D-optimal designs for complex Ornstein-Uhlenbeck processes

TL;DR

This paper investigates D-optimal experimental designs for complex Ornstein-Uhlenbeck processes, revealing that optimal designs are equidistant and exist for both trend and covariance parameters, unlike real OU processes.

Contribution

It demonstrates the existence and structure of D-optimal designs for complex OU processes, including joint estimation of trend and covariance parameters, which was not previously known.

Findings

Optimal designs are equidistant for complex OU processes.

D-optimal designs exist for both trend and covariance parameters.

Contrasts with real OU processes where such designs are not always available.

Abstract

Complex Ornstein-Uhlenbeck (OU) processes have various applications in statistical modelling. They play role e.g. in the description of the motion of a charged test particle in a constant magnetic field or in the study of rotating waves in time-dependent reaction diffusion systems, whereas Kolmogorov used such a process to model the so-called Chandler wobble, small deviation in the Earth's axis of rotation. In these applications parameter estimation and model fitting is based on discrete observations of the underlying stochastic process, however, the accuracy of the results strongly depend on the observation points. This paper studies the properties of D-optimal designs for estimating the parameters of a complex OU process with a trend. We show that in contrast with the case of the classical real OU process, a D-optimal design exists not only for the trend parameter, but also for joint estimation of the covariance parameters, moreover, these optimal designs are equidistant.