Abstract
We propose an optimal experimental design for a curvilinear regression model that minimizes the band-width of simultaneous confidence bands. Simultaneous confidence bands for curvilinear regression are constructed by evaluating the volume of a tube about a curve that is defined as a trajectory of a regression basis vector (Naiman, 1986). The proposed criterion is constructed based on the volume of a tube, and the corresponding optimal design that minimizes the volume of tube is referred to as the tube-volume optimal (TV-optimal) design. For Fourier and weighted polynomial regressions, the problem is formalized as one of minimization over the cone of Hankel positive definite matrices, and the criterion to minimize is expressed as an elliptic integral. We show that the Möbius group keeps our problem invariant, and hence, minimization can be conducted over cross-sections of orbits. We demonstrate that for the weighted polynomial regression and the Fourier regression with three bases, the tube-volume optimal design forms an orbit of the Möbius group containing D-optimal designs as representative elements.
Highlights
Suppose that we observe pairs of explanatory variables xi ∈ X and response variables yi ∈ R, i = 1, . . . , N
We focus on Fourier regression and weighted polynomial regression
Lemma 3.1 means that the Fourier regression model yi = b fF + εi, εi ∼ N (0, 1), is rewritten as the weighted polynomial model yi = b fP (xi) + εi, by letting xi = tan(πti), yi = λ0(xi)−1yi, b = B b, and εi ∼ N (0, λ0(xi)−2)
Summary
When α is small, cα tends to a simpler function This approximation is due to the volumeof-tube method used to construct simultaneous confidence bands in curvilinear regression curves [14, 25, 29, 34]. When comparing two estimated regression curves, Dette, et al [4, 5] proposed to minimize the Lp- or L∞-norm of the variance function of the estimator of the difference between the two curves They demonstrated that their proposal reduces the width substantially compared with the pair of optimized designs for individual regression models. The Mobius group is proved to keep the optimization problem invariant, and can be used to reduce the dimension of the problem
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