Abstract
In this paper we mainly investigate the problem of optimal designs for multi-factor regression models with partially known heteroscedastic structure. The Bayesian \(\varPhi _q\)-optimality criterion proposed by Dette and Wong (Ann Stat 24:2108–2127, 1996), which closely resembles Kiefer’s \(\varPhi _k\)-class of criteria, and the standardized maximin D-optimal criterion are considered. More precisely, for heteroscedastic Kronecker product models, it is shown that the product designs formed from optimal designs for sub-models with a single factor are optimal under the two robust criteria. For additive models with intercept, however, sufficient conditions are given in order to search for Bayesian \(\varPhi _q\)-optimal and standardized maximin D-optimal product designs. Finally, several examples are presented to illustrate the obtained theoretical results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abousaleh H, Zhou J (2021) Minimax \(A\)-, \(c\)- and \(I\)-optimal regression designs for models with heteroscedastic errors. Can J Stat. https://doi.org/10.1002/cjs.11674
Atkinson A, Donev A, Tobias R (2007) Optimum experimental designs, with SAS. Oxford University Press, New York
Biedermann S, Dette H, Woods DC (2011) Optimal design for additive partially nonlinear models. Biometrika 98:449–458
Chaloner K, Verdinelli I (1995) Bayesian experimental design: a review. Stat Sci 10:273–304
Chatterjee S, Hadi AS (2012) Regression analysis by example, 5th edn. Wiley, New York
Chernoff H (1953) Locally optimal designs for estimating parameters. Ann Math Stat 24:586–602
Cook RD, Weisberg S (1983) Diagnostics for heteroscedasticity in regression. Biometrika 70:1–10
Danskin JM (1967) The theory of max–min and its application to weapons allocation problems. Springer, New York
Dette H (1997) Designing experiments with respect to ‘standardized’ optimality criteria. J R Stat Soc B 59:97–110
Dette H, Wong WK (1996) Optimal Bayesian designs for models with partially specified heteroscedastic structure. Ann Stat 24:2108–2127
Dette H, Haines LM, Imhof LA (2005) Bayesian and maximin optimal designs for heteroscedastic regression models. Can J Stat 33:221–241
Dette H, Haines LM, Imhof LA (2007) Maximin and Bayesian optimal designs for regression models. Stat Sin 17:463–480
Fedorov VV (1972) Theory of optimal experiments (translated by W. J. Studden and E. M. Klimko). Academic Press, New York
Hand DJ, Daly F, McConway K, Lunn D, Ostrowski E (1994) A handbook of small data sets. Chapman & Hall, London
He L (2021) Bayesian optimal designs for multi-factor nonlinear models. Stat Methods Appl 30:223–233
He L, Yue R-X (2017) \(R\)-optimal designs for multi-factor model with heteroscedastic errors. Metrika 80:717–732
Imhof LA (2001) Maximin designs for exponential growth models and heteroscedastic polynomial models. Ann Stat 29:561–576
Müller CH, Pázman A (1998) Applications of necessary and sufficient conditions for maximin efficient designs. Metrika 48:1–19
Rodríguez C, Ortiz I (2005) \(D\)-optimum designs in multi-factor models with heteroscedastic errors. J Stat Plan Inference 128:623–631
Rodríguez C, Ortiz I, Martínez I (2015) Locally and maximin optimal designs for multi-factor nonlinear models. Statistics 49:1157–1168
Rodríguez C, Ortiz I, Martínez I (2016) \(A\)-optimal designs for heteroscedastic multifactor regression models. Commun Stat Theory Methods 45:757–771
Venables WN, Ripley BD (2002) Modern applied statistics with S, 4th edn. Springer, New York
Wong WK (1992) A unified approach to the construction of minimax designs. Biometrika 79:611–619
Wong WK (1994) \(G\)-optimal designs for multifactor experiments with heteroscedastic errors. J Stat Plan Inference 40:127–133
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant No. 12101013), the National Social Science Foundation of China (Grant No. 21BTJ034) and the Natural Science Foundation of Anhui Province (Grant Nos. 2008085QA15, 2008085MA08). The authors are grateful to the referees for their helpful comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
He, L., He, D. Bayesian and maximin optimal designs for heteroscedastic multi-factor regression models. Stat Papers 64, 1997–2013 (2023). https://doi.org/10.1007/s00362-022-01368-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-022-01368-y