Ultimate efficiency of designs for processes of Ornstein-Uhlenbeck type

TL;DR

This paper investigates optimal sampling strategies for Ornstein-Uhlenbeck type processes, deriving asymptotic Fisher information and challenging the belief that small samples are most efficient.

Contribution

It establishes the existence of optimal sampling designs with increasing times and extends results to a broader class of Ito SDEs, providing new insights into sampling efficiency.

Findings

Optimal sampling times are strictly increasing.

Asymptotic Fisher information serves as a benchmark.

Small-sample designs may not always be most efficient.

Abstract

For a process governed by a linear Ito stochastic differential equation of the form dX(t)=[a(t)+b(t)X(t)]dt + \sigma(t)dW(t) we prove an existence of optimal sampling designs with strictly increasing sampling times. We derive an asymptotic Fisher information matrix, which we take as a reference in assessing a quality of finite-point sampling designs. The results are extended to a broader class of Ito stochastic differential equations satisfying a certain condition. We give an example based on the Gompertz growth law refuting a generally accepted opinion that small-sample designs lead to a very high level of efficiency.