Contrary to assertions in the literature, we show that the Extended Kalman Filter (EKF) is superior to the Unscented Kalman Filter (UKF) for certain nonlinear estimation problems. In particular, for nonlinearities that are odd functions of the state vector (e.g., x3) the Unscented Kalman Filter usually performs well, whereas for even nonlinearities (e.g., x2), the Extended Kalman Filter is sometimes much better than the Unscented Kalman Filter. This is contrary to the usual engineering folklore, and therefore we have checked our results very thoroughly. In particular, the Unscented Kalman Filter correctly approximates the conditional mean using a 4th order Gauss-Hermite quadrature, in contrast to the Extended Kalman Filter which uses a simple 0th order approximation, but the conditional mean is not the desired estimate in practical applications for strongly bimodal conditional probability densities, which are induced by even nonlinearities, owing to a sign ambiguity. On the other hand, even nonlinearities do not always induce multimodal densities that persist for a significant amount of time, and thus the Unscented Kalman Filter sometimes performs well for such problems. We study the effects of initial uncertainty of the state vector and nonlinearity in measurements.