Optimal designs in regression with correlated errors
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Date
2015
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Abstract
This paper discusses the problem of determining optimal designs for regression models,
when the observations are dependent and taken on an interval. A complete solution
of this challenging optimal design problem is given for a broad class of regression models
and covariance kernels.
We propose a class of estimators which are only slightly more complicated than the ordinary
least-squares estimators. We then demonstrate that we can design the experiments,
such that asymptotically the new estimators achieve the same precision as the best linear
unbiased estimator computed for the whole trajectory of the process. As a by-product
we derive explicit expressions for the BLUE in the continuous time model and analytic
expressions for the optimal designs in a wide class of regression models. We also demonstrate
that for a finite number of observations the precision of the proposed procedure,
which includes the estimator and design, is very close to the best achievable. The results
are illustrated on a few numerical examples.
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Keywords
linear regression, Doob representation, Gaussian processes, BLUE, optimal design, signed measures, correlated observations