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Bernstein polynomial estimation of a spectral density

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  • Yoshihide Kakizawa

Abstract

. We consider an application of Bernstein polynomials for estimating a spectral density of a stationary process. The resulting estimator can be interpreted as a convex combination of the (Daniell) kernel spectral density estimators at m points, the coefficients of which are probabilities of the binomial distribution bin(m − 1, |λ|/π), λ ∈ Π ≡ [−π, π] being the frequency where the spectral density estimation is made. Several asymptotic properties are investigated under conditions of the degree m. We also discuss methods of data‐driven choice of the degree m. For a comparison with the ordinary kernel method, a Monte Carlo simulation illustrates our methodology and examines its performance in small sample.

Suggested Citation

  • Yoshihide Kakizawa, 2006. "Bernstein polynomial estimation of a spectral density," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(2), pages 253-287, March.
    • Handle: RePEc:bla:jtsera:v:27:y:2006:i:2:p:253-287
    DOI: 10.1111/j.1467-9892.2005.00465.x
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    Cited by:

    1. Lu, Lu, 2015. "On the uniform consistency of the Bernstein density estimator," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 52-61.
    2. Bouezmarni, Taoufik & El Ghouch, Anouar, 2011. "Bernstein estimator for unbounded density copula," UC3M Working papers. Economics we1143, Universidad Carlos III de Madrid. Departamento de Economía.
    3. Ouimet, Frédéric, 2021. "Asymptotic properties of Bernstein estimators on the simplex," Journal of Multivariate Analysis, Elsevier, vol. 185(C).
    4. Bouezmarni, Taoufik & Rombouts, Jeroen V.K. & Taamouti, Abderrahim, 2010. "Asymptotic properties of the Bernstein density copula estimator for [alpha]-mixing data," Journal of Multivariate Analysis, Elsevier, vol. 101(1), pages 1-10, January.
    5. Bouezmarni, Taoufik & Rombouts, Jeroen V. K., 2008. "Asymptotic properties of the Bernstein density copula for dependent data," UC3M Working papers. Economics we083619, Universidad Carlos III de Madrid. Departamento de Economía.
    6. Yifu Tang & Claudia Kirch & Jeong Eun Lee & Renate Meyer, 2023. "Posterior consistency for the spectral density of non‐Gaussian stationary time series," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 50(3), pages 1152-1182, September.
    7. Alexandre Leblanc, 2010. "A bias-reduced approach to density estimation using Bernstein polynomials," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(4), pages 459-475.
    8. Alexandre Leblanc, 2009. "Chung–Smirnov property for Bernstein estimators of distribution functions," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(2), pages 133-142.
    9. Kakizawa, Yoshihide, 2007. "Moderate deviations for quadratic forms in Gaussian stationary processes," Journal of Multivariate Analysis, Elsevier, vol. 98(5), pages 992-1017, May.
    10. Bouezmarni Taoufik & Ghouch El & Taamouti Abderrahim, 2013. "Bernstein estimator for unbounded copula densities," Statistics & Risk Modeling, De Gruyter, vol. 30(4), pages 343-360, December.
    11. Brian Hart & Michele Guindani & Stephen Malone & Mark Fiecas, 2022. "A nonparametric Bayesian model for estimating spectral densities of resting‐state EEG twin data," Biometrics, The International Biometric Society, vol. 78(1), pages 313-323, March.

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