# Optimal experimental design that minimizes the width of simultaneous confidence bands

## TL;DR

This paper introduces a novel optimal experimental design method that minimizes the width of simultaneous confidence bands in curvilinear regression models by leveraging geometric and group invariance principles.

## Contribution

It proposes the tube-volume optimal (TV-optimal) design based on the volume of a tube, applicable to Fourier and weighted polynomial regressions, and explores its invariance properties.

## Key findings

- TV-optimal design minimizes confidence band width.
- The design forms an orbit of the M"obius group.
- For specific regressions, TV-optimal designs include D-optimal designs.

## 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 M\"obius 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 M\"obius group containing D-optimal designs as representative elements.