It seems that most optimal estimation literature is divided into either linear Gaussian problems, for which you use Kalman Filter, or non linear and non Gaussian problems for which you use EKF, UKF or Particle filters. How about a linear system with non Gaussian noise, is there a class of filters for that or one should use the non linear filters instead?
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$\begingroup$ Consider a very simple problem -- all observation are independent draws from the same distribution so there's just a single mean and variance to estimate. Note that if you're far from Gaussian the sample mean and variance are not generally efficient estimators of the population mean and variance; indeed all linear estimators may be arbitrarily bad. If you know something about the distribution you may be able to choose better estimators (or at least you may want to ponder ways to avoid the worst impacts). $\endgroup$– Glen_bCommented May 6, 2018 at 5:10
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$\begingroup$ Thanks but what does that inply? $\endgroup$– student1Commented May 6, 2018 at 13:11
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1$\begingroup$ Well, among other things, that it's relevant to an answer what is understood about this distribution. You clearly know something about it. $\endgroup$– Glen_bCommented May 6, 2018 at 15:00
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The Kalman filter can be applied to linear non-Guassian problems. It is just that in the Guassian case, the result is much more beautiful than the case of non-Guassian.
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$\begingroup$ But is it still the optimal linear filter? $\endgroup$– student1Commented May 6, 2018 at 3:00
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$\begingroup$ I believe the result is still optimal, but the evaluation of the estimation might be difficult to obtain. If the noise model is Gaussian, then the state estimation is also Gaussian (state estimate or estimation error is actually the linear combination of the noise samples). When the noise is non-Gaussian, the evaluation, e.g., estimation error, is not that obvious (unknown distribution). $\endgroup$– ZHUANGCommented May 7, 2018 at 1:38
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$\begingroup$ Also, note that we need a) the model perfectly matches the real system, b) the entering noise is white (uncorrelated) and c) the covariances of the noise are exactly known. $\endgroup$– ZHUANGCommented May 7, 2018 at 1:40
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1$\begingroup$ @ZHUANG, please consider incorporating these comments into your answer. The current version of the answer is more of a comment right now. $\endgroup$ Commented Feb 10, 2019 at 17:22
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$\begingroup$ @student1: The book "bayesian forecasting and dynamic linear models" by West and Harrison covers the case of linear filters with non-gaussian noise. The approach used is bayesian. $\endgroup$– mloftonCommented Jun 18, 2020 at 2:51