On frequency estimation for partially observed processes with small noise in observations

TL;DR

This paper investigates the accuracy of frequency estimation for a periodic signal affected by Ornstein-Uhlenbeck noise and white Gaussian noise, demonstrating the estimator's consistency and asymptotic normality in small noise conditions.

Contribution

It introduces a maximum likelihood estimator for frequency in a partially observed system with small noise, based on Kalman-Bucy filter asymptotics, and proves its statistical properties.

Findings

Estimator is consistent in small noise limit

Estimator is asymptotically normal

Method applies to linear partially observed systems

Abstract

We consider the problem of frequency estimation of the periodic signal multiplied by a stationary Gaussian process (Ornstein-Uhlenbeck) and observed in the presence of the white Gaussian noise. We show the consistency and asymptotic normality of the maximum likelihood estimator in the asymptotics of small noise in observations. The model of observations is a linear nonhomogeneous partially observed system and the construction and study of the estimator is essentialy based on the asymptotics of the equations of Kalman-Bucy filtration.