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Rate of convergence of a convolution-type estimator of the marginal density of a MA(1) process


  • Saavedra, Ángeles
  • Cao, Ricardo


In this paper moving-average processes with no parametric assumption on the error distribution are considered. A new convolution-type estimator of the marginal density of a MA(1) is presented. This estimator is closely related to some previous ones used to estimate the integrated squared density and has a structure similar to the ordinary kernel density estimator. For second-order kernels, the rate of convergence of this new estimator is investigated and the rate of the optimal bandwidth obtained. Under limit conditions on the smoothing parameter the convolution-type estimator is proved to be -consistent, which contrasts with the asymptotic behavior of the ordinary kernel density estimator, that is only -consistent.

Suggested Citation

  • Saavedra, Ángeles & Cao, Ricardo, 1999. "Rate of convergence of a convolution-type estimator of the marginal density of a MA(1) process," Stochastic Processes and their Applications, Elsevier, vol. 80(2), pages 129-155, April.
  • Handle: RePEc:eee:spapps:v:80:y:1999:i:2:p:129-155

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    References listed on IDEAS

    1. Hall, Peter & Marron, J. S., 1987. "Estimation of integrated squared density derivatives," Statistics & Probability Letters, Elsevier, vol. 6(2), pages 109-115, November.
    2. Wand, M. P., 1992. "Finite sample performance of density estimators under moving average dependence," Statistics & Probability Letters, Elsevier, vol. 13(2), pages 109-115, January.
    3. Jones, M. C. & Sheather, S. J., 1991. "Using non-stochastic terms to advantage in kernel-based estimation of integrated squared density derivatives," Statistics & Probability Letters, Elsevier, vol. 11(6), pages 511-514, June.
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    Cited by:

    1. Escanciano, Juan Carlos & Jacho-Chávez, David T., 2012. "n-uniformly consistent density estimation in nonparametric regression models," Journal of Econometrics, Elsevier, vol. 167(2), pages 305-316.
    2. Li, Shuo & Tu, Yundong, 2016. "n-consistent density estimation in semiparametric regression models," Computational Statistics & Data Analysis, Elsevier, vol. 104(C), pages 91-109.
    3. Milstein, G.N. & Schoenmakers, J.G.M. & Spokoiny, V., 2007. "Forward and reverse representations for Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 117(8), pages 1052-1075, August.
    4. Chang, Christopher C. & Politis, Dimitris N., 2011. "Bootstrap with larger resample size for root-n consistent density estimation with time series data," Statistics & Probability Letters, Elsevier, vol. 81(6), pages 652-661, June.
    5. Støve, Bård & Tjøstheim, Dag, 2007. "A Convolution Estimator for the Density of Nonlinear Regression Observations," Discussion Papers 2007/25, Norwegian School of Economics, Department of Business and Management Science.


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