# Is the Kalman Filter a Best Linear Unbiased Estimator (BLUE) for Heteroscedastic Noise?

According to the Gauss-Markov Theorem, a ordinary least squares estimator is BLUE if the noise entering a system is uncorrelated with zero mean and is homoscedastic (has a constant finite variance). I am aware that a Kalman Filter applied to a system with additive noise of known mean and variance but non-gaussian distribution is BLUE. Does this imply that the noise has to be homoscedastic ? Or does the KF have a trick up its sleeve ?

• It is the best filter in the sense of minimizing the MSE; However, it is not necessarily unbiased. – Dovid Apr 23 '18 at 14:47
• This is contrast to BLUE, which is by definition unbiased. – Dovid Apr 23 '18 at 14:49

The covariances $R_k$ and $Q_k$ are time-varying.