Optimal Smoothing for a Computationally and Statistically Efficient Single Index Estimator
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.
KeywordsLink 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.
- 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
- Fan, J., & Gijbels, I. (1996). Local polynomial modeling and its applications. London: Chapman & Hall.Google Scholar
- Friedman, J.H. (1984). SMART User’s Guide. Laboratory for Computational Statistics, Stanford University Technical Report No. 1.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