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We propose an optimal experimental design for a curvilinear regression modelthat minimizes the band-width of simultaneous confidence bands. Simultaneousconfidence bands for curvilinear regression are constructed by evaluating thevolume of a tube about a curve that is defined as a trajectory of a regressionbasis vector (Naiman, 1986). The proposed criterion is constructed based on thevolume of a tube, and the corresponding optimal design that minimizes thevolume of tube is referred to as the tube-volume optimal (TV-optimal) design.For Fourier and weighted polynomial regressions, the problem is formalized asone of minimization over the cone of Hankel positive definite matrices, and thecriterion to minimize is expressed as an elliptic integral. We show that theM\"obius group keeps our problem invariant, and hence, minimization can beconducted over cross-sections of orbits. We demonstrate that for the weightedpolynomial regression and the Fourier regression with three bases, thetube-volume optimal design forms an orbit of the M\"obius group containingD-optimal designs as representative elements.

Optimal experimental design that minimizes the width of simultaneous confidence bands