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Estimating Functionals of a Stochastic Process

Part of: Numerical approximation and computational geometry Design of experiments Stochastic processes Inference from stochastic processes

Published online by Cambridge University Press:  01 July 2016

Jacques Istas  and
Catherine Laredo
Jacques Istas*
Affiliation:
Institut National de la Recherche Agronomique
Catherine Laredo*
Affiliation:
Institut National de la Recherche Agronomique
*
∗ Postal address: Laboratoire de Biométrie, Domaine de Vilvert, I.N.R.A., 78350 Jouy-en-Josas, France.
∗ Postal address: Laboratoire de Biométrie, Domaine de Vilvert, I.N.R.A., 78350 Jouy-en-Josas, France.
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Abstract

The problem of estimating the integral of a stochastic process from observations at a finite number N of sampling points has been considered by various authors. Recently, Benhenni and Cambanis (1992) studied this problem for processes with mean 0 and Hölder index K + ½, K ; ℕ These results are here extended to processes with arbitrary Hölder index. The estimators built here are linear in the observations and do not require the a priori knowledge of the smoothness of the process. If the process satisfies a Hölder condition with index s in quadratic mean, we prove that the rate of convergence of the mean square error is N2s+1 as N goes to ∞, and build estimators that achieve this rate.

Keywords

SECOND-ORDER PROCESSFUNCTIONALS ESTIMATIONDISCRETE SAMPLINGSMOOTH PROCESSWAVELETS

MSC classification

Primary: 60G12: General second-order processes 60G35: Signal detection and filtering 65D05: Interpolation
Secondary: 62K05: Optimal designs 62M99: None of the above, but in this section
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 29 , Issue 1 , March 1997 , pp. 249 - 270
Copyright
Copyright © Applied Probability Trust 1997 

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References

Andersson, L., Hall, N., Jawerth, B. and Peters, G. (1993) Wavelets on closed subsets on the real line. In Topics in the Theory and Applications of Wavelets, ed. Schumaker, L. and Webb, G.. Academic Press, New York. pp 1–60.Google Scholar
Benhenni, K. and Cambanis, S. (1992) Sampling designs for estimating integrals of stochastic processes. Ann. Statist. 20, 161–194.Google Scholar
Cohen, A., Daubechies, I., Jawerth, B. and Vial, P. (1992) Multiresolution analysis, wavelets and fast algorithms on the interval. C.R. Acad. Sci. 316, 417–421.Google Scholar
Daubechies, I. (1988) Orthonormal bases of compactly supported wavelets. Commun. Pure Appl. Math. 41, 909–996.Google Scholar
Daubechies, I. (1992) Ten Lectures on Wavelets. SIAM, Philadelphia, PA.Google Scholar
Eubank, R. L., Smith, L. P. and Smith, P. (1982) A note on optimal and asymptotically optimal designs for certain time series models. Ann. Statist. 10, 1295–1301.CrossRefGoogle Scholar
Kerkyacharian, G. and Picard, D. (1993) Density estimation by kernel and wavelet methods. Optimality of Besov spaces. Statist. Prob. Lett. 18, 327–336.CrossRefGoogle Scholar
Meyer, Y. (1990) Ondelettes et Opérateurs. Vol. 1. Hermann, Paris.Google Scholar
Meyer, Y. (1992) Ondelettes et Algorithmes Concurrents. Hermann, Paris.Google Scholar
Sacks, J. and Ylvisaker, D. (1966) Designs for regression problems with correlated errors. Ann. Math. Statist. 37, 66–89.CrossRefGoogle Scholar
Sacks, J. and Ylvisaker, D. (1968) Designs for regression problems with correlated errors II. Ann. Math. Statist. 39, 49–69.Google Scholar
Sacks, J. and Ylvisaker, D. (1970) Designs for regression problems with correlated errors III. Ann. Math. Statist. 41, 2057–2074.Google Scholar
Stein, M. (1995) Predicting integrals of stochastic processes. Ann. Appl. Prob. 5, 158–170.Google Scholar
Wahba, G. (1990) Spline Models for Observational Data. SIAM, Philadelphia, PA.Google Scholar