Optimal exact designs on a circle or a circular arc

Abstract

Fitting a circle to a set of data points on a plane is very common in engineering and science. An important practical problem is how to choose the locations of measurement on a circular feature. So far little attention has been paid to this design issue and only some simulation results are available. In this paper, for Berman's bivariate four-parameter model, $\Phi$-optimality is defined and shown to be equivalent to all $\phi_p$-criteria with $p \epsilon [-\infty, 1)$. Then $\Phi$-optimal exact designs on a circle or a circular arc are derived for any sample size and sampling range. As a by-product, $\Phi$-optimal approximate designs are also obtained. These optimal designs are used to evaluate the efficiency of the equidistant sampling method widely used in practice. These results also provide guidelines for users on sampling method and sample size selection.