Consistent Estimation of Randomly Sampled Ornstein–Uhlenbeck Process Long-Run Mean for Long-Term Target State Prediction

Abstract

In this letter, we study the problem of estimating the long-run mean of the Ornstein–Uhlenbeck (OU) stochastic process and its effect on the long-term prediction of future vessel states, which is a crucial problem for Maritime Situational Awareness (MSA). We employ a sample mean estimator (SME) to estimate the key OU parameter from the observations, computing the closed-form SME covariance error in both the random and constant sampling time regimes, providing a fundamental building block of the overall long-term state prediction covariance. We show also that the SME is: $\sqrt{n}$ -consistent when the sampling time is random; asymptotically efficient when the sampling time is constant; and very close to the Cramer–Rao lower bound in the cases of practical interest for MSA.