Electronic Journal of Statistics

A difference based approach to the semiparametric partial linear model

Lie Wang, Lawrence D. Brown, and T. Tony Cai

Full-text: Open access


A commonly used semiparametric partial linear model is considered. We propose analyzing this model using a difference based approach. The procedure estimates the linear component based on the differences of the observations and then estimates the nonparametric component by either a kernel or a wavelet thresholding method using the residuals of the linear fit. It is shown that both the estimator of the linear component and the estimator of the nonparametric component asymptotically perform as well as if the other component were known. The estimator of the linear component is asymptotically efficient and the estimator of the nonparametric component is asymptotically rate optimal. A test for linear combinations of the regression coefficients of the linear component is also developed. Both the estimation and the testing procedures are easily implementable. Numerical performance of the procedure is studied using both simulated and real data. In particular, we demonstrate our method in an analysis of an attitude data set.

Article information

Electron. J. Statist., Volume 5 (2011), 619-641.

First available in Project Euclid: 27 June 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Asymptotic efficiency difference-based method kernel method wavelet thresholding method partial linear model semiparametric model


Wang, Lie; Brown, Lawrence D.; Cai, T. Tony. A difference based approach to the semiparametric partial linear model. Electron. J. Statist. 5 (2011), 619--641. doi:10.1214/11-EJS621. https://projecteuclid.org/euclid.ejs/1309180102

Export citation


  • [1] Bickel, P. J., Klassen, C. A. J., Ritov, Y. and Wellner, J. A. (1993)., Efficient and Adaptive Estimation for Semiparametric Models. The John Hopkins University Press, Baltimore.
  • [2] Brown, L.D. and Low M.G. (1996). A Constrained Risk Inequality with Applications to Nonparametric Functional Estimations., Ann. Statist. 24, 2524-2535.
  • [3] Cai, T. and Brown, L.D. (1998). Wavelet shrinkage for nonequispaced samples., Ann. Statist. 26, 1783-1799.
  • [4] Cai, T., Levine, M. and Wang, L. (2009). Variance function estimation in multivariate nonparametric regression., Journal of Multivariate Analysis, 100, 126-136.
  • [5] Cai, T. and Wang, L. (2008). Adaptive variance function estimation in heteroscedastic nonparametric regression., Ann. Statist. 36, 2025-C2054.
  • [6] 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.
  • [7] Chang, X. and Qu, L. (2004). Wavelet estimation of partially linear models., Comput. Statist. Data Anal. 47, 31-48.
  • [8] Chatterjee, S. and Price, B. (1977)., Regression Analysis by Example. New York: Wiley.
  • [9] Chen, H. and Shiau, J. H. (1991). A two-stage spline smoothing method for partially linear models., J. Statist. Plann. Inference 27, 187-201.
  • [10] Cuzick, J. (1992). Semiparametric additive regression., J. Roy. Statist. Soc. Ser. B 54, 831-843.
  • [11] Daubechies, I. (1992)., Ten Lectures on Wavelets. SIAM, Philadelphia.
  • [12] Donoho, D.L. and Johnstone, I.M. (1994). Ideal spatial adaptation via wavelet shrinkage., Biometrika 81, 425-55.
  • [13] Engle, R. F., Granger, C. W. J., Rice, J. and Weiss, A. (1986). Nonparametric estimates of the relation between weather and electricity sales., J. Amer. Statist. Assoc. 81, 310-320.
  • [14] Fan, J. and Huang, L. (2001). Goodness-of-Fit Tests for Parametric Regression Models., J. Amer. Statist. Assoc. 96, 640-652.
  • [15] Fan, J. and Zhang, J. (2004). Sieve empirical likelihood ratio tests for nonparametric functions., Ann. Statist. 32, 1858-1907.
  • [16] Fan, J., Zhang, C. and Zhang, J. (2001). Generalized likelihood ratio statistics and Wilks phenomenon., Ann. Statist. 29, 153-193.
  • [17] Gannaz, I. (2007). Robust estimation and wavelet thresholding in partially linear models., Statist. Comput. 17, 293-310.
  • [18] Hall, P., Kay, J. and Titterington, D. M. (1990). Asymptotically optimal difference-based estimation of variance in nonparametric regression., Biometrika 77, 521-528.
  • [19] Hamilton, S. and Truong, Y. (1997). Local linear estimation in partly linear models., J. Multivariate Anal. 60, 1-C19.
  • [20] Härdle, W. (1991)., Smoothing Techniques. Berlin: Springer-Verlag.
  • [21] Horowitz, J. and Spokoiny, V. (2001). An adaptive rate-optimal test of a parametric mean-regression model against a nonparametric alternative., Econometrica 69, 599-631.
  • [22] Lam, C. and Fan, J. (2007). Profile-Kernel Likelihood Inference With Diverging Number of Parameters., Ann. Statist. , to appear.
  • [23] Müller, M. (2001). Estimation and testing in generalized partial linear models – A comparative study., J. Statist. Comput. 11, 299-309.
  • [24] Munk, A., Bissantz, N., Wagner, T. and Freitag, G. (2005). On difference based variance estimation in nonparametric regression when the covariate is high dimensional., J. Roy. Statist. Soc. B 67, 19-41.
  • [25] Rice, J. A. (1984). Bandwidth choice for nonparametric regression., Ann. Statist. 12, 1215-1230.
  • [26] Ritov, Y. and Bickel, P. J. (1990). Achieving information bounds in semi and non parametric models., Ann. Statist. 18, 925-938.
  • [27] Robinson, P. M. (1988). Root-N consistent semiparametric regression., Econometrica. 56, 931-954.
  • [28] Scott, D. W. (1992)., Multivariate Density Estimation. New York: Wiley.
  • [29] Severini, T. A. and Wong, W. H. (1992). Generalized profile likelihood and conditional parametric models., Ann. Statist. 20, 1768-1802.
  • [30] Speckman, P. (1988). Kernel smoothing in partial linear models., Jour. Roy. Statist. Soc.Ser. B 50, 413-436.
  • [31] Schimek, M. (2000). Estimation and inference in partially linear models with smoothing splines., J. Statist. Plann. Inference 91, 525-C540.
  • [32] Strang, G. (1989). Wavelet and dilation equations: A brief introduction., SIAM Rev. 31, 614-627.
  • [33] Wahba, G. (1984). Cross validated spline methods for the estimation of multivariate functions from data on functionals., Statistics: An Appraisal. Proceedings 50th Anniversary Conference Iowa State Statistical Laboratory (H. A. David, ed.). Iowa State Univ. Press.
  • [34] Yatchew, A. (1997). An elementary estimator of the partial linear model., Economics Letters 57, 135-143.
  • [35] Yatchew, A. (2003)., Semiparametric regression for the applied econometrician. New York: Cambridge University press.