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In this paper high order vector filter equations are developed for estimation in non-Gaussian noise. The difference between the filters developed here and the standard Kalman filter is that the filter equation contains nonlinear functions of the innovations process. These filters are general in that the initial state covariance, the measurement noise covariance, and the process noise covariance can all have non-Gaussian distributions. Two filter structures are developed. The first filter is designed for systems with asymmetric probability densities. The second is designed for systems with symmetric probability densities. Experimental evaluation of these filters for estimation in non-Gaussian noise, formed from Gaussian sum distributions, shows that these filters perform much better than the standard Kalman filter, and close to the optimal Bayesian estimator.