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A new approach to optimal design for linear models with correlated observations. (English) Zbl 1390.62151

Summary: We consider the problem of designing experiments for regression in the presence of correlated observations with the location model as the main example. For a fixed correlation structure approximate optimal designs are determined explicitly, and it is demonstrated that under the model assumptions made by P. J. Bickel and A. M. Herzberg [Ann. Stat. 7, 77–95 (1979; Zbl 0403.62051)] for the determination of asymptotic optimal design, the designs derived in this article converge weakly to the measures obtained by these authors. {
} We also compare the asymptotic optimal design concepts of J. Sacks and D. Ylvisaker [Ann. Math. Stat. 37, 66–89 (1966; Zbl 0152.17503); Ann. Math. Stat. 39, 49–69 (1968; Zbl 0165.21505)] and Bickel and Herzberg [loc. cit.] and point out some inconsistencies of the latter. Finally, we combine the best features of both concepts to develop a new approach for the design of experiments for correlated observations, and it is demonstrated that the resulting design problems are related to the (logarithmic) potential theory.

MSC:

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