Skip to main content
SpringerLink
Log in
Book cover

Exploring Research Frontiers in Contemporary Statistics and Econometrics pp 229–261Cite as

Optimal Smoothing for a Computationally and Statistically Efficient Single Index Estimator

  • Chapter
  • First Online:

  • 955 Accesses

Abstract

In semiparametric models it is a common approach to under-smooth the nonparametric functions in order that estimators of the finite dimensional parameters can achieve root-n consistency. The requirement of under-smoothing may result, as we show, from inefficient estimation methods or technical difficulties. Xia et al. (J. Roy. Statist. Soc. B. 64:363–410, 2002) proposed an adaptive method for the multiple-index model, called MAVE. In this chapter we further refine the estimation method. Under some conditions, our estimator of the single-index is asymptotically normal and most efficient in the semi-parametric sense. Moreover, we derive higher-order expansions for our estimator and use them to define an optimal bandwidth for the purposes of index estimation. As a result we obtain a practically more relevant method and we show its superior performance in a variety of applications.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  • Bickel, P., Klaassen, A.J., Ritov, Y., & Wellner, J.A. (1993). Efficient and adaptive inference in semiparametric models. Baltimore: Johns Hopkins University Press.

    Google Scholar 

  • Carroll, R.J., Fan. J., Gijbels, I., & Wand, M.P. (1997). Generalized partially linear single-index models. Journal of American Statistical Association, 92, 477–489.

    Article  MathSciNet  MATH  Google Scholar 

  • de la Peña, V.H. (1999) A general class of exponential inequalities for martingales and ratios. The Annals of Probability, 27, 537–564.

    Article  MathSciNet  MATH  Google Scholar 

  • Delecroix, M., Härdle, W., & Hristache, M. (2003). Efficient estimation in conditional single-index regression. Journal of Multivariate Analysis, 86, 213–226.

    Article  MathSciNet  MATH  Google Scholar 

  • Delecroix, M., Hristache, M., & Patilea, V. (2006). On semiparametric M-estimation in single-index regression. Journal of Statistical Planning and Inference, 136, 730–769.

    Article  MathSciNet  MATH  Google Scholar 

  • Fan, J., & Gijbels, I. (1996). Local polynomial modeling and its applications. London: Chapman & Hall.

    Google Scholar 

  • Fan, J., & Yao, Q. (2003). Nonlinear time series: nonparametric and parametric methods. New York: Springer.

    Book  MATH  Google Scholar 

  • Friedman, J.H. (1984). SMART User’s Guide. Laboratory for Computational Statistics, Stanford University Technical Report No. 1.

    Google Scholar 

  • Hall, P. (1989). On projection pursuit regression. Annals of Statistics, 17, 573–588.

    Article  MathSciNet  MATH  Google Scholar 

  • Härdle, W., Hall, P., & Ichimura, H. (1993). Optimal smoothing in single-index models. Annals of Statistics, 21, 157–178.

    Article  MathSciNet  MATH  Google Scholar 

  • Härdle, W., & Stoker, T.M. (1989) Investigating smooth multiple regression by method of average derivatives. Journal of American Statistical Association, 84 986–995.

    MATH  Google Scholar 

  • Härdle, W., & Tsybakov, A.B. (1993). How sensitive are average derivatives? Journal of Econometrics, 58, 31–48.

    Article  MathSciNet  MATH  Google Scholar 

  • Horowitz, J.L. & Härdle, W. (1996) Direct semiparametric estimation of single-index models with discrete covariates. Journal of American Statistical Association, 91, 1632–1640.

    Article  MATH  Google Scholar 

  • Hristache, M., Juditsky, A., & Spokoiny, V. (2001) Direct estimation of the single-index coefficients in single-index models. Annals of Statistics29, 1–32.

    Google Scholar 

  • Hristache, M., Juditsky, A., & Spokoiny, V. (2002). Direct estimation of the index coefficient in a single-index model. Annals of Statistics, 29, 593–623.

    Google Scholar 

  • Ichimura, H. (1993). Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. Journal of Econometrics, 58, 71–120.

    Article  MathSciNet  MATH  Google Scholar 

  • Li, K.C. (1991). Sliced inverse regression for dimension reduction (with discussion). Journal of American Statistical Association, 86, 316–342.

