Statistically Efficient Estimation of Noise Variances for a Wiener Process Observed with Measurement Noise

In this work, we consider a discrete-time scalar Wiener process driven by a zero-mean Gaussian white noise with unknown variance, observed with an additive Gaussian white measurement noise, also with unknown variance. The estimators of these noise variances are obtained using Maximum Likelihood Estimation (MLE). We demonstrate that the Log-Likelihood Function (LLF) of the Kalman filter gain and innovation variance in this system can be expressed as a quadratic function of the measurements. This quadratic formulation of the LLF, derived from the measurements' probability density function (pdf) as a product of the pdf of the innovations, allows for an analytical expression of the LLF in terms of the filter gain and innovation variance. This approach facilitates the evaluation of the Cramer-Rao Lower Bound (CRLB) and makes it possible to confirm the statistical efficiency of the MLE for the filter gain and innovation variance, i.e., achieving the CRLB and thus demonstrating optimality. The practical application of this methodology is shown for an inertial navigation sensor, characterized by a Wiener process drift and measurement noise.