Optimal Smoothing for a Computationally and Statistically Efficient Single Index Estimator

  • Yingcun Xia
  • Wolfgang Karl Härdle
  • Oliver Linton


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.


Link Function Optimal Bandwidth High Order Property Slice Inverse Regression Good Bandwidth 



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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Yingcun Xia
    • 1
  • Wolfgang Karl Härdle
    • 2
  • Oliver Linton
    • 3
  1. 1.Department of Statistics and Applied Probability and Risk Management InstituteNational University of SingaporeSingaporeSingapore
  2. 2.C.A.S.E. Centre for Applied Statistics and Economics, School of Business and EconomicsHumboldt-Universität zu BerlinBerlinGermany
  3. 3.CambridgeUK

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