The Annals of Statistics

Oracally efficient estimation of autoregressive error distribution with simultaneous confidence band

Jiangyan Wang, Rong Liu, Fuxia Cheng, and Lijian Yang

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Abstract

We propose kernel estimator for the distribution function of unobserved errors in autoregressive time series, based on residuals computed by estimating the autoregressive coefficients with the Yule–Walker method. Under mild assumptions, we establish oracle efficiency of the proposed estimator, that is, it is asymptotically as efficient as the kernel estimator of the distribution function based on the unobserved error sequence itself. Applying the result of Wang, Cheng and Yang [J. Nonparametr. Stat. 25 (2013) 395–407], the proposed estimator is also asymptotically indistinguishable from the empirical distribution function based on the unobserved errors. A smooth simultaneous confidence band (SCB) is then constructed based on the proposed smooth distribution estimator and Kolmogorov distribution. Simulation examples support the asymptotic theory.

Article information

Source
Ann. Statist. Volume 42, Number 2 (2014), 654-668.

Dates
First available in Project Euclid: 20 May 2014

Permanent link to this document
http://projecteuclid.org/euclid.aos/1400592173

Digital Object Identifier
doi:10.1214/13-AOS1197

Mathematical Reviews number (MathSciNet)
MR3210982

Zentralblatt MATH identifier
1308.62096

Subjects
Primary: 62G15: Tolerance and confidence regions
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Keywords
$\mathrm{AR}(p)$ bandwidth error kernel oracle efficiency residual

Citation

Wang, Jiangyan; Liu, Rong; Cheng, Fuxia; Yang, Lijian. Oracally efficient estimation of autoregressive error distribution with simultaneous confidence band. Ann. Statist. 42 (2014), no. 2, 654--668. doi:10.1214/13-AOS1197. http://projecteuclid.org/euclid.aos/1400592173.


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References

  • [1] Bachmann, D. and Dette, H. (2005). A note on the Bickel–Rosenblatt test in autoregressive time series. Statist. Probab. Lett. 74 221–234.
  • [2] Bai, J. (1996). Testing for parameter constancy in linear regressions: An empirical distribution function approach. Econometrica 64 597–622.
  • [3] Bai, J. and Ng, S. (2001). A consistent test for conditional symmetry in time series models. J. Econometrics 103 225–258.
  • [4] Bickel, P. J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates. Ann. Statist. 1 1071–1095.
  • [5] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • [6] Bosq, D. (1998). Nonparametric Statistics for Stochastic Processes: Estimation and Prediction, 2nd ed. Lecture Notes in Statistics 110. Springer, New York.
  • [7] Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
  • [8] Cheng, F. (2002). Consistency of error density and distribution function estimators in nonparametric regression. Statist. Probab. Lett. 59 257–270.
  • [9] Cheng, F. (2005). Asymptotic distributions of error density estimators in first-order autoregressive models. Sankhyā 67 553–567.
  • [10] Cheng, F. (2005). Asymptotic distributions of error density and distribution function estimators in nonparametric regression. J. Statist. Plann. Inference 128 327–349.
  • [11] Dette, H., Neumeyer, N. and Van Keilegom, I. (2007). A new test for the parametric form of the variance function in non-parametric regression. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 903–917.
  • [12] Fernholz, L. T. (1991). Almost sure convergence of smoothed empirical distribution functions. Scand. J. Stat. 18 255–262.
  • [13] Huang, W.-M. (1986). A characterization of limiting distributions of estimators in an autoregressive process. Ann. Inst. Statist. Math. 38 137–144.
  • [14] Kiwitt, S. and Neumeyer, N. (2012). Estimating the conditional error distribution in non-parametric regression. Scand. J. Stat. 39 259–281.
  • [15] Lee, S. and Na, S. (2002). On the Bickel–Rosenblatt test for first-order autoregressive models. Statist. Probab. Lett. 56 23–35.
  • [16] Liu, R. and Yang, L. (2008). Kernel estimation of multivariate cumulative distribution function. J. Nonparametr. Stat. 20 661–677.
  • [17] Müller, U. U., Schick, A. and Wefelmeyer, W. (2004). Estimating functionals of the error distribution in parametric and nonparametric regression. J. Nonparametr. Stat. 16 525–548.
  • [18] Müller, U. U., Schick, A. and Wefelmeyer, W. (2012). Estimating the error distribution function in semiparametric additive regression models. J. Statist. Plann. Inference 142 552–566.
  • [19] Neumeyer, N. and Dette, H. (2007). Testing for symmetric error distribution in nonparametric regression models. Statist. Sinica 17 775–795.
  • [20] Neumeyer, N., Dette, H. and Nagel, E.-R. (2006). Bootstrap tests for the error distribution in linear and nonparametric regression models. Aust. N. Z. J. Stat. 48 129–156.
  • [21] Neumeyer, N. and Selk, L. (2013). A note on non-parametric testing for Gaussian innovations in AR–ARCH models. J. Time Series Anal. 34 362–367.
  • [22] Neumeyer, N. and Van Keilegom, I. (2009). Change-point tests for the error distribution in non-parametric regression. Scand. J. Stat. 36 518–541.
  • [23] Neumeyer, N. and Van Keilegom, I. (2010). Estimating the error distribution in nonparametric multiple regression with applications to model testing. J. Multivariate Anal. 101 1067–1078.
  • [24] Pham, T. D. and Tran, L. T. (1985). Some mixing properties of time series models. Stochastic Process. Appl. 19 297–303.
  • [25] Rootzén, H. (1986). Extreme value theory for moving average processes. Ann. Probab. 14 612–652.
  • [26] van der Vaart, A. (1994). Weak convergence of smoothed empirical processes. Scand. J. Stat. 21 501–504.
  • [27] Wang, J., Cheng, F. and Yang, L. (2013). Smooth simultaneous confidence bands for cumulative distribution functions. J. Nonparametr. Stat. 25 395–407.
  • [28] Wang, J., Liu, R., Cheng, F. and Yang, L. (2014). Supplement to “Oracally efficient estimation of autoregressive error distribution with simultaneous confidence band.” DOI:10.1214/13-AOS1197SUPP.
  • [29] Yamato, H. (1973). Uniform convergence of an estimator of a distribution function. Bull. Math. Statist. 15 69–78.

Supplemental materials

  • Supplementary material: Supplement to “Oracally efficient estimation of autoregressive error distribution with simultaneous confidence band”. This supplement contains additional technical proofs and some supporting numerical results.