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UNIFORM BAHADUR REPRESENTATION FOR LOCAL POLYNOMIAL ESTIMATES OF M-REGRESSION AND ITS APPLICATION TO THE ADDITIVE MODEL

Published online by Cambridge University Press:  05 March 2010

Efang Kong ,
Oliver Linton  and
Yingcun Xia
Efang Kong
Affiliation:
Technische Universiteit Eindhoven
Oliver Linton*
Affiliation:
London School of Economics
Yingcun Xia
Affiliation:
Nanjing University, China and National University of Singapore
*
*Address for correspondence to Oliver Linton, Department of Economics, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom; e-mail: o.linton@lse.ac.uk.
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Abstract

We use local polynomial fitting to estimate the nonparametric M-regression function for strongly mixing stationary processes {(Yi, Xi)}. We establish a strong uniform consistency rate for the Bahadur representation of estimators of the regression function and its derivatives. These results are fundamental for statistical inference and for applications that involve plugging such estimators into other functionals where some control over higher order terms is required. We apply our results to the estimation of an additive M-regression model.

Type
ARTICLES
Information
Econometric Theory , Volume 26 , Issue 5 , October 2010 , pp. 1529 - 1564
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Andrews, D.W.K. (1994) Asymptotics for semiparametric econometric models via stochastic equicontinuity. Econometrica 62, 43–72.CrossRefGoogle Scholar
Bahadur, R.R. (1966) A note on quantiles in large samples. Annals of Mathematical Statistics 37, 577–580.CrossRefGoogle Scholar
Bosq, D. (1998) Nonparametric Statistics for Stochastic Processes. Springer-Verlag.CrossRefGoogle Scholar
Chen, X., Linton, O., & Van Keilegom, I. (2003) Estimation of semiparametric models when the criterion is not smooth. Econometrica 71, 1591–1608.CrossRefGoogle Scholar
Fan, J. & Gijbels, I. (1996) Local Polynomial Modelling and Its Applications. Chapman & Hall.Google Scholar
Fan, J., Heckman, N.E., & Wand, M.P. (1995) Local polynomial kernel regression for generalized linear models and quasi-likelihood functions. Journal of the American Statistical Association 90, 141–150.CrossRefGoogle Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Application. NewYork: Academic Press.Google Scholar
Härdle, W. (1990) Applied Nonparametric Regression. Cambridge University Press.CrossRefGoogle Scholar
Hengartner, N.W. & Sperlich, S. (2005) Rate optimal estimation with the integration method in the presence of many covariates. Journal of Multivariate Analysis 95, 246–272.CrossRefGoogle Scholar
Hong, S. (2003) Bahadur representation and its application for local polynomial estimates in nonparametric M-regression. Journal of Nonparametric Statistics 15, 237–251.CrossRefGoogle Scholar
Horowitz, J.L. & Lee, S. (2005) Nonparametric estimation of an additive quantile regression model. Journal of the American Statistical Association 100, 1238–1249.CrossRefGoogle Scholar
Huber, P.J. (1973) Robust regression. Annals of Statistics 1, 799–821.CrossRefGoogle Scholar
Kiefer, J. (1967) On Bahadur’s representation of sample quantiles. Annals of Mathematical Statistics. 38, 1323–1342.CrossRefGoogle Scholar
Linton, O. (2001) Estimating additive nonparametric models by partial Lq norm: The curse of fractionality. Econometric Theory 17, 1037–1350.CrossRefGoogle Scholar
Linton, O. & Härdle, W. (1996) Estimation of additive regression models with known links. Biometrika 83, 529–540.CrossRefGoogle Scholar
Linton, O. & Nielsen, J.P. (1995) A kernel method of estimating structured nonparametric regression based on marginal integration. Biometrika 82, 93–100.CrossRefGoogle Scholar
Linton, O., Sperlich, S., & Van Keilegom, I. (2008) Estimation of a semiparametric transformation model by minimum distance. Annals of Statistics 36, 686–718.CrossRefGoogle Scholar
Masry, E. (1996) Multivariate local polynomial regression for time series: Uniform strong consistency and rates. Journal of Time Series Analysis 17, 571–599.CrossRefGoogle Scholar
Peng, L. & Yao, Q. (2003) Least absolute deviation estimation for ARCH and GARCH models. Biometrika 90, 967–975.CrossRefGoogle Scholar
Powell, J.L. (1994) Estimation in semiparametric models. In Engle, R.F. & McFadden, D.L. (eds.), The Handbook of Econometrics, vol. 4, pp. 2444–2521. North Holland.Google Scholar
Sperlich, S., Linton, O., & Härdle, W. (1998) A simulation comparison between the backfitting and integration methods of estimating separable nonparametric models. Test 8, 419–458.CrossRefGoogle Scholar
Stone, C.J. (1982) Optimal global rates of convergence for nonparametric regression. Annals of Statistics 10, 1040–1053.CrossRefGoogle Scholar
Stone, C.J. (1986) The dimensionality reduction principle for generalized additive models. Annals of Statistics 14, 592–606.CrossRefGoogle Scholar
Wu, W.B. (2005) On the Bahadur representation of sample quantiles for dependent sequences. Annals of Statistics 33, 1934–1963.CrossRefGoogle Scholar