Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-28T05:35:08.987Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximating integrals of stochastic processes: extensions

Part of: Approximations and expansions Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Karim Benhenni
Karim Benhenni*
Affiliation:
Université Pierre Mendès France
*
∗Postal address: LABSAD, Bâtiment des Sciences Humaines et Mathématiques, Université Pierre Mendès France, BP 47 X, 38040 Grenoble, France. Email address: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the problem of predicting integrals of second order processes whose covariances satisfy some Hölder regularity condition of order α > 0. When α is an odd integer, linear estimators based on regular sampling designs were constructed and asymptotic results for the approximation error were derived. We extend this result to any α > 0. When 2K < α ≤ 2K + 2, K a non-negative integer, we use an appropriate predictor based on the Euler-MacLaurin formula of order K with regular sampling designs. We give the corresponding result for the mean square error.

Keywords

Euler-MacLaurin formulaHölder regularity conditionsampling designRiemann zeta functionrate of convergence

MSC classification

Primary: 62M20: Prediction
Secondary: 41A25: Rate of convergence, degree of approximation 41A55: Approximate quadratures
Type
Research Papers
Information
Journal of Applied Probability , Volume 35 , Issue 4 , December 1998 , pp. 843 - 855
Copyright
Copyright © Applied Probability Trust 1998 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramovitz, M., and Stegun, I. (1965). Handbook of Mathematical Functions. Dover, New York.Google Scholar
Benhenni, K., and Cambanis, S. (1992a). Sampling designs for estimating integrals of stochastic processes. Ann. Statist. 20, 161–194.Google Scholar
Benhenni, K., and Cambanis, S. (1992b). Sampling designs for estimating integrals of stochastic processes using quadratic mean derivatives. In Approximation Theory, ed. Anastassiou, G. A., Dekker, New York, pp. 93–128.Google Scholar
Benhenni, K., and Cambanis, S. (1996). The effect of quantization on the performance of sampling designs. Technical Report No. 477, Center for Stochastic Processes, University of North Carolina, Chapel Hill.Google Scholar
Berndt, B. C., and Joshi, P. T. (1983). Chapter 9 of Ramanujan's Second Notebook: Infinite Series Identities, Transformations, and Evaluations. Contemporary Mathematics 23. American Mathematical Society, Providence, RI.Google Scholar
Cambanis, S. (1985). Sampling designs for time series. Time series in time domain. In Handbook of Statistics, ed. Hannan, E. J., Krishnaiah, P. R., and Rao, M. M., North-Holland, Amsterdam, pp. 337–362.Google Scholar
Eubank, R. L, Smith, P. L., and Smith, P. W. (1982). A note on optimal and asymptotically optimal designs for certain time series models. Ann. Statist. 10, 1295–1301.Google Scholar
Istas, J., and Loredo, C. (1997). Estimating functionals of a stochastic process. Adv. Appl. Prob. 10, 249–270.Google Scholar
Krylov, V. I. (1962). Approximate Calculation of Integrals. Macmillan, New York.Google Scholar
Pitt, L. D., Robeva, R., and Wang, D. Y. (1995). An error analysis for the numerical calculation of certain random integrals: Part 1. Ann. Appl. Prob. 5, 171–197.Google Scholar
Ritter, K. (1996). Asymptotic optimality of regular sequence designs. Ann. Statist. 24, 2081–2096.Google Scholar
Sacks, J., and Ylvisaker, D. (1966). Designs for regression problems with correlated errors. Ann. Math. Statist. 37, 68–89.Google 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. (1970a). Designs for regression problems with correlated errors III. Ann. Math. Statist. 41, 2057–2074.Google Scholar
Sacks, J., and Ylvisaker, D. (1970b). Statistical designs and integral approximations. In Proc. Twelfth Biennial Seminar of the Canadian Math. Congress, Canadian Math. Congress, Montreal, pp. 115–136.Google Scholar
Stein, M. (1995). Predicting integrals of stochastic processes. Ann. Appl. Prob. 5, 158–170.CrossRefGoogle Scholar
Titchmarsh, E. C. (1951). The Theory of the Riemann Zeta-Function. Clarendon Press, Oxford.Google Scholar