A required assumption of a Kalman filter, the most-widely-used state estimator in avionic systems, is the Gaussian and whiteness of process and measurement noise. If the assumption fails, the performance of the Kalman filter degrades, and its estimation results are no longer optimal. In fact, many avionic applications produce colored noise, and the parameters of colored noise models are typically unknown beforehand without additional information about the noise statistics. In addition, the functions of each underlying model - nonlinear dynamic and measurement models - are sometimes improper or partially unknown. To estimate the states of systems with unknown correlations of each instance of noise and uncertain modeling errors of parametric models, we propose a novel approach that incorporates the kernel embedding of distributions into the extended Kalman filter. In our approach, kernel embedding maps process and measurement residuals, defined by differences between outputs of approximate system models and collected training data, into a reproducing kernel Hilbert space to generate nonparametric models in the functional space. Results from Monte Carlo simulations demonstrate that the proposed method, compared to existing methods (e.g., extended Kalman filter and Gaussian process-based filter), improves the accuracy of state estimation under colored noise conditions.