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Bischoff, Wolfgang; Miller, Frank

Optimal designs which are efficient for lack of fit tests. (English) Zbl 1246.62175

Ann. Stat. 34, No. 4, 2015-2025 (2006).
Summary: Linear regression models are among the models most used in practice, although the practitioners are often not sure whether their assumed linear regression model is at least approximately true. In such situations, only designs for which the linear model can be checked are accepted in practice. For important linear regression models, such as polynomial regression, optimal designs do not have this property. To get practically attractive designs, we suggest the following strategy. One part of the design points is used to allow one to carry out a lack of fit test with good power for practically interesting alternatives. The rest of the design points are determined in such a way that the whole design is optimal for inference on the unknown parameter in case the lack of fit test does not reject the linear regression model. To solve this problem, we introduce efficient lack of fit designs. Then we explicitly determine the \(e_{k}\)-optimal design in the class of efficient lack of fit designs for polynomial regression of degree \(k-1\).

Cited in 11 Documents

MSC:

62K05 Optimal statistical designs
62J05 Linear regression; mixed models

Keywords:

linear regression models; efficient maximin power; polynomial regression of degree k-1; estimation of the highest coefficient
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References:

[1] Biedermann, S. and Dette, H. (2001). Optimal designs for testing the functional form of a regression via nonparametric estimation techniques. Statist. Probab. Lett. 52 215–224. · Zbl 1012.62079 · doi:10.1016/S0167-7152(00)00244-3
[2] Bischoff, W. (1998). A functional central limit theorem for regression models. Ann. Statist. 26 1398–1410. · Zbl 0936.62072 · doi:10.1214/aos/1024691248
[3] Bischoff, W. (2002). The structure of residual partial sums limit processes of linear regression models. Theory Stoch. Processes 8 23–28. · Zbl 1027.60029
[4] Bischoff, W. and Miller, F. (2000). Asymptotically optimal tests and optimal designs for testing the mean in regression models with applications to change-point problems. Ann. Inst. Statist. Math. 52 658–679. · Zbl 0978.62060 · doi:10.1023/A:1017521225616
[5] Bischoff, W. and Miller, F. (2006). Efficient lack of fit designs that are optimal to estimate the highest coefficient of a polynomial. J. Statist. Plann. Inference 136 4239–4249. · Zbl 1098.62098 · doi:10.1016/j.jspi.2005.06.007
[6] Biswas, A. and Chaudhuri, P. (2002). An efficient design for model discrimination and parameter estimation in linear models. Biometrika 89 709–718. JSTOR: · Zbl 1037.62070 · doi:10.1093/biomet/89.3.709
[7] Box, G. E. P. and Draper, N. (1959). A basis for the selection of a response surface design. J. Amer. Statist. Assoc. 54 622–654. JSTOR: · Zbl 0116.36804 · doi:10.2307/2282542
[8] Dette, H. (1993). Bayesian \(D\)-optimal and model robust designs in linear regression models. Statistics 25 27–46. · Zbl 0816.62054 · doi:10.1080/02331889308802429
[9] Dette, H. and Studden, W. J. (1997). The Theory of Canonical Moments With Applications in Statistics , Probability , and Analysis. Wiley, New York. · Zbl 0886.62002
[10] El-Krunz, S. M. and Studden, W. J. (1991). Bayesian optimal designs for linear regression models. Ann. Statist. 19 2183–2208. · Zbl 0755.62056 · doi:10.1214/aos/1176348392
[11] Ermakov, S. M. and Melas, V. B. (1995). Design and Analysis of Simulation Experiments. Kluwer, London. · Zbl 0925.62346
[12] Federov, V. V. (1972). Theory of Optimal Experiments. Academic Press, New York.
[13] Kiefer, J. and Wolfowitz, J. (1959). Optimum designs in regression problems. Ann. Math. Statist. 30 271–294. · Zbl 0090.11404 · doi:10.1214/aoms/1177706252
[14] Krafft, O. (1978). Lineare statistische Modelle und optimale Versuchspläne. Vandenhoeck and Ruprecht, Göttingen. · Zbl 0386.62048
[15] Miller, F. (2002). Optimale Versuchspläne bei Einschränkungen in der Versuchspunktwahl. Ph.D. dissertation, Fakultät für Mathematik, Univ. Karlsruhe. Available at www.ubka.uni-karlsruhe.de/cgi-bin/psview?document=2002/mathematik/10.
[16] Montepiedra, G. and Yeh, A. B. (1998). A two-stage strategy for the construction of \(D\)-optimal experimental designs. Comm. Statist. Simulation Comput. 27 377–401. · Zbl 0931.62062 · doi:10.1080/03610919808813486
[17] Pukelsheim, F. (1993). Optimal Design of Experiments. Wiley, New York. · Zbl 0834.62068
[18] Pukelsheim, F. and Rosenberger, J. L. (1993). Experimental designs for model discrimination. J. Amer. Statist. Assoc. 88 642–649. JSTOR: · Zbl 0773.62052 · doi:10.2307/2290346
[19] Rivlin, T. J. (1990). Chebyshev Polynomials. From Approximation Theory to Algebra and Number Theory , 2nd ed. Wiley, New York. · Zbl 0734.41029
[20] Sacks, J. and Ylvisaker, D. (1966). Design for regression problems with correlated errors. Ann. Math. Statist. 37 66–89. · Zbl 0152.17503 · doi:10.1214/aoms/1177699599
[21] Schwarz, H. R. (1988). Numerische Mathematik , 2nd ed. Teubner, Stuttgart. · Zbl 0669.65002
[22] Silvey, S. D. (1980). Optimal Design. Chapman and Hall, London. · Zbl 0468.62070
[23] Studden, W. J. (1968). Optimal designs on Tchebycheff points. Ann. Math. Statist. 39 1435–1447. · Zbl 0174.22404
[24] Wiens, D. P. (1991). Designs for approximately linear regression: Two optimality properties of uniform designs. Statist. Probab. Lett. 12 217–221. · Zbl 0736.62066 · doi:10.1016/0167-7152(91)90081-2
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