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October 1996 Optimal designs for rational models
Zhuoqiong He, William J. Studden, Dongchu Sun
Ann. Statist. 24(5): 2128-2147 (October 1996). DOI: 10.1214/aos/1069362314

Abstract

In this paper, experimental designs for a rational model, $Y = P(x)/Q(x) + \varepsilon$, are investigated, where $P(x) = \theta_0 + \theta_1 x + \dots + \theta_p x^p$ and $Q(x) = 1 + \theta_{p + 1} x + \dots + \theta_{p+q} x^q$ are polynomials and $\varepsilon$ is a random error. Two approaches, Bayesian D-optimal and Bayesian optimal design for extrapolation, are examined. The first criterion maximizes the expected increase in Shannon information provided by the experiment asymptotically, and the second minimizes the asymptotic variance of the maximum likelihood estimator (MLE) of the mean response at an extrapolation point $x_e$. Corresponding locally optimal designs are also discussed. Conditions are derived under which a $p + q + 1$-point design is a locally D-optimal design. The Bayesian D-optimal design is shown to be independent of the parameters in $P(x)$ and to be equally weighted at each support point if the number of support points is the same as the number of parameters in the model. The existence and uniqueness of the locally optimal design for extrapolation are proven. The number of support points for the locally optimal design for extrapolation is exactly $p + q + 1$. These $p + q + 1$ design points are proved to be independent of the extrapolation point x e and the parameters in $P(x)$. The corresponding weights are also independent of the parameters in $P(x)$, but depend on $x_e$ and are not equal.

Citation

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Zhuoqiong He. William J. Studden. Dongchu Sun. "Optimal designs for rational models." Ann. Statist. 24 (5) 2128 - 2147, October 1996. https://doi.org/10.1214/aos/1069362314

Information

Published: October 1996
First available in Project Euclid: 20 November 2003

zbMATH: 0867.62063
MathSciNet: MR1421165
Digital Object Identifier: 10.1214/aos/1069362314

Subjects:
Primary: 62K05
Secondary: 62F15, 62L99

Rights: Copyright © 1996 Institute of Mathematical Statistics

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Vol.24 • No. 5 • October 1996
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