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The paper deals with state estimation of nonlinear non-Gaussian systems with a special focus on the Gaussian sum filters. To achieve a higher estimate quality, state and measurement predictive moments appearing in the filters are computed by the randomized unscented transform, which provides asymptotically exact estimates of the moments. The use of the Gaussian sum filter employing the randomized unscented transform is introduced and the proposed algorithm is illustrated in a numerical example. The analysis of the numerical example involves a comparison of several filters using a number of performance metrics both absolute and relative, assessing the point estimate quality, the estimate error quality, and the density estimate quality.