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Nonparametric functional central limit theorem for time series regression with application to self-normalized confidence interval

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  • Kim, Seonjin
  • Zhao, Zhibiao
  • Shao, Xiaofeng

Abstract

This paper is concerned with the inference of nonparametric mean function in a time series context. The commonly used kernel smoothing estimate is asymptotically normal and the traditional inference procedure then consistently estimates the asymptotic variance function and relies upon normal approximation. Consistent estimation of the asymptotic variance function involves another level of nonparametric smoothing. In practice, the choice of the extra bandwidth parameter can be difficult, the inference results can be sensitive to bandwidth selection and the normal approximation can be quite unsatisfactory in small samples leading to poor coverage. To alleviate the problem, we propose to extend the recently developed self-normalized approach, which is a bandwidth free inference procedure developed for parametric inference, to construct point-wise confidence interval for nonparametric mean function. To justify asymptotic validity of the self-normalized approach, we establish a functional central limit theorem for recursive nonparametric mean regression function estimates under primitive conditions and show that the limiting process is a Gaussian process with non-stationary and dependent increments. The superior finite sample performance of the new approach is demonstrated through simulation studies.

Suggested Citation

  • Kim, Seonjin & Zhao, Zhibiao & Shao, Xiaofeng, 2015. "Nonparametric functional central limit theorem for time series regression with application to self-normalized confidence interval," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 277-290.
  • Handle: RePEc:eee:jmvana:v:133:y:2015:i:c:p:277-290
    DOI: 10.1016/j.jmva.2014.09.017
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    References listed on IDEAS

    as
    1. Wei Biao Wu & Zhibiao Zhao, 2007. "Inference of trends in time series," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 69(3), pages 391-410.
    2. Lobato I. N., 2001. "Testing That a Dependent Process Is Uncorrelated," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1066-1076, September.
    3. Xiaofeng Shao, 2010. "Corrigendum: A self-normalized approach to confidence interval construction in time series," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(5), pages 695-696.
    4. Kiefer, Nicholas M. & Vogelsang, Timothy J., 2005. "A New Asymptotic Theory For Heteroskedasticity-Autocorrelation Robust Tests," Econometric Theory, Cambridge University Press, vol. 21(06), pages 1130-1164, December.
    5. Nicholas M. Kiefer & Timothy J. Vogelsang & Helle Bunzel, 2000. "Simple Robust Testing of Regression Hypotheses," Econometrica, Econometric Society, vol. 68(3), pages 695-714, May.
    6. Kiefer, Nicholas M. & Vogelsang, Timothy J., 2002. "Heteroskedasticity-Autocorrelation Robust Testing Using Bandwidth Equal To Sample Size," Econometric Theory, Cambridge University Press, vol. 18(06), pages 1350-1366, December.
    7. Xiaofeng Shao, 2010. "A self-normalized approach to confidence interval construction in time series," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(3), pages 343-366.
    8. Zhou Zhou & Xiaofeng Shao, 2013. "Inference for linear models with dependent errors," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(2), pages 323-343, March.
    9. Xiaofeng Shao, 2012. "Parametric Inference in Stationary Time Series Models with Dependent Errors," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 39(4), pages 772-783, December.
    10. Toshio Honda, 2000. "Nonparametric Estimation of a Conditional Quantile for α-Mixing Processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 52(3), pages 459-470, September.
    11. Shao, Xiaofeng & Zhang, Xianyang, 2010. "Testing for Change Points in Time Series," Journal of the American Statistical Association, American Statistical Association, vol. 105(491), pages 1228-1240.
    12. Seonjin Kim & Zhibiao Zhao, 2013. "Unified inference for sparse and dense longitudinal models," Biometrika, Biometrika Trust, vol. 100(1), pages 203-212.
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