    Article  MATH  Google Scholar 

  • Li, K.C. (1992). On principal Hessian directions for data visualization and dimension reduction: another application of Stein’s lemma. Journal of American Statistical Association, 87, 1025–1039.

    Article  MATH  Google Scholar 

  • Linton, O. (1995) Second order approximation in the partially linear regression model. Econometrica, 63, 1079–1112.

    Article  MathSciNet  MATH  Google Scholar 

  • Nishiyama, Y., & Robinson, P.M. (2000). Edgeworth expansions for semiparametric average derivatives. Econometrica, 68, 931–980.

    Article  MATH  Google Scholar 

  • Nishiyama, Y., & Robinson, P.M. (2005). The bootstrap and the Edgeworth correction for semiparametric average derivatives. Econometrica, 73, 903–948.

    Article  MathSciNet  MATH  Google Scholar 

  • Powell, J.L., Stock, J.H., & Stoker, T.M. (1989). Semiparametric estimation of index coefficients. Econometrica, 57, 1403–1430.

    Article  MathSciNet  MATH  Google Scholar 

  • Powell, J.L., & Stoker, T.M. (1996). Optimal bandwidth choice for density weighted averages. Journal of Econometrics, 755, 291–316.

    Article  MathSciNet  Google Scholar 

  • Robinson, P. (1988). Root-N consistent semi-parametric regression. Econometrica, 156, 931–954.

    Article  Google Scholar 

  • Ruppert, D., Sheather, J., & Wand, P.M. (1995). An effective bandwidth selector for local least squares regression. Journal of American Statistical Association, 90, 1257–1270.

    Article  MathSciNet  MATH  Google Scholar 

  • Schott, J.R. (1997) Matrix analysis for statistics. New York: Wiley.

    MATH  Google Scholar 

  • Weisberg, S., & Welsh, A.H. (1994). Estimating the missing link functions, Annals of Statistics, 22, 1674–1700.

    Article  MathSciNet  MATH  Google Scholar 

  • Xia, Y., Tong, H., Li, W.K., & Zhu, L. (2002). An adaptive estimation of dimension reduction space (with discussion). Journal of the Royal Statistical Society Series B, 64, 363–410.

    MathSciNet  MATH  Google Scholar 

  • Xia, Y. (2006). Asymptotic distributions for two estimators of the single-index model. Econometric Theory, 22, 1112–1137.

    Article  MathSciNet  MATH  Google Scholar 

  • Xia, Y., & Li, W.K. (1999) On single-index coefficient regression models. Journal of American Statistical Association, 94, 1275–1285.

    Article  MathSciNet  MATH  Google Scholar 

  • Yin, X., & Cook, R.D. (2005). Direction estimation in single-index regressions. Biometrika, 92, 371–384.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first author is most grateful to Professor V. Spokoiny for helpful discussions and NUS RMI for support. The second author thanks the Deutsche Forschungsgemeinschaft SFB 649 “Ökonomisches Risiko” for financial support. The third author thanks the ESRC for financial support.

Author information

Authors and Affiliations

  1. Department of Statistics and Applied Probability and Risk Management Institute, National University of Singapore, Singapore, Singapore

    Yingcun Xia

  2. C.A.S.E. Centre for Applied Statistics and Economics, School of Business and Economics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Wolfgang Karl Härdle

  3. Austin Robinson Building, Sidgwick Avenue, Cambridge, CB3 9DD, UK

    Oliver Linton (Faculty of Economics)

Authors
  1. Yingcun Xia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wolfgang Karl Härdle
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Oliver Linton
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yingcun Xia .

Editor information

Editors and Affiliations

  1. , Institut de statistique,, Université catholique de Louvain, Voie du Roman Pays 20, Louvain-la-Neuve, 1348, Belgium

    Ingrid Van Keilegom

  2. Dept. Economics, Clemson University, Sirrene Hall 222, Clemson, 29634-1309, South Carolina, USA

    Paul W. Wilson

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Xia, Y., Härdle, W.K., Linton, O. (2011). Optimal Smoothing for a Computationally and Statistically Efficient Single Index Estimator. In: Van Keilegom, I., Wilson, P. (eds) Exploring Research Frontiers in Contemporary Statistics and Econometrics. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-2349-3_11

Download citation

Publish with us

Policies and ethics

Search

Navigation