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ASYMPTOTIC DISTRIBUTIONS FOR TWO ESTIMATORS OF THESINGLE-INDEX MODEL

Published online by Cambridge University Press:  03 November 2006

Yingcun Xia
Yingcun Xia
Affiliation:
National University of Singapore
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Abstract

The single-index model is one of the most popular semiparametricmodels in applied quantitative sciences. Two new estimation methodshave been proposed recently by Hristache, Juditski, and Spokoiny(2001, Annals ofStatistics 29, 595–623) and Xia, Tong, Li, and Zhu(2002, Journal of the RoyalStatistical Society, Series B 64, 363–410),respectively. However, their asymptotic distributions have not beeninvestigated yet. In this paper, alternative versions for themethods are investigated. Asymptotic distributions of the estimatorsare derived. Efficiency comparisons between the estimation methodsare made.The author is most gratefulto Professor O. Linton and Professor W. Härdle for helpfuldiscussions. Valuable comments from two anonymous reviewers haveimproved the presentation of the paper substantially. Theresearch has been partially supported by NUS research grantR-155-000-048-112, National University of Singapore, Singapore,and the Alexander von Humboldt Foundation, Germany.

Information

Type
Research Article
Information
Econometric Theory , Volume 22 , Issue 6 , December 2006 , pp. 1112 - 1137
Copyright
© 2006 Cambridge University Press

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References

REFERENCES

Bai, Z.D., B.Q. Miao, & C.R. Rao (1991) Estimation of directions of arrival of signals: Asymptotic results. In S. Haykin (ed.), Advances in Spectrum Analysis and Array Processing, vol. 1, 327–347. Prentice Hall.
Bickel, P., A.J. Klaassen, Y. Ritov, & J.A. Wellner (1993) Efficient and Adaptive Inference in Semiparametric Models. Johns Hopkins University Press.
Carroll, R.J., J. Fan, I. Gijbels, & M.P. Wand (1997) Generalized partially linear single-index models. Journal of the American Statistical Association 92, 477–489.Google Scholar
Chow, Y.S. & H. Teicher (1978) Probability Theory: Independence, Interchangeability, Martingales. Springer-Verlag.
Delecroix, M., M. Hristache, & V. Patilea (2006) On semiparametric M-estimation in single-index regression. Journal of Statistic Planning and Inference 136, 730–769.Google Scholar
Härdle, W., P. Hall, & H. Ichimura (1993) Optimal smoothing in single-index models. Annals of Statistics 21, 157–178.Google Scholar
Härdle, W. & T.M. Stoker (1989) Investigating smooth multiple regression by method of average derivatives. Journal of the American Statistical Association 84, 986–995.Google Scholar
Horowitz, J.L. & W. Härdle (1996) Direct semiparametric estimation of single-index models with discrete covariates. Journal of the American Statistical Association 91, 1632–1640.Google Scholar
Hristache, M., A. Juditski, J. Polzehl, & V. Spokoiny (2001) Structure adaptive approach for dimension reduction. Annals of Statistics 29, 1537–1566.Google Scholar
Hristache, M., A. Juditski, & V. Spokoiny (2001) Direct estimation of the index coefficients in a single-index model. Annals of Statistics 29, 595–623.Google Scholar
Ichimura, H. (1993) Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. Journal of Econometrics 58, 71–120.Google Scholar
Klein, R.W. & R.H. Spady (1993) An efficient semiparametric estimator for binary response models. Econometrica 61, 387–421.Google Scholar
Mack, Y.P. & B.W. Silverman (1982) Weak and strong uniform consistency of kernel regression estimates. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 61, 405–415.Google Scholar
Nishiyama, Y. & P.M. Robinson (2005) The bootstrap and the Edgeworth correction for semiparametric averaged derivatives. Econometrica 73, 903–948.Google Scholar
Powell, J.L., J.H. Stock, & T.M. Stoker (1989) Semiparametric estimation of index coefficients. Econometrica 57, 1403–1430.Google Scholar
Schott, J.R. (1997) Matrix Analysis for Statistics. Wiley.
Sherman, R.P. (1994) Maximal inequalities for degenerate U-processes with applications to optimization estimators. Annals of Statistics 22, 439–459.Google Scholar
Xia, Y., O. Linton, & W. Härdle (2004) Computationally and Statistically Efficient Single Index Estimation. Manuscript, Department of Statistics and Applied Probability, National University of Singapore.
Xia, Y., H. Tong, W.K. Li, & L. Zhu (2002) An adaptive estimation of dimension reduction space (with discussions). Journal of the Royal Statistical Society, Series B 64, 363–410.Google Scholar
Yin, X. & D.R. Cook (2005) Direction estimation in single-index regressions. Biometrika 92, 371–384.Google Scholar
Yu, Y. & D. Ruppert (2002) Penalized spline estimation for partially linear single-index models. Journal of the American Statistical Association 97, 1042–1054.Google Scholar