What are the advantages of higher order Kalman Filters like EKF, UKF? Kalman Filter provides the optimal estimate of the states of a stochastic dynamical system if the system is linear, the measurements are also linear functions of states and the errors in system modeling and the measurements are Gaussian white noise. However, for a nonlinear system, the use of Extended kalman Filter (EKF) or Unscented Kalman Filter(UKF) provides a sub-optimal estimate. In general, for a nonlinear system, EKF gives a less accurate measure of covariance than UKF. In order to correct the covariance, higher order EKF have been proposed. Similarly, higher order UKF also leads to a more accurate covariance. My queries are: 


Apart from improvement in the covariance, do higher order filters provide any other advantage?
Can we capture skewness or kurtosis using these high order filters?
They should improve the convergence properties of the state estimate, i.e. if you have an initial value far from the trajectory, they could improve the region of attraction. They should also reduce the bias in the estimate.
In the EKF and UKF, there are no direct ways of including higher order moments of distributions, in both cases the state distributions are assumed to be Gaussian. Presumably, you can approximate non-Gaussian distributions with additional modeling. For (some) higher order approximations, it should be possible to include higher order moments. [However, as far as I understand, the additional computational complexity is usually not worth the performance gain (if any), since there are so many sources of uncertainty when modeling a system.] The go-to solution for determining the distributions of state-estimates is Monte-Carlo simulation.