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Assessing the number of mean square derivatives of a Gaussian process

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  • Blanke, Delphine
  • Vial, Céline

Abstract

We consider a real Gaussian process X with unknown smoothness where the mean square derivative X(r0) is supposed to be Hölder continuous in quadratic mean. First, from selected sampled observations, we study the reconstruction of X(t), t[set membership, variant][0,1], with a piecewise polynomial interpolation of degree r>=1. We show that the mean square error of the interpolation is a decreasing function of r but becomes stable as soon as r>=r0. Next, from an interpolation-based empirical criterion and n sampled observations of X, we derive an estimator of r0 and prove its strong consistency by giving an exponential inequality for . Finally, we establish the strong consistency of with an almost optimal rate.

Suggested Citation

  • Blanke, Delphine & Vial, Céline, 2008. "Assessing the number of mean square derivatives of a Gaussian process," Stochastic Processes and their Applications, Elsevier, vol. 118(10), pages 1852-1869, October.
    • Handle: RePEc:eee:spapps:v:118:y:2008:i:10:p:1852-1869
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    References listed on IDEAS

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    1. D. Blanke & B. Pumo, 2003. "Optimal sampling for density estimation in continuous time," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(1), pages 1-23, January.
    2. Müller-Gronbach, Thomas & Ritter, Klaus, 1997. "Uniform reconstruction of Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 69(1), pages 55-70, July.
    3. Blanke, D. & Bosq, D., 1997. "Accurate rates of density estimators for continuous-time processes," Statistics & Probability Letters, Elsevier, vol. 33(2), pages 185-191, April.
    4. Lasinger, Rudolf, 1993. "Integration of covariance kernels and stationarity," Stochastic Processes and their Applications, Elsevier, vol. 45(2), pages 309-318, April.
    5. Susanne Ditlevsen & Michael Sørensen, 2004. "Inference for Observations of Integrated Diffusion Processes," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 31(3), pages 417-429, September.
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    1. Delphine Blanke & Céline Vial, 2011. "Estimating the order of mean-square derivatives with quadratic variations," Statistical Inference for Stochastic Processes, Springer, vol. 14(1), pages 85-99, February.
    2. Karim Benhenni & Mustapha Rachdi & Yingcai Su, 2013. "The effect of the regularity of the error process on the performance of kernel regression estimators," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(6), pages 765-781, August.

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