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Time Series Residuals With Application To Probability Density Estimation

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  • P. M. Robinson

Abstract

. A linear stationary and invertible process yt models the second‐order properties of T observations on a discrete time series, up to finitely many unknown parameters θ. Two estimators of the residuals or innovations ɛt of yt are presented, based on a θ estimator which is root‐T consistent with respect to a wide class of ɛt distributions, such as a Gaussian estimator. One sets unobserved yt equal to their mean, the other treats yt as a circulant and may be best computed via two passes of the fast Fourier transform. The convergence of both estimators to ɛt is investigated. We apply the estimated ɛt to estimate the probability density function of ɛt. Kernel density estimators are shown to converge uniformly in probability to the true density. A new sub‐class of linear time series models is motivated.

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  • P. M. Robinson, 1987. "Time Series Residuals With Application To Probability Density Estimation," Journal of Time Series Analysis, Wiley Blackwell, vol. 8(3), pages 329-344, May.
    • Handle: RePEc:bla:jtsera:v:8:y:1987:i:3:p:329-344
    DOI: 10.1111/j.1467-9892.1987.tb00445.x
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    Cited by:

    1. J. H. Wright, 1995. "Stochastic Orders Of Magnitude Associated With Two‐Stage Estimators Of Fractional Arima Systems," Journal of Time Series Analysis, Wiley Blackwell, vol. 16(1), pages 119-125, January.
    2. Marc Hallin & Lanh Tran, 1996. "Kernel density estimation for linear processes: Asymptotic normality and optimal bandwidth derivation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 48(3), pages 429-449, September.
    3. Sophie Dabo-Niang & Anne-Françoise Yao, 2013. "Kernel spatial density estimation in infinite dimension space," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(1), pages 19-52, January.
    4. Lu, Zudi & Chen, Xing, 2004. "Spatial kernel regression estimation: weak consistency," Statistics & Probability Letters, Elsevier, vol. 68(2), pages 125-136, June.
    5. Carbon, Michel & Garel, Bernard & Tran, Lanh Tat, 1997. "Frequency polygons for weakly dependent processes," Statistics & Probability Letters, Elsevier, vol. 33(1), pages 1-13, April.
    6. Robinson, P.M. & Vidal Sanz, J., 2006. "Modified Whittle estimation of multilateral models on a lattice," Journal of Multivariate Analysis, Elsevier, vol. 97(5), pages 1090-1120, May.
    7. Kengo Kato, 2012. "Asymptotic normality of Powell’s kernel estimator," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 64(2), pages 255-273, April.
    8. Hallin, Marc & Lu, Zudi & Tran, Lanh T., 2004. "Kernel density estimation for spatial processes: the L1 theory," Journal of Multivariate Analysis, Elsevier, vol. 88(1), pages 61-75, January.
    9. Peter Robinson & J. Vidal Sanz Vidal Sanz, 2003. "Modified whittle estimation of multilateral spatial models," CeMMAP working papers 18/03, Institute for Fiscal Studies.

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