härdle, W. and Stoker, T. M. (1989). Investigating smooth multiple regression by the method of average derivatives., J. Amer. Statist. Assoc. 84 986–995.
  • [13] Härdle, W., Müller, M., Sperlich, S. and Werwatz, A. (2004)., Nonparametric and Semiparametric Models. Springer-Verlag, New York.
  • [14] Hall, P. (1989). On projection pursuit regression., Ann. Statist. 17 573–588.
  • [15] Hall, P. and Van Keilegom, I. (2003). Using difference-based methods for inference in nonparametric regression with time series errors., J. R. Stat. Soc. Ser. B Stat. Methodol. 65 443–456.
  • [16] Hastie, T. J. and Tibshirani, R. J. (1990)., Generalized Additive Models. Chapman and Hall, London.
  • [17] Horowitz, J. L. and Härdle, W. (1996). Direct semiparametric estimation of single-index models with discrete covariates., J. Amer. Statist. Assoc. 91 1632–1640.
  • [18] Huang, J. and Yang, L. (2004). Identification of nonlinear additive autoregressive models., J. R. Stat. Soc. Ser. B Stat. Methodol. 66 463–477.
  • [19] Huber, P. J. (1985). Projection pursuit (with discussion)., Ann. Statist. 13 435–525.
  • [20] Ichimura, H. (1993). Semiparametric least squares (SLS) and weighted SLS estimation of single-index models., J. Econometrics 58 71–120.
  • [21] Jiang, R., Zhou, Z. G., Qian, W. M. and Chen, Y. (2013). Two step composite quantile regression for single-index models., Comput. Statist. Data Anal. 64 180–191.
  • [22] Klein, R. W. and Spady. R. H. (1993). An efficient semiparametric estimator for binary response models., Econometrica 61 387–421.
  • [23] Krivobokova, T., Kneib, T. and Claeskens, G. (2010). Simultaneous confidence bands for penalized spline estimators., J. Amer. Statist. Assoc. 105 852–863.
  • [24] Linton, O. B. (1997). Efficient estimation of additive nonparametric regression models., Biometrika 84 469–473.
  • [25] Liu, R. and Yang, L. (2010). Spline-backfitted kernel smoothing of additive coefficient model., Econometric Theory 26 29–59.
  • [26] Liu, R., Yang, L. and Härdle, W. (2013). Oracally efficient two-step estimation of generalized additive model., J. Amer. Statist. Assoc. 108 619–631.
  • [27] Mammen, E., Linton, O. and Nielsen, J. (1999). The existence and asymptotic properties of a backfitting projection algorithm under weak conditions., Ann. Statist. 27 1443–1490.
  • [28] Ma, S. and Yang, L. (2011). Spline-backfitted kernel smoothing of partially linear additive model., J. Statist. Plann. Inference 141 204–219.
  • [29] Ma, S., Yang, L. and Carroll, R. (2012). A simultaneous confidence band for sparse longitudinal regression., Statist. Sinica. 22 95–122.
  • [30] Ma, S., Liang, H. and Tsai, C.L. (2014a). Partially linear single index models for repeated measurements., J. Multivariate Anal. 130 354–375.
  • [31] Ma, S., Zhang, J., Sun, Z. and Liang, H. (2014b). Integrated conditional moment test for partially linear single index models incorporating dimension-reduction., Elect. J. Statist. 8 523–542.
  • [32] Opsomer, J. D. and Ruppert, D. (1998). A fully automated bandwidth selection method for fitting additive models., J. Amer. Statist. Assoc. 93 605–619.
  • [33] Powell, J. L., Stock, J. H. and Stoker, T. M. (1989). Semiparametric estimation of index coefficients., Econometrica. 57 1403–1430.
  • [34] Silverman, B. W. (1986)., Density Estimation. Chapman and Hall, London.
  • [35] Sperlich, S., Härdle, W. and Spokoiny, V. (1997). Semiparametric single index versus fixed link function modelling., Ann. Statist. 25 212–243.
  • [36] Wang, J., Liu, R., Cheng, F. and Yang, L. (2014). Oracally efficient estimation of autoregressive error distribution with simultaneous confidence band., Ann. Statist. 42 654–668.
  • [37] Wang, J. and Yang, L. (2009a). Efficient and fast spline-backfitted kernel smoothing of additive models., Ann. Inst. Stat. Math. 61 663–690.
  • [38] Wang, L. and Yang, L. (2007). Spline-backfitted kernel smoothing of nonlinear additive autoregression model., Ann. Statist. 35 2474–2503.
  • [39] Wang, L. and Yang, L. (2009b). Spline estimation of single index model., Statist. Sinica. 19 765–783.
  • [40] Wu, T. Z., Yu, K. and Yu, Y. (2010). Single-index quantile regression., J. Multivariate Anal. 101 1607–1621.
  • [41] Xia, Y. (1998). Bias-corrected confidence bands in nonparametric regression., J. R. Stat. Soc. Ser. B Stat. Methodol. 60 797–811.
  • [42] Xia, Y. and Li, W. K. (1999). On single-index coefficient regression models., J. Amer. Statist. Assoc. 94 1275–1285.
  • [43] Xia, Y., Li, W. K., Tong, H. and Zhang, D. (2004). A goodness-of-fit test for single-index models., Statist. Sinica. 14 1–39.
  • [44] Xue, L. and Yang, L. (2006a). Estimation of semiparametric additive coefficient model., J. Statist. Plann. Inference 136 2506–2534.
  • [45] Xue, L. and Yang, L. (2006b). Additive coefficient modeling via polynomial spline., Statist. Sinica. 16 1423–1446.
  • [46] Yu, K. and Lu, Z. (2004). Local linear additive quantile regression., Scand. J. Statist. 31 333–346.
  • [47] Zheng, S., Yang, L. and Härdle, W. (2014). A smooth simultaneous confidence corridor for the mean of sparse functional data., J. Amer. Statist. Assoc. 109 661–673.
  • [48] Zhu, H., Li, R. and Kong, L. (2012). Multivariate varying coefficient model for functional responses., Ann. Statist. 40 2634–2666.