In a linear Kalman filter, we assume that the state and measurement noise are white noise N(0,Q) and N(0,R) respectively. Is it common practice to test these hypothesis? And what are the most common statistical tests used in this case?
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1$\begingroup$ Could you give some context? 1) As is it would fit better on CrossValidated but 2) I'm guessing they would also require more context. $\endgroup$– Bob Jansen ♦Nov 27, 2018 at 9:14
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The linear Kalman filter still minimises the mean squared error of the estimate even if noise are non-Gaussian white, it's just that in the Gaussian case you will have the full posterior distribution from the mean and covariance estimates alone.
So in practice it wouldn't be absolutely necessary to test for Gaussian noise, and have the Kalman filter still 'work'.
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$\begingroup$ Hi: Assume for a moment that the KF just has an observation equation. ( system dynamics are known ). Then, if the noise in the observation equation is Gaussian, this implies that , in addition to having the full posterior, you can also be sure that your updated estimate $\hat{Y}_{t}$ represents the conditional expectation of $Y_{t}$ given the data up to $(t-1)$. $\endgroup$ Apr 26, 2019 at 17:33