Abstract
The problem of estimating a continuous-time random process from its observations at appropriately designed sampling points is considered. The quality of an estimator is measured by its integrated mean square error (IMSE). Here, sampling points are designed stepwisely to minimize the IMSE and the best linear unbiased estimator (BLUE) is so determined that the earlier calculations do not have to be repeated with addition of one or more new samples. For random processes whose covariance has a sharp corner at the diagonal, it is shown that essentially, an optimal one-step forward sampling location is one of the midpoints of intervals determined by the current and previous sampling points. Both analytical and numerical examples are considered.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, T. W.: Some stochastic process models for intelligence test scores, in K. J. Arrow et al. (eds), Mathematical Methods in the Social Sciences, Stanford Univ. Press, Stanford, CA, 1960, 205–220.
Benhenni, K., Cambanis, S.: Sampling designs for estimating integrals of stochastic processes, Ann. Math. Statist. 20 (1992a), 161–194.
Benhenni, K., Cambanis, S.: Sampling designs for estimating integrals of stochastic processes using quadratic mean derivatives, in G. A. Anastassiou (ed.), Approximation Theory: Proceedings of the Sixth Southeastern Approximation Theorists Annual Conference, Dekker, New York, 1992b, pp. 93–123.
Cambanis, S.: Sampling designs for time series, in E. J. Hannan, P. R. Krishnaiah and M. M. Rao (eds), Handbook of Statistics, 5: Time Series in Time Domain, North-Holland, Amsterdam, 1985, pp. 337–362.
Bucklew, J. A. and Cambanis, S.: Estimating random integrals from moisy observations: sampling designs and their performance, IEEE Trans. Inform. Theory 34(1) (1988), 111–127.
Jones, A. E.: Systematic sampling of continuous parameter populations, Biometrika 35 (1948), 283–290.
Kendall, M. G.: Continuation of Dr. Jones's paper, Biometrika 35 (1948), 291–296.
Morrison, D. F.: The optimal spacing of repeated measurements, Biometrics 26 (1970), 281–290.
Müller-Gronbach, T.: Optimal designs for approximating the path of a stochastic process, J. Statist. Planning Inference 49 (1996), 371–385.
Müller-Gronbach, T. and Ritter, K.: Uniform reconstruction of Gaussian processes, Stochastic Processes Appl. 69 (1997), 55–70.
Ritter, K.: Asymptotic optimality of regular sequence designs, Ann. Math. Statist. 24 (1996), 2081–2096.
Sacks, J., Welch, W. J., Mitchell T. J. and Wynn H. P.: Designs and analysis of computer experiments, Statistical Sci. 4 (1989), 409–435.
Sacks, J. and Ylvisaker, D.: Designs for regression problems with correlated errors, Ann. Math. Statist 37 (1995), 66–89.
Stein, M. L.: Predicting integrals of stochastic processes, Ann. Appl. Probab. 5 (1995), 158–170.
Su, Y. C. and Cambanis, S.: Sampling designs for estimation of a random process, Stochastic Processes Appl. 41 (1993), 47–89.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Su, Y. Stepwise Estimation of Random Processes. Statistical Inference for Stochastic Processes 2, 287–308 (1999). https://doi.org/10.1023/A:1009999702098
Issue Date:
DOI: https://doi.org/10.1023/A:1009999702098